text:
Maxwell, Lorentz, gauges and gauge transformations

I’ve done quite a few posts already on electromagnetism. They were all focused on the math one needs to understand Maxwell’s equations. Maxwell’s equations are a set of (four) differential equations, so they relate some function with its derivatives. To be specific, they relate E and B, i.e. the electric and magnetic field vector respectively, with their derivatives in space and in time. [Let me be explicit here: E and B have three components, but depend on both space as well as time, so we have three dependent and four independent variables for each function: E = (Ex, Ey, Ez) = E(x, y, z, t) and B = (Bx, By, Bz) = B(x, y, z, t).] That’s simple enough to understand, but the dynamics involved are quite complicated, as illustrated below.

Maxwell interactionI now want to do a series on the more interesting stuff, including an exploration of the concept of gauge in field theory, and I also want to show how one can derive the wave equation for electromagnetic radiation from Maxwell’s equations. Before I start, let’s recall the basic concept of a field.

The reality of fields

I said a couple of time already that (electromagnetic) fields are real. They’re more than just a mathematical structure. Let me show you why. Remember the formula for the electrostatic potential caused by some charge q at the origin:

E 4

We know that the (negative) gradient of this function, at any point in space, gives us the electric field vector at that point: E = –Φ. [The minus sign is there because of convention: we take the reference point Φ = 0 at infinity.] Now, the electric field vector gives us the force on a unit charge (i.e. the charge of a proton) at that point. If q is some positive charge, the force will be repulsive, and the unit charge will accelerate away from our q charge at the origin. Hence, energy will be expended, as force over distance implies work is being done: as the charges separate, potential energy is converted into kinetic energy. Where does the energy come from? The energy conservation law tells us that it must come from somewhere.

It does: the energy comes from the field itself. Bringing in more or bigger charges (from infinity, or just from further away) requires more energy. So the new charges change the field and, therefore, its energy. How exactly? That’s given by Gauss’ Law: the total flux out of a closed surface is equal to:

Gauss Law

You’ll say: flux and energy are two different things. Well… Yes and no. The energy in the field depends on E. Indeed, the formula for the energy density in space (i.e. the energy per unit volume) is

D 6

Getting the energy over a larger space is just another integral, with the energy density as the integral kernel:

energy integral

Feynman’s illustration below is not very sophisticated but, as usual, enlightening. 🙂

energy in the field

Gauss’ Theorem connects both the math as well as the physics of the situation and, as such, underscores the reality of fields: the energy is not in the electric charges. The energy is in the fields they produce. Everything else is just the principle of superposition of fields –  i.e. E = E+ E– coming into play. I’ll explain Gauss’ Theorem in a moment. Let me first make some additional remarks.

First, the formulas are valid for electrostatics only (so E and B only vary in space, not in time), so they’re just a piece of the larger puzzle. 🙂 As for now, however, note that, if a field is real (or, to be precise, if its energy is real), then the flux is equally real.

Second, let me say something about the units. Field strength (E or, in this case, its normal component En = E·n) is measured in newton (N) per coulomb (C), so in N/C. The integral above implies that flux is measured in (N/C)·m2. It’s a weird unit because one associates flux with flow and, therefore, one would expect flux is some quantity per unit time and per unit area, so we’d have the m2 unit (and the second) in the denominator, not in the numerator. But so that’s true for heat transfer, for mass transfer, for fluid dynamics (e.g. the amount of water flowing through some cross-section) and many other physical phenomena. But for electric flux, it’s different. You can do a dimensional analysis of the expression above: the sum of the charges is expressed in coulomb (C), and the electric constant (i.e. the vacuum permittivity) is expressed in C2/(N·m2), so, yes, it works: C/[C2/(N·m2)] = (N/C)·m2. To make sense of the units, you should think of the flux as the total flow, and of the field strength as a surface density, so that’s the flux divided by the total area, so (field strength) = (flux)/(area). Conversely, (flux) = (field strength)×(area). Hence, the unit of flux is [flux] = [field strength]×[area] = (N/C)·m2.

OK. Now we’re ready for Gauss’ Theorem. 🙂 I’ll also say something about its corollary, Stokes’ Theorem. It’s a bit of a mathematical digression but necessary, I think, for a better understanding of all those operators we’re going to use.

Gauss’ Theorem

The concept of flux is related to the divergence of a vector field through Gauss’ Theorem. Gauss’s Theorem has nothing to do with Gauss’ Law, except that both are associated with the same genius. Gauss’ Theorem is:

Gauss Theorem

The ·C in the integral on the right-hand side is the divergence of a vector field. It’s the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

Huh? What’s a volume density? Good question. Just substitute C for E in the surface and volume integral above (the integral on the left is a surface integral, and the one on the right is a volume integral), and think about the meaning of what’s written. To help you, let me also include the concept of linear density, so we have (1) linear, (2) surface and (3) volume density. Look at that representation of a vector field once again: we said the density of lines represented the magnitude of E. But what density? The representation hereunder is flat, so we can think of a linear density indeed, measured along the blue line: so the flux would be six (that’s the number of lines), and the linear density (i.e. the field strength) is six divided by the length of the blue line.

Linear density

However, we defined field strength as a surface density above, so that’s the flux (i.e. the number of field lines) divided by the surface area (i.e. the area of a cross-section): think of the square of the blue line, and field lines going through that square. That’s simple enough. But what’s volume density? How do we count the number of lines inside of a box? The answer is: mathematicians actually define it for an infinitesimally small cube by adding the fluxes out of the six individual faces of an infinitesimally small cube:

Volume density for small cube 1

So, the truth is: volume density is actually defined as a surface density, but for an infinitesimally small volume element. That, in turn, gives us the meaning of the divergence of a vector field. Indeed, the sum of the derivatives above is just ·C (i.e. the divergence of C), and ΔxΔyΔz is the volume of our infinitesimal cube, so the divergence of some field vector C at some point P is the flux – i.e. the outgoing ‘flow’ of Cper unit volume, in the neighborhood of P, as evidenced by writing

Volume density for small cube

Indeed, just bring ΔV to the other side of the equation to check the ‘per unit volume’ aspect of what I wrote above. The whole idea is to determine whether the small volume is like a sink or like a source, and to what extent. Think of the field near a point charge, as illustrated below. Look at the black lines: they are the field lines (the dashed lines are equipotential lines) and note how the positive charge is a source of flux, obviously, while the negative charge is a sink.

Now, the next step is to acknowledge that the total flux from a volume is the sum of the fluxes out of each part. Indeed, the flux through the part of the surfaces common to two parts will cancel each other out. Feynman illustrates that with a rough drawing (below) and I’ll refer you to his Lecture on it for more detail.

So… Combining all of the gymnastics above – and integrating the divergence over an entire volume, indeed –  we get Gauss’ Theorem:

Gauss Theorem

Stokes’ Theorem

There is a similar theorem involving the circulation of a vector, rather than its flux. It’s referred to as Stokes’ Theorem. Let me jot it down:

Stokes Theorem

We have a contour integral here (left) and a surface integral (right). The reasoning behind is quite similar: a surface bounded by some loop Γ is divided into infinitesimally small squares, and the circulation around Γ is the sum of the circulations around the little loops. We should take care though: the surface integral takes the normal component of ×C, so that’s (×C)n = (×Cn. The illustrations below should help you to understand what’s going on.

Stokes Theorem 1Stokes Theorem 2

The electric versus the magnetic force

There’s more than just the electric force: we also have the magnetic force. The so-called Lorentz force is the combination of both. The formula, for some charge q in an electromagnetic field, is equal to:

Lorentz force

Hence, if the velocity vector v is not equal to zero, we need to look at the magnetic field vector B too! The simplest situation is magnetostatics, so let’s first have a look at that.

Magnetostatics imply that that the flux of E doesn’t change, so Maxwell’s third equation reduces to c2×B = j0. So we just have a steady electric current (j): no accelerating charges. Maxwell’s fourth equation, B = 0, remains what is was: there’s no such thing as a magnetic charge. The Lorentz force also remains what it is, of course: F = q(E+v×B) = qE +qv×B. Also note that the v, j and the lack of a magnetic charge all point to the same: magnetism is just a relativistic effect of electricity.

What about units? Well… While the unit of E, i.e. the electric field strength, is pretty obvious from the F = qE term  – hence, E = F/q, and so the unit of E must be [force]/[charge] = N/C – the unit of the magnetic field strength is more complicated. Indeed, the F = qv×B identity tells us it must be (N·s)/(m·C), because 1 N = 1C·(m/s)·(N·s)/(m·C). Phew! That’s as horrendous as it looks, and that’s why it’s usually expressed using its shorthand, i.e. the tesla: 1 T = 1 (N·s)/(m·C). Magnetic flux is the same concept as electric flux, so it’s (field strength)×(area). However, now we’re talking magnetic field strength, so its unit is T·m= (N·s·m)/(m·C) = (N·s·m)/C, which is referred to as the weber (Wb). Remembering that 1 volt = 1 N·m/C, it’s easy to see that a weber is also equal to 1 Wb = 1 V·s. In any case, it’s a unit that is not so easy to interpret.

Magnetostatics is a bit of a weird situation. It assumes steady fields, so the ∂E/∂t and ∂B/∂t terms in Maxwell’s equations can be dropped. In fact, c2×B = j0 implies that ·(c2×B ·(j0) and, therefore, that ·= 0. Now, ·= –∂ρ/∂t and, therefore, magnetostatics is a situation which assumes ∂ρ/∂t = 0. So we have electric currents but no change in charge densities. To put it simply, we’re not looking at a condenser that is charging or discharging, although that condenser may act like the battery or generator that keeps the charges flowing! But let’s go along with the magnetostatics assumption. What can we say about it? Well… First, we have the equivalent of Gauss’ Law, i.e. Ampère’s Law:

Ampere Law

We have a line integral here around a closed curve, instead of a surface integral over a closed surface (Gauss’ Law), but it’s pretty similar: instead of the sum of the charges inside the volume, we have the current through the loop, and then an extra c2 factor in the denominator, of course. Combined with the B = 0 equation, this equation allows us to solve practical problems. But I am not interested in practical problems. What’s the theory behind?

The magnetic vector potential

TheB = 0 equation is true, always, unlike the ×E = 0 expression, which is true for electrostatics only (no moving charges). It says the divergence of B is zero, always, and, hence, it means we can represent B as the curl of another vector field, always. That vector field is referred to as the magnetic vector potential, and we write:

·B = ·(×A) = 0 and, hence, B×A

In electrostatics, we had the other theorem: if the curl of a vector field is zero (everywhere), then the vector field can be represented as the gradient of some scalar function, so if ×= 0, then there is some Ψ for which CΨ. Substituting C for E, and taking into account our conventions on charge and the direction of flow, we get E = –Φ. Substituting E in Maxwell’s first equation (E = ρ/ε0) then gave us the so-called Poisson equation: ∇2Φ = ρ/ε0, which sums up the whole subject of electrostatics really! It’s all in there!

Except magnetostatics, of course. Using the (magnetic) vector potential A, all of magnetostatics is reduced to another expression:

2A= −j0, with ·A = 0

Note the qualifier: ·A = 0. Why should the divergence of A be equal to zero? You’re right. It doesn’t have to be that way. We know that ·(×C) = 0, for any vector field C, and always (it’s a mathematical identity, in fact, so it’s got nothing to do with physics), but choosing A such that ·A = 0 is just a choice. In fact, as I’ll explain in a moment, it’s referred to as choosing a gauge. The·A = 0 choice is a very convenient choice, however, as it simplifies our equations. Indeed, c2×B = j0 = c2×(×A), and – from our vector calculus classes – we know that ×(×C) = (·C) – ∇2C. Combining that with our choice of A (which is such that ·A = 0, indeed), we get the ∇2A= −j0 expression indeed, which sums up the whole subject of magnetostatics!

The point is: if the time derivatives in Maxwell’s equations, i.e. ∂E/∂t and ∂B/∂t, are zero, then Maxwell’s four equations can be nicely separated into two pairs: the electric and magnetic field are not interconnected. Hence, as long as charges and currents are static, electricity and magnetism appear as distinct phenomena, and the interdependence of E and B does not appear. So we re-write Maxwell’s set of four equations as:

  1. ElectrostaticsE = ρ/ε0 and ×E = 0
  2. Magnetostatics: ×B = j/c2ε0 and B = 0

Note that electrostatics is a neat example of a vector field with zero curl and a given divergence (ρ/ε0), while magnetostatics is a neat example of a vector field with zero divergence and a given curl (j/c2ε0).

But reality is usually not so simple. With time-varying fields, Maxwell’s equations are what they are, and so there is interdependence, as illustrated in the introduction of this post. Note, however, that the magnetic field remains divergence-free in dynamics too! That’s because there is no such thing as a magnetic charge: we only have electric charges. So ·B = 0 and we can define a magnetic vector potential A and re-write B as B×A, indeed.

I am writing a vector potential field because, as I mentioned a couple of times already, we can choose A. Indeed, as long as ·A = 0, it’s fine, so we can add curl-free components to the magnetic potential: it won’t make a difference. This condition is referred to as gauge invariance. I’ll come back to that, and also show why this is what it is.

While we can easily get B from A because of the B×A, getting E from some potential is a different matter altogether. It turns out we can get E using the following expression, which involves both Φ (i.e. the electric or electrostatic potential) as well as A (i.e. the magnetic vector potential):

E = –Φ – ∂A/∂t

Likewise, one can show that Maxwell’s equations can be re-written in terms of Φ and A, rather than in terms of E and B. The expression looks rather formidable, but don’t panic:

Equations 2

Just look at it. We have two ‘variables’ here (Φ and A) and two equations, so the system is fully defined. [Of course, the second equation is three equations really: one for each component x, y and z.] What’s the point? Why would we want to re-write Maxwell’s equations? The first equation makes it clear that the scalar potential (i.e. the electric potential) is a time-varying quantity, so things are not, somehow, simpler. The answer is twofold. First, re-writing Maxwell’s equations in terms of the scalar and vector potential makes sense because we have (fairly) easy expressions for their value in time and in space as a function of the charges and currents. For statics, these expressions are:

Integrals staticsSo it is, effectively, easier to first calculate the scalar and vector potential, and then get E and B from them. For dynamics, the expressions are similar:

Integrals dynamics

Indeed, they are like the integrals for statics, but with “a small and physically appealing modification”, as Feynman notes: when doing the integrals, we must use the so-called retarded time t′ = t − r12/ct’. The illustration below shows how it works: the influences propagate from point (2) to point (1) at the speed c, so we must use the values of ρ and j at the time t′ = t − r12/ct’ indeed!

Retarded timeThe second aspect of the answer to the question of why we’d be interested in Φ and A has to do with the topic I wanted to write about here: the concept of a gauge and a gauge transformation.

Gauges and gauge transformations in electromagnetics

Let’s see what we’re doing really. We calculate some A and then solve for B by writing: B = ×A. Now, I say some A because any A‘ = AΨ, with Ψ any scalar field really. Why? Because the curl of the gradient of Ψ – i.e. curl(gradΨ) = ×(Ψ) – is equal to 0. Hence, ×(AΨ) = ×A×Ψ = ×A.

So we have B, and now we need E. So the next step is to take Faraday’s Law, which is Maxwell’s second equation: ×E = –∂B/∂t. Why this one? It’s a simple one, as it does not involve currents or charges. So we combine this equation and our B = ×A expression and write:

×E = –∂(∇×A)/∂t

Now, these operators are tricky but you can verify this can be re-written as:

×(E + ∂A/∂t) = 0

Looking carefully, we see this expression says that E + ∂A/∂t is some vector whose curl is equal to zero. Hence, this vector must be the gradient of something. When doing electrostatics, When we worked on electrostatics, we only had E, not the ∂A/∂t bit, and we said that E tout court was the gradient of something, so we wrote E = −Φ. We now do the same thing for E + ∂A/∂t, so we write:

E + ∂A/∂t = −Φ

So we use the same symbol Φ but it’s a bit of a different animal, obviously. However, it’s easy to see that, if the ∂A/∂t would disappear (as it does in electrostatics, where nothing changes with time), we’d get our ‘old’ −Φ. Now, E + ∂A/∂t = −Φ can be written as:

E = −Φ – ∂A/∂t

So, what’s the big deal? We wrote B and E as a function of Φ and A. Well, we said we could replace A by any A‘ = AΨ but, obviously, such substitution would not yield the same E. To get the same E, we need some substitution rule for Φ as well. Now, you can verify we will get the same E if we’d substitute Φ for Φ’ = Φ – ∂Ψ/∂t. You should check it by writing it all out:

E = −Φ’–∂A’/∂t = −(Φ–∂Ψ/∂t)–∂(A+Ψ)/∂t

= −Φ+(∂Ψ/∂t)–∂A/∂t–∂(Ψ)/∂t = −Φ – ∂A/∂t = E

Again, the operators are a bit tricky, but the +(∂Ψ/∂t) and –∂(Ψ)/∂t terms do cancel out. Where are we heading to? When everything is said and done, we do need to relate it all to the currents and the charges, because that’s the real stuff out there. So let’s take Maxwell’s E = ρ/ε0 equation, which has the charges in it, and let’s substitute E for E = −Φ – ∂A/∂t. We get:

That equation can be re-written as:

equation 1

So we have one equation here relating Φ and A to the sources. We need another one, and we also need to separate Φ and A somehow. How do we do that?

Maxwell’s fourth equation, i.e. c2×B = j+ ∂E/∂t can, obviously, be written as c2×− E/∂t = j0. Substituting both E and B yields the following monstrosity:

equation 3

We can now apply the general ∇×(×C) = (·C) – ∇2C identity to the first term to get:

equation 4

It’s equally monstrous, obviously, but we can simplify the whole thing by choosing Φ and A in a clever way. For the magnetostatic case, we chose A such that ·A = 0. We could have chosen something else. Indeed, it’s not because B is divergence-free, that A has to be divergence-free too! For example, I’ll leave it to you to show that choosing ·A such that

equation 5also respects the general condition that any A and Φ we choose must respect the A‘ = AΨ and Φ’ = Φ – ∂Ψ/∂t equalities. Now, if we choose ·A such that ·A = −c–2·∂Φ/∂t indeed, then the two middle terms in our monstrosity cancel out, and we’re left with a much simpler equation for A:

equation 6

In addition, doing the substitution in our other equation relating Φ and A to the sources yields an equation for Φ that has the same form:

equation 7

What’s the big deal here? Well… Let’s write it all out. The equation above becomes:

wave equation

That’s a wave equation in three dimensions. In case you wonder, just check one of my posts on wave equations. The one-dimensional equivalent for a wave propagating in the x direction at speed c (like a sound wave, for example) is ∂2Φ/∂xc–2·∂2Φ/∂t2, indeed. The equation for A yields above yields similar wave functions for A‘s components Ax, Ay, and Az.

So, yes, it is a big deal. We’ve written Maxwell’s equations in terms of the scalar (Φ) and vector (A) potential and in a form that makes immediately apparent that we’re talking electromagnetic waves moving out at the speed c. Let me copy them again:

Equations 2

You may, of course, say that you’d rather have a wave equation for E and B, rather than for A and Φ. Well… That can be done. Feynman gives us two derivations that do so. The first derivation is relatively simple and assumes the source our electromagnetic wave moves in one direction only. The second derivation is much more complicated and gives an equation for E that, if you’ve read the first volume of Feynman’s Lectures, you’ll surely remember:

equation for E

The links are there, and so I’ll let you have fun with those Lectures yourself. I am finished here, indeed, in terms of what I wanted to do in this post, and that is to say a few words about gauges in field theory. It’s nothing much, really, and so we’ll surely have to discuss the topic again, but at least you now know what a gauge actually is in classical electromagnetic theory. Let’s quickly go over the concepts:

  1. Choosing the ·A is choosing a gauge, or a gauge potential (because we’re talking scalar and vector potential here). The particular choice is also referred to as gauge fixing.
  2. Changing A by adding ψ is called a gauge transformation, and the scalar function Ψ is referred to as a gauge function. The fact that we can add curl-free components to the magnetic potential without them making any difference is referred to as gauge invariance.
  3. Finally, the ·A = −c–2·∂Φ/∂t gauge is referred to as a Lorentz gauge.

Just to make sure you understand: why is that Lorentz gauge so special? Well… Look at the whole argument once more: isn’t it amazing we get such beautiful (wave) equations if we stick it in? Also look at the functional shape of the gauge itself: it looks like a wave equation itself! […] Well… No… It doesn’t. I am a bit too enthusiastic here. We do have the same 1/c2 and a time derivative, but it’s not a wave equation. 🙂 In any case, it all confirms, once again, that physics is all about beautiful mathematical structures. But, again, it’s not math only. There’s something real out there. In this case, that ‘something’ is a traveling electromagnetic field. 🙂

But why do we call it a gauge? That should be equally obvious. It’s really like choosing a gauge in another context, such as measuring the pressure of a tyre, as shown below. 🙂

Gauges and group theory

You’ll usually see gauges mentioned with some reference to group theory. For example, you will see or hear phrases like: “The existence of arbitrary numbers of gauge functions ψ(r, t) corresponds to the U(1) gauge freedom of the electromagnetic theory.” The U(1) notation stands for a unitary group of degree n = 1. It is also known as the circle group. Let me copy the introduction to the unitary group from the Wikipedia article on it:

In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group.

The unitary group U(n) is a real Lie group of of dimension n2. The Lie algebra of U(n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes) consists of all matrices A such that A*A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix.

Phew! Does this make you any wiser? If anything, it makes me realize I’ve still got a long way to go. 🙂 The Wikipedia article on gauge fixing notes something that’s more interesting (if only because I more or less understand what it says):

Although classical electromagnetism is now often spoken of as a gauge theory, it was not originally conceived in these terms. The motion of a classical point charge is affected only by the electric and magnetic field strengths at that point, and the potentials can be treated as a mere mathematical device for simplifying some proofs and calculations. Not until the advent of quantum field theory could it be said that the potentials themselves are part of the physical configuration of a system. The earliest consequence to be accurately predicted and experimentally verified was the Aharonov–Bohm effect, which has no classical counterpart.

This confirms, once again, that the fields are real. In fact, what this says is that the potentials are real: they have a meaningful physical interpretation. I’ll leave it to you to expore that Aharanov-Bohm effect. In the meanwhile, I’ll study what Feynman writes on potentials and all that as used in quantum physics. It will probably take a while before I’ll get into group theory though. :-/

Indeed, it’s probably best to study physics at a somewhat less abstract level first, before getting into the more sophisticated stuff.

Fields and charges (I)

My previous posts focused mainly on photons, so this one should be focused more on matter-particles, things that have a mass and a charge. However, I will use it more as an opportunity to talk about fields and present some results from electrostatics using our new vector differential operators (see my posts on vector analysis).

Before I do so, let me note something that is obvious but… Well… Think about it: photons carry the electromagnetic force, but have no electric charge themselves. Likewise, electromagnetic fields have energy and are caused by charges, but so they also carry no charge. So… Fields act on a charge, and photons interact with electrons, but it’s only matter-particles (notably the electron and the proton, which is made of quarks) that actually carry electric charge. Does that make sense? It should. 🙂

Another thing I want to remind you of, before jumping into it all head first, are the basic units and relations that are valid always, regardless of what we are talking about. They are represented below:

Let me recapitulate the main points:

  • The speed of light is always the same, regardless of the reference frame (inertial or moving), and nothing can travel faster than light (except mathematical points, such as the phase velocity of a wavefunction).
  • This universal rule is the basis of relativity theory and the mass-energy equivalence relation E = mc2.
  • The constant speed of light also allows us to redefine the units of time and/or distance such that c = 1. For example, if we re-define the unit of distance as the distance traveled by light in one second, or the unit of time as the time light needs to travel one meter, then c = 1.
  • Newton’s laws of motion define a force as the product of a mass and its acceleration: F = m·a. Hence, mass is a measure of inertia, and the unit of force is 1 newton (N) = 1 kg·m/s2.
  • The momentum of an object is the product of its mass and its velocity: p = m·v. Hence, its unit is 1 kg·m/s = 1 N·s. Therefore, the concept of momentum combines force (N) as well as time (s).
  • Energy is defined in terms of work: 1 Joule (J) is the work done when applying a force of one newton over a distance of one meter: 1 J = 1 N·m. Hence, the concept of energy combines force (N) and distance (m).
  • Relativity theory establishes the relativistic energy-momentum relation pc = Ev/c, which can also be written as E2 = p2c+ m02c4, with mthe rest mass of an object (i.e. its mass when the object would be at rest, relative to the observer, of course). These equations reduce to m = E and E2 = p2 + m0when choosing time and/or distance units such that c = 1. The mass is the total mass of the object, including its inertial mass as well as the equivalent mass of its kinetic energy.
  • The relationships above establish (a) energy and time and (b) momentum and position as complementary variables and, hence, the Uncertainty Principle can be expressed in terms of both. The Uncertainty Principle, as well as the Planck-Einstein relation and the de Broglie relation (not shown on the diagram), establish a quantum of action, h, whose dimension combines force, distance and time (h ≈ 6.626×10−34 N·m·s). This quantum of action (Wirkung) can be defined in various ways, as it pops up in more than one fundamental relation, but one of the more obvious approaches is to define h as the proportionality constant between the energy of a photon (i.e. the ‘light particle’) and its frequency: h = E/ν.

Note that we talked about forces and energy above, but we didn’t say anything about the origin of these forces. That’s what we are going to do now, even if we’ll limit ourselves to the electromagnetic force only.

According to Wikipedia, electrostatics deals with the phenomena and properties of stationary or slow-moving electric charges with no acceleration. Feynman usually uses the term when talking about stationary charges only. If a current is involved (i.e. slow-moving charges with no acceleration), the term magnetostatics is preferred. However, the distinction does not matter all that much because  – remarkably! – with stationary charges and steady currents, the electric and magnetic fields (E and B) can be analyzed as separate fields: there is no interconnection whatsoever! That shows, mathematically, as a neat separation between (1) Maxwell’s first and second equation and (2) Maxwell’s third and fourth equation:

  1. Electrostatics: (i) ∇•E = ρ/ε0 and (ii) ×E = 0.
  2. Magnetostatics: (iii) c2∇×B = j0 and (iv) B = 0.

Electrostatics: The ρ in equation (i) is the so-called charge density, which describes the distribution of electric charges in space: ρ = ρ(x, y, z). To put it simply: ρ is the ‘amount of charge’ (which we’ll denote by Δq) per unit volume at a given point. As for ε0, that’s a constant which ensures all units are ‘compatible’. Equation (i) basically says we have some flux of E, the exact amount of which is determined by the charge density ρ or, more in general, by the charge distribution in space. As for equation (ii), i.e. ×E = 0, we can sort of forget about that. It means the curl of E is zero: everywhere, and always. So there’s no circulation of E. Hence, E is a so-called curl-free field, in this case at least, i.e. when only stationary charges and steady currents are involved.

Magnetostatics: The j in (iii) represents a steady current indeed, causing some circulation of B. The cfactor is related to the fact that magnetism is actually only a relativistic effect of electricity, but I can’t dwell on that here. I’ll just refer you to what Feynman writes about this in his Lectures, and warmly recommend to read it. Oh… Equation (iv), B = 0, means that the divergence of B is zero: everywhere, and always. So there’s no flux of B. None. So B is a divergence-free field.

Because of the neat separation, we’ll just forget about B and talk about E only.

The electric potential

OK. Let’s try to go through the motions as quickly as we can. As mentioned in my introduction, energy is defined in terms of work done. So we should just multiply the force and the distance, right? 1 Joule = 1 newton × 1 meter, right? Well… Yes and no. In discussions like this, we talk potential energy, i.e. energy stored in the system, so to say. That means that we’re looking at work done against the force, like when we carry a bucket of water up to the third floor or, to use a somewhat more scientific description of what’s going on, when we are separating two masses. Because we’re doing work against the force, we put a minus sign in front of our integral:

formula 1

Now, the electromagnetic force works pretty much like gravity, except that, when discussing gravity, we only have positive ‘charges’ (the mass of some object is always positive). In electromagnetics, we have positive as well as negative charge, and please note that two like charges repel (that’s not the case with gravity). Hence, doing work against the electromagnetic force may involve bringing like charges together or, alternatively, separating opposite charges. We can’t say. Fortunately, when it comes to the math of it, it doesn’t matter: we will have the same minus sign in front of our integral. The point is: we’re doing work against the force, and so that’s what the minus sign stands for. So it has nothing to do with the specifics of the law of attraction and repulsion in this case (electromagnetism as opposed to gravity) and/or the fact that electrons carry negative charge. No.

Let’s get back to the integral. Just in case you forgot, the integral sign ∫ stands for an S: the S of summa, i.e. sum in Latin, and we’re using these integrals because we’re adding an infinite number of infinitesimally small contributions to the total effort here indeed. You should recognize it, because it’s a general formula for energy or work. It is, once again, a so-called line integral, so it’s a bit different than the ∫f(x)dx stuff you learned from high school. Not very different, but different nevertheless. What’s different is that we have a vector dot product F•ds after the integral sign here, so that’s not like f(x)dx. In case you forgot, that f(x)dx product represents the surface of an infinitesimally rectangle, as shown below: we make the base of the rectangle smaller and smaller, so dx becomes an infinitesimal indeed. And then we add them all up and get the area under the curve. If f(x) is negative, then the contributions will be negative.

But so we don’t have little rectangles here. We have two vectors, F and ds, and their vector dot product, F•ds, which will give you… Well… I am tempted to write: the tangential component of the force along the path, but that’s not quite correct: if ds was a unit vector, it would be true—because then it’s just like that h•n product I introduced in our first vector calculus class. However, ds is not a unit vector: it’s an infinitesimal vector, and, hence, if we write the tangential component of the force along the path as Ft, then F•d= |F||ds|cosθ = F·cosθ·ds = Ft·ds. So this F•ds is a tangential component over an infinitesimally small segment of the curve. In short, it’s an infinitesimally small contribution to the total amount of work done indeed. You can make sense of this by looking at the geometrical representation of the situation below.

illustration 1

I am just saying this so you know what that integral stands for. Note that we’re not adding arrows once again, like we did when calculating amplitudes or so. It’s all much more straightforward really: a vector dot product is a scalar, so it’s just some real number—just like any component of a vector (tangential, normal, in the direction of one of the coordinates axes, or in whatever direction) is not a vector but a real number. Hence, W is also just some real number. It can be positive or negative because… Well… When we’d be going down the stairs with our bucket of water, our minus sign doesn’t disappear. Indeed, our convention to put that minus sign there should obviously not depend on what point a and b we’re talking about, so we may actually be going along the direction of the force when going from a to b.

As a matter of fact, you should note that’s actually the situation which is depicted above. So then we get a negative number for W. Does that make sense? Of course it does: we’re obviously not doing any work here as we’re moving along the direction, so we’re surely not adding any (potential) energy to the system. On the contrary, we’re taking energy out of the system. Hence, we are reducing its (potential) energy and, hence, we should have a negative value for W indeed. So, just think of the minus sign being there to ensure we add potential energy to the system when going against the force, and reducing it when going with the force.

OK. You get this. You probably also know we’ll re-define W as a difference in potential between two points, which we’ll write as Φ(b) – Φ(a). Now that should remind you of your high school integral ∫f(x)dx once again. For a definite integral over a line segment [a, b], you’d have to find the antiderivative of f(x), which you’d write as F(x), and then you’d take the difference F(b) – F(a) too. Now, you may or may not remember that this antiderivative was actually a family of functions F(x) + k, and k could be any constant – 5/9, 6π, 3.6×10124, 0.86, whatever! – because such constant vanishes when taking the derivative.

Here we have the same, we can define an infinite number of functions Φ(r) + k, of which the gradient will yield… Stop! I am going too fast here. First, we need to re-write that W function above in order to ensure we’re calculating stuff in terms of the unit charge, so we write:

unit chage

Huh? Well… Yes. I am using the definition of the field E here really: E is the force (F) when putting a unit charge in the field. Hence, if we want the work done per unit charge, i.e. W(unit), then we have to integrate the vector dot product E·ds over the path from a to b. But so now you see what I want to do. It makes the comparison with our high school integral complete. Instead of taking a derivative in regard to one variable only, i.e. dF(x)/dx) = f(x), we have a function Φ here not in one but in three variables: Φ = Φ(x, y, z) = Φ(r) and, therefore, we have to take the vector derivative (or gradient as it’s called) of Φ to get E:

Φ(x, y, z) = (∂Φ/∂x, ∂Φ/∂y, ∂Φ/∂z) = –E(x, y, z)

But so it’s the same principle as what you learned how to use to solve your high school integral. Now, you’ll usually see the expression above written as:

E = –Φ

Why so short? Well… We all just love these mysterious abbreviations, don’t we? 🙂 Jokes aside, it’s true some of those vector equations pack an awful lot of information. Just take Feynman’s advice here: “If it helps to write out the components to be sure you understand what’s going on, just do it. There is nothing inelegant about that. In fact, there is often a certain cleverness in doing just that.” So… Let’s move on.

I should mention that we can only apply this more sophisticated version of the ‘high school trick’ because Φ and E are like temperature (T) and heat flow (h): they are fields. T is a scalar field and h is a vector field, and so that’s why we can and should apply our new trick: if we have the scalar field, we can derive the vector field. In case you want more details, I’ll just refer you to our first vector calculus class. Indeed, our so-called First Theorem in vector calculus was just about the more sophisticated version of the ‘high school trick’: if we have some scalar field ψ (like temperature or potential, for example: just substitute the ψ in the equation below for T or Φ), then we’ll always find that:

First theorem

The Γ here is the curve between point 1 and 2, so that’s the path along which we’re going, and ψ must represent some vector field.

Let’s go back to our W integral. I should mention that it doesn’t matter what path we take: we’ll always get the same value for W, regardless of what path we take. That’s why the illustration above showed two possible paths: it doesn’t matter which one we take. Again, that’s only because E is a vector field. To be precise, the electrostatic field is a so-called conservative vector field, which means that we can’t get energy out of the field by first carrying some charge along one path, and then carrying it back along another. You’ll probably find that’s obvious,  and it is. Just note it somewhere in the back of your mind.

So we’re done. We should just substitute E for Φ, shouldn’t we? Well… Yes. For minus Φ, that is. Another minus sign. Why? Well… It makes that W(unit) integral come out alright. Indeed, we want a formula like W = Φ(b) – Φ(a), not like Φ(a) – Φ(b). Look at it. We could, indeed, define E as the (positive) gradient of some scalar field ψ = –Φ, and so we could write E = ψ, but then we’d find that W = –[ψ(b) – ψ(a)] = ψ(a) – ψ(b).

You’ll say: so what? Well… Nothing much. It’s just that our field vectors would point from lower to higher values of ψ, so they would be flowing uphill, so to say. Now, we don’t want that in physics. Why? It just doesn’t look good. We want our field vectors to be directed from higher potential to lower potential, always. Just think of it: heat (h) flows from higher temperature (T) to lower, and Newton’s apple falls from greater to lower height. Likewise, when putting a unit charge in the field, we want to see it move from higher to lower electric potential. Now, we can’t change the direction of E, because that’s the direction of the force and Nature doesn’t care about our conventions and so we can’t choose the direction of the force. But we can choose our convention. So that’s why we put a minus sign in front of Φ when writing E = –Φ. It makes everything come out alright. 🙂 That’s why we also have a minus sign in the differential heat flow equation: h = –κT.

So now we have the easy W(unit) = Φ(b) – Φ(a) formula that we wanted all along. Now, note that, when we say a unit charge, we mean a plus one charge. Yes: +1. So that’s the charge of the proton (it’s denoted by e) so you should stop thinking about moving electrons around! [I am saying this because I used to confuse myself by doing that. You end up with the same formulas for W and Φ but it just takes you longer to get there, so let me save you some time here. :-)]

But… Yes? In reality, it’s electrons going through a wire, isn’t? Not protons. Yes. But it doesn’t matter. Units are units in physics, and they’re always +1, for whatever (time, distance, charge, mass, spin, etcetera). AlwaysFor whatever. Also note that in laboratory experiments, or particle accelerators, we often use protons instead of electrons, so there’s nothing weird about it. Finally, and most fundamentally, if we have a –e charge moving through a neutral wire in one direction, then that’s exactly the same as a +e charge moving in the other way.

Just to make sure you get the point, let’s look at that illustration once again. We already said that we have F and, hence, E pointing from a to b and we’ll be reducing the potential energy of the system when moving our unit charge from a to b, so W was some negative value. Now, taking into account we want field lines to point from higher to lower potential, Φ(a) should be larger than Φ(b), and so… Well.. Yes. It all makes sense: we have a negative difference Φ(b) – Φ(a) = W(unit), which amounts, of course, to the reduction in potential energy.

The last thing we need to take care of now, is the reference point. Indeed, any Φ(r) + k function will do, so which one do we take? The approach here is to take a reference point Pat infinity. What’s infinity? Well… Hard to say. It’s a place that’s very far away from all of the charges we’ve got lying around here. Very far away indeed. So far away we can say there is nothing there really. No charges whatsoever. 🙂 Something like that. 🙂 In any case. I need to move on. So Φ(P0) is zero and so we can finally jot down the grand result for the electric potential Φ(P) (aka as the electrostatic or electric field potential):

So now we can calculate all potentials, i.e. when we know where the charges are at least. I’ve shown an example below. As you can see, besides having zero potential at infinity, we will usually also have one or more equipotential surfaces with zero potential. One could say these zero potential lines sort of ‘separate’ the positive and negative space. That’s not a very scientifically accurate description but you know what I mean.

Let me make a few final notes about the units. First, let me, once again, note that our unit charge is plus one, and it will flow from positive to negative potential indeed, as shown below, even if we know that, in an actual electric circuit, and so now I am talking about a copper wire or something similar, that means the (free) electrons will move in the other direction.

1280px-Current_notationIf you’re smart (and you are), you’ll say: what about the right-hand rule for the magnetic force? Well… We’re not discussing the magnetic force here but, because you insist, rest assured it comes out alright. Look at the illustration below of the magnetic force on a wire with a current, which is a pretty standard one.

terminalSo we have a given B, because of the bar magnet, and then v, the velocity vector for the… Electrons? No. You need to be consistent. It’s the velocity vector for the unit charges, which are positive (+e). Now just calculate the force F = qv×B = ev×B using the right-hand rule for the vector cross product, as illustrated below. So v is the thumb and B is the index finger in this case. All you need to do is tilt your hand, and it comes out alright.

But… We know it’s electrons going the other way. Well… If you insist. But then you have to put a minus sign in front of the q, because we’re talking minus e (–e). So now v is in the other direction and so v×B is in the other direction indeed, but our force F = qv×B = –ev×is not. Fortunately not, because physical reality should not depend on our conventions. 🙂 So… What’s the conclusion. Nothing. You may or may not want to remember that, when we say that our current j current flows in this or that direction, we actually might be talking electrons (with charge minus one) flowing in the opposite direction, but then it doesn’t matter. In addition, as mentioned above, in laboratory experiments or accelerators, we may actually be talking protons instead of electrons, so don’t assume electromagnetism is the business of electrons only.

To conclude this disproportionately long introduction (we’re finally ready to talk more difficult stuff), I should just make a note on the units. Electric potential is measured in volts, as you know. However, it’s obvious from all that I wrote above that it’s the difference in potential that matters really. From the definition above, it should be measured in the same unit as our unit for energy, or for work, so that’s the joule. To be precise, it should be measured in joule per unit charge. But here we have one of the very few inconsistencies in physics when it comes to units. The proton is said to be the unit charge (e), but its actual value is measured in coulomb (C). To be precise: +1 e = 1.602176565(35)×10−19 C. So we do not measure voltage – sorry, potential difference 🙂 – in joule but in joule per coulomb (J/C).

Now, we usually use another term for the joule/coulomb unit. You guessed it (because I said it): it’s the volt (V). One volt is one joule/coulomb: 1 V = 1 J/C. That’s not fair, you’ll say. You’re right, but so the proton charge e is not a so-called SI unit. Is the Coulomb an SI unit? Yes. It’s derived from the ampere (A) which, believe it or not, is actually an SI base unit. One ampere is 6.241×1018 electrons (i.e. one coulomb) per second. You may wonder how the ampere (or the coulomb) can be a base unit. Can they be expressed in terms of kilogram, meter and second, like all other base units. The answer is yes but, as you can imagine, it’s a bit of a complex description and so I’ll refer you to the Web for that.

The Poisson equation

I started this post by saying that I’d talk about fields and present some results from electrostatics using our ‘new’ vector differential operators, so it’s about time I do that. The first equation is a simple one. Using our E = –Φ formula, we can re-write the ∇•E = ρ/ε0 equation as:

∇•E = ∇•∇Φ = ∇2Φ = –ρ/ε0

This is a so-called Poisson equation. The ∇2 operator is referred to as the Laplacian and is sometimes also written as Δ, but I don’t like that because it’s also the symbol for the total differential, and that’s definitely not the same thing. The formula for the Laplacian is given below. Note that it acts on a scalar field (i.e. the potential function Φ in this case).

LaplacianAs Feynman notes: “The entire subject of electrostatics is merely the study of the solutions of this one equation.” However, I should note that this doesn’t prevent Feynman from devoting at least a dozen of his Lectures on it, and they’re not the easiest ones to read. [In case you’d doubt this statement, just have a look at his lecture on electric dipoles, for example.] In short: don’t think the ‘study of this one equation’ is easy. All I’ll do is just note some of the most fundamental results of this ‘study’.

Also note that ∇•E is one of our ‘new’ vector differential operators indeed: it’s the vector dot product of our del operator () with E. That’s something very different than, let’s say, Φ. A little dot and some bold-face type make an enormous difference here. 🙂 You may or may remember that we referred to the ∇• operator as the divergence (div) operator (see my post on that).

Gauss’ Law

Gauss’ Law is not to be confused with Gauss’ Theorem, about which I wrote elsewhere. It gives the flux of E through a closed surface S, any closed surface S really, as the sum of all charges inside the surface divided by the electric constant ε(but then you know that constant is just there to make the units come out alright).

Gauss' Law

The derivation of Gauss’ Law is a bit lengthy, which is why I won’t reproduce it here, but you should note its derivation is based, mainly, on the fact that (a) surface areas are proportional to r2 (so if we double the distance from the source, the surface area will quadruple), and (b) the magnitude of E is given by an inverse-square law, so it decreases as 1/r2. That explains why, if the surface S describes a sphere, the number we get from Gauss’ Law is independent of the radius of the sphere. The diagram below (credit goes to Wikipedia) illustrates the idea.

The diagram can be used to show how a field and its flux can be represented. Indeed, the lines represent the flux of E emanating from a charge. Now, the total number of flux lines depends on the charge but is constant with increasing distance because the force is radial and spherically symmetric. A greater density of flux lines (lines per unit area) means a stronger field, with the density of flux lines (i.e. the magnitude of E) following an inverse-square law indeed, because the surface area of a sphere increases with the square of the radius. Hence, in Gauss’ Law, the two effect cancel out: the two factors vary with distance, but their product is a constant.

Now, if we describe the location of charges in terms of charge densities (ρ), then we can write Qint as:

Q int

Now, Gauss’ Law also applies to an infinitesimal cubical surface and, in one of my posts on vector calculus, I showed that the flux of E out of such cube is given by E·dV. At this point, it’s probably a good idea to remind you of what this ‘new’ vector differential operator •, i.e. our ‘divergence’ operator, stands for: the divergence of E (i.e. • applied to E, so that’s E) represents the volume density of the flux of E out of an infinitesimal volume around a given point. Hence, it’s the flux per unit volume, as opposed to the flux out of the infinitesimal cube itself, which is the product of and dV, i.e. E·dV.

So what? Well… Gauss’ Law applied to our infinitesimal volume gives us the following equality:

ES 1

That, in turn, simplifies to:

ES 2

So that’s Maxwell’s first equation once again, which is equivalent to our Poisson equation: E = ∇2Φ = –ρ/ε0. So what are we doing here? Just listing equivalent formulas? Yes. I should also note they can be derived from Coulomb’s law of force, which is probably the one you learned in high school. So… Yes. It’s all consistent. But then that’s what we should expect, of course. 🙂

The energy in a field

All these formulas look very abstract. It’s about time we use them for something. A lot of what’s written in Feynman’s Lectures on electrostatics is applied stuff indeed: it focuses, among other things, on calculating the potential in various circumstances and for various distributions of charge. Now, funnily enough, while that E = –ρ/ε0 equation is equivalent to Coulomb’s law and, obviously, much more compact to write down, Coulomb’s law is easier to start with for basic calculations. Let me first write Coulomb’s law. You’ll probably recognize it from your high school days:

Coulomb's law

Fis the force on charge q1, and Fis the force on charge q2. Now, qand q2. may attract or repel each other but, in both cases, the forces will be equal and opposite. [In case you wonder, yes, that’s basically the law of action and reaction.] The e12 vector is the unit vector from qto q1, not from qto q2, as one might expect. That’s because we’re not talking gravity here: like charges do not attract but repel and, hence, we have to switch the order here. Having said that, that’s basically the only peculiar thing about the equation. All the rest is standard:

  1. The force is inversely proportional to the square of the distance and so we have an inverse-square law here indeed.
  2. The force is proportional to the charge(s).
  3. Finally, we have a proportionality constant, 1/4πε0, which makes the units come out alright. You may wonder why it’s written the way it’s written, i.e. with that 4π factor, but that factor (4π or 2π) actually disappears in a number of calculations, so then we will be left with just a 1/ε0 or a 1/2ε0 factor. So don’t worry about it.

We want to calculate potentials and all that, so the first thing we’ll do is calculate the force on a unit charge. So we’ll divide that equation by q1, to calculate E(1) = F1/q1:

E 1

Piece of cake. But… What’s E(1) really? Well… It’s the force on the unit charge (+e), but so it doesn’t matter whether or not that unit charge is actually there, so it’s the field E caused by a charge q2. [If that doesn’t make sense to you, think again.] So we can drop the subscripts and just write:

E 3

What a relief, isn’t it? The simplest formula ever: the (magnitude) of the field as a simple function of the charge q and its distance (r) from the point that we’re looking at, which we’ll write as P = (x, y, z). But what origin are we using to measure x, y and z. Don’t be surprised: the origin is q.

Now that’s a formula we can use in the Φ(P) integral. Indeed, the antiderivative is ∫(q/4πε0r2)dr. Now, we can bring q/4πε0 out and so we’re left with ∫(1/r2)dr. Now ∫(1/r2)dr is equal to –1/r + k, and so the whole antiderivative is –q/4πε0r + k. However, the minus sign cancels out with the minus sign in front of the Φ(P) = Φ(x, y, z)  integral, and so we get:

E 4

You should just do the integral to check this result. It’s the same integral but with P0 (infinity) as point a and P as point b in the integral, so we have ∞ as start value and r as end value. The integral then yields Φ(P) – Φ(P0) = –q/4πε0[1/r – 1/∞). [The k constant falls away when subtracting Φ(P0) from Φ(P).] But 1/∞ = 0, and we had a minus sign in front of the integral, which cancels the sign of –q/4πε0. So, yes, we get the wonderfully simple result above. Also please do quickly check if it makes sense in terms of sign: the unit charge is +e, so that’s a positive charge. Hence, Φ(x, y, z) will be positive if the sign of q is also positive, but negative if q would happen to be negative. So that’s OK.

Also note that the potential – which, remember, represents the amount of work to be done when bringing a unit charge (e) from infinity to some distance r from a charge q – is proportional to the charge of q. We also know that the force and, hence, the work is proportional to the charge that we are bringing in (that’s how we calculated the work per unit in the first place: by dividing the total amount of work by the charge). Hence, if we’d not bring some unit charge but some other charge q2, the work done would also be proportional to q2. Now, we need to make sure we understand what we’re writing and so let’s tidy up and re-label our first charge once again as q1, and the distance r as r12, because that’s what r is: the distance between the two charges. We then have another obvious but nice result: the work done in bringing two charges together from a large distance (infinity) is

U 1Now, one of the many nice properties of fields (scalar or vector fields) and the associated energies (because that’s what we are talking about here) is that we can simply add up contributions. For example, if we’d have many charges and we’d want to calculate the potential Φ at a point which we call 1, we can use the same Φ(r) = q/4πε0r formula which we had derived for one charge only, for all charges, and then we simply add the contributions of each to get the total potential:

P 1

Now that we’re here, I should, of course, also give the continuum version of this formula, i.e. the formula used when we’re talking charge densities rather than individual charges. The sum then becomes an infinite sum (i.e. an integral), and qj (note that j goes from 2 to n) becomes a variable which we write as ρ(2). We get:

U 2

Going back to the discrete situation, we get the same type of sum when bringing multiple pairs of charges qi and qj together. Hence, the total electrostatic energy U is the sum of the energies of all possible pairs of charges:

U 3It’s been a while since you’ve seen any diagram or so, so let me insert one just to reassure you it’s as simple as that indeed:

U system

Now, we have to be aware of the risk of double-counting, of course. We should not be adding qiqj/4πε0rij twice. That’s why we write ‘all pairs’ under the ∑ summation sign, instead of the usual i, j subscripts. The continuum version of this equation below makes that 1/2 factor explicit:

U 4

Hmm… What kind of integral is that? It’s a so-called double integral because we have two variables here. Not easy. However, there’s a lucky break. We can use the continuum version of our formula for Φ(1) to get rid of the ρ(2) and dV2 variables and reduce the whole thing to a more standard ‘single’ integral. Indeed, we can write:

U 5Now, because our point (2) no longer appears, we can actually write that more elegantly as:

U 6That looks nice, doesn’t it? But do we understand it? Just to make sure. Let me explain it. The potential energy of the charge ρdV is the product of this charge and the potential at the same point. The total energy is therefore the integral over ϕρdV, but then we are counting energies twice, so that’s why we need the 1/2 factor. Now, we can write this even more beautifully as:

U 7

Isn’t this wonderful? We have an expression for the energy of a field, not in terms of the charges or the charge distribution, but in terms of the field they produce.

I am pretty sure that, by now, you must be suffering from ‘formula overload’, so you probably are just gazing at this without even bothering to try to understand. Too bad, and you should take a break then or just go do something else, like biking or so. 🙂

First, you should note that you know this EE expression already: EE is just the square of the magnitude of the field vector E, so EE = E2. That makes sense because we know, from what we know about waves, that the energy is always proportional to the square of an amplitude, and so we’re just writing the same here but with a little proportionality constant (ε0).

OK, you’ll say. But you probably still wonder what use this formula could possibly have. What is that number we get from some integration over all space? So we associate the Universe with some number and then what? Well… Isn’t that just nice? 🙂 Jokes aside, we’re actually looking at that EE = Eproduct inside of the integral as representing an energy density (i.e. the energy per unit volume). We’ll denote that with a lower-case symbol and so we write:

D 6

Just to make sure you ‘get’ what we’re talking about here: u is the energy density in the little cube dV in the rather simplistic (and, therefore, extremely useful) illustration below (which, just like most of what I write above, I got from Feynman).

Now that should make sense to you—I hope. 🙂 In any case, if you’re still with me, and if you’re not all formula-ed out you may wonder how we get that ε0EE = ε0E2 expression from that ρΦ expression. Of course, you know that E = –∇Φ, and we also have the Poisson equation ∇2Φ = –ρ/ε0, but that doesn’t get you very far. It’s one of those examples where an easy-looking formula requires a lot of gymnastics. However, as the objective of this post is to do some of that, let me take you through the derivation.

Let’s do something with that Poisson equation first, so we’ll re-write it as ρ = –ε02Φ, and then we can substitute ρ in the integral with the ρΦ product. So we get:

U 8

Now, you should check out those fancy formulas with our new vector differential operators which we listed in our second class on vector calculus, but, unfortunately, none of them apply. So we have to write it all out and see what we get:

D 1

Now that looks horrendous and so you’ll surely think we won’t get anywhere with that. Well… Physicists don’t despair as easily as we do, it seems, and so they do substitute it in the integral which, of course, becomes an even more monstrous expression, because we now have two volume integrals instead of one! Indeed, we get:

D 2But if Φ is a vector field (it’s minus E, remember!), then ΦΦ is a vector field too, and we can then apply Gauss’ Theorem, which we mentioned in our first class on vector calculus, and which – mind you! – has nothing to do with Gauss’ Law. Indeed, Gauss produced so much it’s difficult to keep track of it all. 🙂 So let me remind you of this theorem. [I should also show why ΦΦ still yields a field, but I’ll assume you believe me.] Gauss’ Theorem basically shows how we can go from a volume integral to a surface integral:

Gauss Theorem-2If we apply this to the second integral in our U expression, we get:

D 4

So what? Where are we going with this? Relax. Be patient. What volume and surface are we talking about here? To make sure we have all charges and influences, we should integrate over all space and, hence, the surface goes to infinity. So we’re talking a (spherical) surface of enormous radius R whose center is the origin of our coordinate system. I know that sounds ridiculous but, from a math point of view, it is just the same like bringing a charge in from infinity, which is what we did to calculate the potential. So if we don’t difficulty with infinite line integrals, we should not have difficulty with infinite surface and infinite volumes. That’s all I can, so… Well… Let’s do it.

Let’s look at that product ΦΦ•n in the surface integral. Φ is a scalar and Φ is a vector, and so… Well… Φ•is a scalar too: it’s the normal component of Φ = –E. [Just to make sure, you should note that the way we define the normal unit vector n is such that ∇Φ•n is some positive number indeed! So n will point in the same direction, more or less, as ∇Φ = –E. So the θ angle  between ∇Φ = –E and n is surely less than ± 90° and, hence, the cosine factor in the ∇Φ•= |∇Φ||n|cosθ = |∇Φ|cosθ is positive, and so the whole vector dot product is positive.]

So, we have a product of two scalars here.  What happens with them if R goes to infinity? Well… The potential varies as 1/r as we’re going to infinity. That’s obvious from that Φ = (q/4πε0)(1/r) formula: just think of q as some kind of average now, which works because we assume all charges are located within some finite distance, while we’re going to infinity. What about Φ•n? Well… Again assuming that we’re reasonably far away from the charges, we’re talking the density of flux lines here (i.e. the magnitude of E) which, as shown above, follows an inverse-square law, because the surface area of a sphere increases with the square of the radius. So Φ•n varies not as 1/r but as 1/r2. To make a long story short, the whole product ΦΦ•n falls of as 1/r goes to infinity. Now, we shouldn’t forget we’re integrating a surface integral here, with r = R, and so it’s R going to infinity. So that surface integral has to go to zero when we include all space. The volume integral still stands however, so our formula for U now consists of one term only, i.e. the volume integral, and so we now have:

D 5

Done !

What’s left?

In electrostatics? Lots. Electric dipoles (like polar molecules), electrolytes, plasma oscillations, ionic crystals, electricity in the atmosphere (like lightning!), dielectrics and polarization (including condensers), ferroelectricity,… As soon as we try to apply our theory to matter, things become hugely complicated. But the theory works. Fortunately! 🙂 I have to refer you to textbooks, though, in case you’d want to know more about it. [I am sure you don’t, but then one never knows.]

What I wanted to do is to give you some feel for those vector and field equations in the electrostatic case. We now need to bring magnetic field back into the picture and, most importantly, move to electrodynamics, in which the electric and magnetic field do not appear as completely separate things. No! In electrodynamics, they are fully interconnected through the time derivatives ∂E/∂t and ∂B/∂t. That shows they’re part and parcel of the same thing really: electromagnetism.

But we’ll try to tackle that in future posts. Goodbye for now!


Content-Type: application/http; msgtype=response

WARC-Identified-Payload-Type: text/html

WARC-Type: response

text:
< New Search

Sorry we are experiencing system issues. Please try again.
Contact The Agent
“All we can say is WOW! Steve was always available for questions, and made us feel like we were the only client. Our home sold within 12 days and we found our dream home within one week. We were totally...
Get More Information
Please use a valid email.
Please use a valid phone.
Please fill in all fields.
view now
My Dashboard

warcinfo:
isPartOf: CC-MAIN-2019-04
publisher: Common Crawl
description: Wide crawl of the web for January 2019
operator: Common Crawl Admin (info@commoncrawl.org)
hostname: ip-10-145-104-113.ec2.internal
software: Apache Nutch 1.15 (modified, https://github.com/commoncrawl/nutch/)
robots: checked via crawler-commons 0.11-SNAPSHOT (https://github.com/crawler-commons/crawler-commons)
format: WARC File Format 1.1
conformsTo: http://iipc.github.io/warc-specifications/specifications/warc-format/warc-1.1/


text:
Wedding Wednesday Ceremony Deets

I wish I could describe walking down the aisle but there are literally no words. I can tell you I had only string quartet music playing because I selected specific music for every part of the ceremony and our DJ did an amazing job with it. I can tell you that my son walked me down the aisle and that not tripping over my dress was honestly a concern. That thing was huge! I was a big ivory cupcake! I remember catching my grooms eye and nothing else in the world mattered. The ceremony was traditional and short. The ceremony was officiated by a friend of my husband’s.  We didn’t write our own vows or anything like that. Just the normal love honor cherish till death do us part kinda thing. The ceremony was a little chilly so we handed out the wedding favor of throw blankets early to keep everyone warm and they were a grand hit. And now here are some pics! I hope you enjoy. I’m also including a link to two of the songs used for the ceremony. My groom entrance and then of course my entrance! It was definitely one of the most amazing moments of my life. I got so emotional but I made it through without crying barely.

Groom entrance
Bride entrance
Happy wedding planning etc my loves!

Wedding Wednesday – First Look

There is a lot of controversy over the first look. Should you have one? Can you kiss? There are variables and opinions upon opinions to be had. My entire wedding party (well all the girls and my mom) were completely against it. But we held our ground and stayed firm to wanting to see each other before the ceremony. Our day had been kinda hectic and not going quite as planned before the first look and I think by the time we were to that moment in the timeline for the days events all we wanted was each other. We met in the garden at the Hermitage and even though it was fall it was still so beautiful. Here are some of our favorite pics from that moment and short period of time.

That collage above was all real moment. We were so happy to see each other we literally forgot the camera was there. We both enjoyed this moment together so very much. We still talk about it these 4 and a half months later. It was just powerful. Yes I realize there is the moment that the groom sees the bride when she’s walking down the aisle and it is magical but in our opinion this “first look” did not take away from that moment at all. And then of course we ended with some posed shots! Man this was really just the best day ever!

Wedding Wednesday – Getting Ready

So you may be wondering if in fact there was a wedding and well yes there was. I’m going to post some pics today about the getting ready time and I’ll add more details and fun stuff to future posts but here’s a good start. The night before me and one of my girls stayed at the hotel. My future husband had all of his boys with him at his house. I don’t have enough photos of that girl time or even us getting our makeup on, but if I find some they are totally going to be shared.

This last shot is me on the way to our first look.  I couldn’t wait to hug him. It had been quite the morning. And my future husband was waiting to see what all the fuss was about this gigantic cupcake dress!

Until next week, Happy Wedding Wednesday!

Travel Thursday – Louisville

Louisville was such a fun place to visit. And I can’t wait to return. One of my favorite thing to do is explore historical places. And that place for me on this particular trip was Conrad Caldwell House.

 The dress in the photo was the dress on the couch below. I was more impressed the frame was original.

A dream place setting to say the least. Monogram is my jam!
 Original doorways. Check the intricacy of the woodwork.

Oh the staircase. This is a piece of beauty that you should visit if there is any way. It’s worth the trip.

  My cookbook fascination went nuts when I came across these little babies!

 This painting was commissioned by one of the owners of the house. I wish I had paid better attention but it’s amazing and huge!

Again fascinating detailed woodwork.

Stain glassed window if you please!

 Well of course it’s a wedding dress because I’m obsessed. Lol!

 An amazing tour! This historical home is a must see. Check out their website Conrad Caldwell House and if you are in Louisville you should visit. It’s worth it!


text:
Search Site

• • • • • • • • •

Recent stuff

[an error occurred while processing this directive]

• • • • • • • • •

Our email address is editor @realization.org.

Copyright 2001 Realization.org.

The Upanishads

THE UPANISHADS are philosophical texts that describe the teachings of Hindu sages who lived two to three thousand years ago.

They were the last texts added to the Vedas, the vast collection of writings that constitute Hinduism's most sacred literature.

Tradition recognizes 108 Vedic Upanishads, of which 10 to 13 are usually regarded as major.

The Upanishads differ from earlier Vedic texts in their lack of emphasis on physical sacrifice and ritual. Instead they speculate about the nature of reality and develop the doctrine of Brahman. It is in the Upanishads that the nondualistic equation of atman (self) and Brahman (God) emerges.

The predominant school of Hindu philosophy, Advaita Vedanta, is based on the Upanishads.

Aitareya Upanishad
This is one of the shortest of the major Upanishads. It contains a creation myth.

Isa Upanishad
This is one of the shortest of the major Upanishads. It usually appears first in traditional compilations.

Katha Upanishad

One of the most famous and widely-translated Upanishads. The King of Death explains the nature of the Self and how to realize it.

Kena Upanishad
The theme of this short Upanishad is the paradoxical nature of knowing God. If you consider yourself to be the performer of actions, such knowledge is impossible.

Mandukya Upanishad
Consisting of only twelve verses, this is the shortest Upanishad. According to Sankaracharya, this Upanishad, together with Gaudapada's commentary on it, "contains the epitome of the substance of the import of Vedanta."

Mundaka Upanishad
This short, poetic Upanishad makes the startling assertion that the knowledge contained in the Vedas, the holiest Hindu scriptures, is only a lower kind of knowledge. It prescribes a higher knowledge, the apprehension of Brahman.

Prasna Upanishad
One of the major Upanishads. It describes the five types of prana. Translated by F. Max Müller.

Taittiriya Upanishad
This Upanishad is of special importance to students of Advaita Vedanta and Jnana Yoga because it's the only one that sets forth the doctrine of the five kosas or sheaths that conceal the true Self.

  Upanisads: A New Translation
By Patrick Olivelle

This is the best English translation we've seen of the twelve major Upanishads. It's the only one that's fully informed by modern scholarship, and the style is lucid and pleasant to read. To top it off, the book happens to be dirt-cheap. Patrick Olivelle is professor of Sanskrit and Indian Religions at the University of Texas at Austin.

Order from ABE

The Ten Principal Upanishads
Translated by Shree Purohit Swami and W.B. Yeats

There are translations for the heart and for the head; those that recreate the poetic, literary greatness of the original, and those that aim at academic fidelity. This is the best English translation of the first type that has ever been made of the Upanishads. Shree
Purohit Swami was an enormously talented yogi who came to London in 1930, and W.B. Yeats was one of the greatest English poets of the twentieth century. Unfortunately, this book is out of print, so you'll have to buy a used copy. It's worth it.

Sanskrit Documents List
List of links to texts of Upanishads on the Internet, mostly in Sanskrit.

Sanskrit Texts and Stotras
Includes some Upanishads in Sanskrit with English translations. Files are in PDF format.

The Upanishads

Translated by Max Müller.

Upanishad Links
Maintained by the Sanskrit Documents List website.

Upanishad Page
Part of the Global Hindu Electronic Network (HinduNet).

Illustration c. 1680, courtesy Victoria and Albert Museum, accession number IS.42-1953.

This page was published on May 30, 2000 and last revised on May 12, 2001.

Copyright 2001 Realization.org. All rights reserved.


text:
Hollywood Moms Out Of Control

Okay National Enquirer, you got me with that headline. You got my $3.99US (a random $5.49 if I were in Canada.) You win. Smut wins the day. Curiosity killed this cat, a mother cat desperately looking for a feeling that someone was doing a worse job than she in the grooming, discipline and care of her kittens.

I wanted those Hollywood moms to be out of control damn it! The cover promised “Boozing, Drugs, Stripping, Suicide Attempts and More.” More? You say. More than all that? Well I haven’t done any of those things in the last twenty-four to forty-eight hours, so I will read on.

Turns out Kelly Ripa pretended to strip for a comedy event, told some jokes and went home to the hot husband she’s still faithful to. Also Kirstie Alley eats too much; I believe when she named her own show “Fat Actress” she was pretty much copping to that one. Janice Dickinson is addicted to plastic surgery and also has children, which is slightly disturbing, but she’s marrying a shrink, which is allegedly good news. Marie Osmond was depressed in 2006, Kate Moss did coke in 2005, and get this, you guys, because I know you are going to FREAK OUT, but Courtney Love, contrary to her squeaky clean image, has been a habitual drug user and may still be one.

It’s a shame they don’t wrap fish and chips in newspaper here, as at least this garbage would actually be useful for something.

Although still reeling from the other revelations in the Enquirer, like that Tori Spelling, Goldie Hawn and Lisa Marie Presley each had too much to drink ONCE, I will attempt to fashion some sense as to why this piece of Pulitzer Prize winning journalism would have appealed to me in the first place.

I am having one of those days where I feel that as a mother I am not good enough. Also, as an actress, not successful enough (though I just yesterday ran into a rabid fan who “almost died” watching my videos, which is, I guess a good thing.) While I have no bitterness about the creative work I have produced in the last decade or two, and in turn the difference that has made to many, for some reason I’m not thrilled that I still have no measurable income. Add that to the shitty motherhood complex and you have a recipe for “Low Self-Esteem”, the long anticipated sequel to “That Time I thought About Killing Myself When I Had Two Children Under Two And The Nanny Quit Without Warning.”

Although since Heather Locklear attempted suicide, it kind of lost its subversive allure.

I need to know that no one is having it all today, as I feel I’m having barely any of it. I want to be there for my kids, whose need for me is evidenced by six hang-up calls, one conversation and one message left by my older son insisting that I HAD to bring his iPad to after school care. The message he left an hour later was him crying, “Mommy, I’m so tired I want to go home.” My heart broke even though, I was an hour and a half away engaged in tedious therapy so I could be a better mother for both him and his brother.
While the iPad part made me consider sending him on a six-month sojourn to the Peace Corps in Uganda where they neither have iPads, nor even iPad cases to put them in. I am tempted to have a little chat about entitlement, and then let’s send his privileged white ass off to live in a hut, shall we?

But then, of course another part of me feels guilty for not being there when he needs me, for the fact that I keep them both in childcare until six o’clock, because it is so tough to handle them and otherwise I would never write a thing. A part of me worries that I have done it all wrong and need to start again as a stay-at-home mom who bakes gluten free banana bread, and does fun crafts and never loses her temper. And that part of me that I have to live with is the Jewish part. And it whispers in my ear that maybe the National Enquirer might be onto something, that sometimes a little guilt can be a good thing.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s


text:
Виртуал аялал

FOR SALE: Amazing property in the first ocean row in Puertito de Los Silos, Tenerife!

Two bedroom penthouse apartment on the 8th floor with panoramic ocean views!

The apartment has high ceilings, amazing terrace and windows in every room. Low community fees and a well-maintained complex with swimming pool, elevators and a football field! Free parking by the front door!

The property is perfectly located:

  • Puertito de Los Silos – the little port where you can keep your boat in just 50 meters.
  • Little sandy beach inside the port – 50 meters.
  • Bigger beach and natural pools – 350 meters.
  • Surf spots, walking trails, restaurant, bar, playground for kids and big municipal swimming pool – all within 350 meters from the complex.

The apartment requires renovation. Our company can assist you with it! With additional €10.000 – €15.000 you can convert it into a penthouse of your dreams, renovated and decorated to your taste!

Please contact us to arrange you visit!


text:
A Meaning for the Mess

…yeah, apparently I have a thing for sunsets… and quotes…

...yeah I apparently have a thing for sunsets...
“Finish every day and be done with it. You have done what you could. Some blunders and absurdities no doubt crept in; forget them as soon as you can. Tomorrow is a new day; begin it well and serenely and with too high a spirit to be cumbered with your old nonsense. This day is all that is good and fair.. It is too dear with its hopes and invitations, to waste a moment on the yesterdays.” ~Ralph Waldo Emerson

Today I have a few pictures and 2 lovely posts from other people I’m feeling led to share…! I don’t know how they know what I’m thinking or what I need to hear but I will defer to them and their beautiful words…and maybe it’s something you need to hear as well…

First, this lady is Hi-larious. For real. I read her book “7” and I know people looked at me strangely in waiting rooms as I laughed out loud more than once because she was so fantastically amusing…

Secondly this gal is awesome. Again, I’ve read a couple of her books and each one has me taking notes and really thinking through and unpacking my reactions to it for days on end…

 On another seemingly random, but totally related note if you read that post by Lysa Terkeurst that I linked to above…

I have driven by this sad run down place for 7+ years….

"It's all messy: The hair. The bed. The words. The heart. Life" ~William Leal
“It’s all messy: The hair. The bed. The words. The heart. Life” ~William Leal

but what I failed to notice until yesterday was this:

"There's joy to be found in every.single.day. It's all good" ~Anonymous
“There’s joy to be found in every.single.day. It’s all good” ~Anonymous

Yep that door was blue, well, technically it still is blue… I had to look past the mess on the front. Past the rotted boards. Past the general not prettiness to see it but it was there waiting for me over on the side…

What joys are you noticing today?

P.S. Looking for more heartfelt quotes, old pictures and all around joy in the mundane? It’s my pleasure to create connection, community and sometimes cute stuff just for you! And I’d love it if you’d join our little tribe. Click right here to get the scoop and a set of printable scripture cards! Looking forward to chatting with you soon. Thanks so much for being here.

10 thoughts on “A Meaning for the Mess

  1. Can I borrow the first quote for myself??? AHHHH I need this hung in my living room for real! That makes 2 great phrases this week, 1 from here and 1 from FaithBarista (a scripture). I seriously think I need to just plaster positives all around and post my journey in it all! I am about to check out the links too. I am excited about the funny one – I need more funny in my life hehe. Thanks for sharing!!

    Liked by 1 person

I ♥ when YOU comment! (then I know I'm not just talking to myself..!)

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s


text:
sell rental property for cash
Tag Archive

Rental Property

Tired Landlords, do you Need to Sell your Rental Property?

As a landlord, you wear a lot of hats. You are the property owner so all the financial responsibility is legally on you. You might be the plumber, handyman, rent collector and property manager.…

Read more

Ready for your offer?


text:
SOLD – VW T25 High Top Campervan

1981 2ltr Petrol Air cooled with Drive Away Awning – MOT September 2015

VW with Awning

Features include:

  • Drive away awning.
  • Captains chair (passenger seat).
  • 2 ring gas cooker and grill.
  • 12 volt/gas fridge. Runs off battery when travelling, gas when parked up.
  • Sink with electric pump.
  • Large on board water tank (housed under rear seats)
  • Rock ‘n roll bed.
  • Additional double bed in high top.
  • Leisure battery – Requires replacing.
  • CD player/radio with 4 speakers (2 front, 2 rear) and line in.
  • Over cab storage.
  • Hook up lead with 4 bank adapter.
  • New curtains.
  • Taxed

Recently MOT’d has had a replacement rear swinging arm fitted.  The body work is sound and runs well.  Took us to Glastonbury and on for a tour of Devon this year with numerous weekend jaunts into Wales.  Not in the first flush of youth but ready to go.  Cosmetically leaves you plenty of scope to make your own mark.

Select thumbnail below for a whole gallery of images!


text:
Red Right 88

In Cleveland, hope dies last

Archive for the category “Cleveland Indians”

From the editor’s notebook …

jd martinez indiansIf the Cleveland Indians come up short this season in their quest to return to the playoffs, there will be plenty of moments and miss opportunities for Tribe fans to point to as they review the season.

The biggest one may be J.D. Martinez of the Detroit Tigers.

Friday night, Martinez had a three-run homer and a triple as the Tigers beat the Indians for the fourth time in the past five meetings. On the season, Martinez is batting .327 against the Tribe, with six home runs and 18 RBI.

It’s not just the Indians that Martinez has been tormenting as he has clearly taken a liking to the American League Central. He’s batting .432 against Kansas City, with five home runs and 15 RBI; .324 against Minnesota; and .319 against the White Sox.

Read more…

From the editor’s notebook …

browns final rosterThe final roster cuts have been made, the practice squad players have been signed and the Cleveland Browns have begun preparations for Sunday’s season-opening game against Pittsburgh.

We’re less than a week away from Browns football, whether anyone is ready for it or not.

The only real surprise among the player cuts was MarQueis Gray, who seemed like he might be able to help the team as a fullback. But the Browns apparently decided that if they were going with a fullback on the roster, why not have Ray Agnew – someone who has actually played the position – over trying to convert a play like Gray to the spot. It’s a radical idea, we know, but it just may be crazy enough to work.

The Browns didn’t waste any time parting ways with cornerback Leon McFadden -who was selected about three rounds too early in the 2013 NFL Draft – and guard Garrett Gilkey, leaving Barkevious Mingo and Armonty Bryant as the only players remaining from the one and only draft from Joe Banner and Mike Lombardi. So much for that duo’s continued claims that they set the team up for long-term success.

Read more…

From the editor’s notebook …

hoyer starting quarterbackCleveland Browns head coach Mike Pettine made if official earlier today, naming Brian Hoyer as the team’s starting quarterback.

The decision ends what we can only hope was not an ill-fated quarterback competition between Hoyer and rookie Johnny Manziel. Now Hoyer can work exclusively to get not only himself but the offense ready for the season-opener on Sept. 7 against Pittsburgh.

“(Brian) was the clear leader from the beginning,” Pettine said in announcing the decision. “We’ve maintained all along that if it was close, I would prefer to go with the more experienced player. Brian has done a great job in the meeting rooms and with his teammates on the practice field and in the locker room.”

Pettine probably made the right call (we’re pretty sure he did, but we won’t know until the season starts). Even though it is only a few games, Hoyer has more experience running an NFL offense than Manziel. Hopefully having him play most of Saturday’s preseason friendly against St. Louis, combined with giving him all the first-team reps in practice from here on out, will help him shake off the rust from not having played since injuring his knee last October.

Read more…

How did that happen?

tribe playoff raceIt has been an interesting summer so far, with plenty of attention on the World Cup; the return of LeBron James to Cleveland, and the possibility of a deal for Kevin Love; and all things Johnny Manziel as the Cleveland Browns work their way through training camp.

The Cleveland Indians have pretty much just been an afterthought, sitting on the periphery of Cleveland’s collective sporting conscience with a .500 record.

But take a closer look at the standings in the American League and, heading into Tuesday night’s game against Cincinnati, we realized that the Tribe is just 2.5 games out of a Wild Card spot.

Wait … what?

We’re not sure how they are doing it, but manager Terry Francona once again has the Tribe in the race for a playoff appearance.

In light of this surprising development, it seems like a good time for a quick Thumbs Up/Thumbs Down on the Tribe so far.

Read more…

From the editor’s notebook …

kyrie reacts to injuryWhen we heard the news that Indiana’s Paul George had suffered a major leg injury during Friday’s scrimmage with USA Basketball it reinforced our belief of club over country.

We understand that injuries can happen at any time (see Winslow, Kellen) and that this is the first  major injury suffered by an NBA player since they started playing on the national team in 1992, but we’ve never been worked up if a Cleveland player is passed over for an All-Star game or for an international competition. It certainly is a nice honor for a player, but from a fan perspective we never saw the value in it.

Why do we need to see Kyrie Irving (who has enough trouble staying on the court as it is) or LeBron James spend the off-season putting extra miles on their bodies? Or watch Joe Thomas or Alex Mack at the Pro Bowl, just waiting for someone to accidentally roll up their leg and blow out a knee? How did playing in the World Baseball Classic work out for Vinnie Pestano?

Read more…

From the editor’s notebook …

david blatt cavsThe Cleveland Cavaliers landed their man on Friday, naming David Blatt as the team’s latest head coach.

We admit to knowing next to nothing about Blatt before the Cavs undertook their third head coaching search in the past four years. But the more we learn about him the more we are intrigued by the hiring.

Blatt comes to the Cavs fresh off leading Maccabi Tel Aviv to the Euroleague championship. His coaching job in the tournament led one NBA general manager to observe that “Maccabi was outgunned at every position except coach. David took down two Goliaths in a weekend. He belongs in the NBA.” (h/t to @WFNYJacob for the link to that story.)

Over the past four seasons, Blatt’s teams went 222-55 and, in addition to this year’s Euroleague title, won the Israeli Cup and Israeli league title. He’s been a head coach in Europe for 15 years and coached the Russian national team from 2006 to 2012. (So at least he will be used to the harsh winter weather in Northeast Ohio.)

Read more…

The Tribe’s $56-million question

Nick+Swisher+Cleveland+Indians+v+Minnesota+DIPqAEf9VfllAs the Cleveland Indians continue to try and keep pace in the American League Central Division, manager Terry Francona faces an important question.

What to do about the Tribe’s $56-million man, Nick Swisher?

Since signing the largest free agent contract in franchise history prior to the 2013 season (more on that in a moment), Swisher has been average at best and pretty horrible at worst, considering that he has batted .225 or less in five of the nine months he has been a member of the team.

Read more…

Lonnie Chisenhall has a night to remember in Texas

lonnie chisenhall big nightRaise your hand if you saw this coming from Lonnie Chisenhall.

Chisenhall entered the season with a lifetime batting average of just .194 against left-handed pitching (and a pedestrian .254 against righties) but something has clicked for the Cleveland Indians’ third baseman this year – in a major way.

Chisenhall has seen his average go up each month of the season, and since he hit .362 in April, that’s saying something. He batted .373 in May and through the first nine days of June he is hitting .452. More importantly, he is ripping left-handed pitching, batting .520 on the season. He may only have 25 at bats against lefties, but that’s still impressive.

His season may have reached its high point Monday night in Texas, as Chisenhall had what may have been the best offensive night in the history of baseball. It was certainly the best in Tribe history.

Read more…

From the editor’s notebook …

Emre Can liverpoolLiverpool continued its summer makeover with the news on Thursday that they have agreed on a transfer deal for Emre Can from the German club Bayer Leverkusen.

The 20-year-old Can for four goals in 39 games with Leverkusen last season after joining the club from Bayern Munich. He was a key figure as Leverkusen finished in fourth place in the Bundesliga.

“Emre Can has developed very quickly at Bayer 04 Leverkusen over the past 12 months,” Michael Schade, the club’s chief executive, said in a statement. “We wanted to keep Emre at Leverkusen but the release clause is part of his contract. So we wish him all the best and lots of success in his time in the Premier League.”

Read more…

Thumbs up, thumbs down as we try to get a handle on the Tribe

tribe up and downWith a little more than a third of the Major League Baseball season now in the books, it is still hard to get a handle on the current edition of the Cleveland Indians.

Are they the team that was swept at home by Oakland, being outscored 30-6 in the three games? Or the one that followed that series by winning five-of-six, including a three-game sweep of Detroit?

Did we see the true Tribe during last week’s sweep at the hands of the Chicago White Sox? Or was it this past weekend when they swept the Colorado Rockies?

Maybe the truth lies somewhere in the middle, which would explain the Tribe’s current record of 27-30, which leaves the Indians a (somewhat) manageable six games out of first place in the A.L. Central Division and three games out of the Wild Card (although their are six teams ahead of them).

As the Indians host Boston before heading out on a 1o-game road trip – just in time for school to let out and the weather to warm up – that will take them to Texas, Kansas City and Boston, let’s look at what’s gone right and what’s gone wrong so far this season.

Read more…

Post Navigation


text:
Galatians: Called by God

Galatians 1:11-24

This week Ben Fleming preaches the Good News from Paul’s letter to the Galatians.  How many people in your life praise God because of you?  That’s what happens when the Gospel takes hold on our lives.  We see it in the life of Saul turned Paul.  Do we see it in our lives, as agents of God’s grace?

Sermon Audio

Written by Brian Culbertson

Sinner turned Saint because of Jesus.

Leave a Reply


text:
Billing and Payment (4)

Billing and Payment

Domain Names (3)

Domain Names

Domain Status (0)

Status of domain name flagged as expired despite domain not expired

Renewals (0)

Renewing a domain name

Les plus consultés

An organization (like, accredited by the relevant registries in assigning and...

The action that involves reactivating a domain name after it has been suspended for non payment...

 Billing Contact

The billing contact of a domain name is only authorized to access invoices of the domain name,...

A name server is responsible for handling requests for a domain name. Name servers are what allow...

A Domain Registrant is the owner of a domain name.


text:
Laser Mole Removal Can be removed easily with a simple surgical procedure called Laser Mole Removal.

Laser Mole Removal

There are different types of moles on our body.

  • Skin tags are small skin growths usually dark in color and painless.
  • Raised moles usually skin colored, but occasionally dark in color.
  • Flat mole can be any dark spot or irregularity in the skin.

Some moles we are born with and others form as we are exposed to the sun over time. We can have different reasons to want them removed, but most can be removed easily with a simple surgical procedure called Laser Mole Removal.

The treatment

After the doctor has examined the mole an anesthetic will in some cases be used either by topical cream or injection. Some moles are so small and not deep enough, that this is not necessary. A laser is used to heat the cells or tissue in order to cut  them or burst them. The laser beam is absorbed by the pigment of the mole, which then is broken up and absorbed by the body during healing. Because the laser seals the blood vessels, there is no need for stitches. Immediately after the treatment the area usually turns darker, this fades away quickly and the mole will “fall off” after 4-7 days.

Repeat treatments

In most cases only one treatment is needed. But in some rare cases, especially if the mole is deep, a follow up treatment is required. The interval between treatments depends on the healing time, but usually 3-4 weeks is needed in between.

Side effects & after-care

Make up can be worn after 3-4 days after treatment or after the wound has dried. Sunscreen with a SPF 30+ is required for  4-6 weeks after the treatment. You can read more about after care and side effects by following this link.

Equipment used

At Rejuvie Aesthetic & Anti-Aging we use the Surgitron Laser from Ellman. If you would like to know more about this device, please follow this link to Ellman’s own website.

More information

If you would like to see a short video on how a mole is removed, please view this video from YouTube:

The price

The price of a Laser Mole Removal starts from Rp 300.000 per mole

View price list

You might want to consider

Let us help you achieve your wellness and beauty goals by booking a free consultation with us today.

Contact & Booking