B.Thide - Electromagnetic Field Theory

Squaring the latter and combining with the former, one obtains the second order algebraic equation (in 2)

As a consequence, the solution of the time-independent telegrapher's equation,

Equation (2.41) on the preceding page, can be written

With the aid of Equation (2.40) on the facing page we can calculate the associated magnetic field, and find that it is given by

where we have, in the last step, rewritten +i in the amplitude-phase form jAjexpfi g. From the above, we immediately see that E, and consequently also B, is damped, and that E and B in the wave are out of phase.

In the case that "0! , we can approximate K as follows:

From this analysis we conclude that when the wave impinges perpendicularly upon the medium, the fields are given, inside this medium, by

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Hence, both fields fall off by a factor 1=e at a distance

s

2

= 0 ! (2.53)

This distance is called the skin depth.

2.3 Observables and Averages

In the above we have used complex notation quite extensively. This is for mathematical convenience only. For instance, in this notation differentiations are almost trivial to perform. However, every physical measurable quantity is always real valued. I.e., “Ephysical = Re fEmathematicalg.” It is particularly important to remember this when one works with products of physical quantities. For instance, if we have two physical vectors F and G which both are time-harmonic, i.e., can be represented by Fourier components proportional to expfi!tg, then we must make the following interpretation

Furthermore, letting denotes complex conjugate, we can express the real part of the complex vector F as

and similarly for G. Hence, the physically acceptable interpretation of the scalar product of two complex vectors, representing physical observables, is

F(t;x) G(t;x) = Re F0(x) e i!t Re G0(x) e i!t

=12 [F0(x) e i!t +F0(x) ei!t] 12 [G0(x) e i!t +G0(x) ei!t]

=14 F0 G0 +F0 G0 +F0 G0 e 2i!t +F0 G0 e2i!t

= Re F0 e i t G ei t +F0 G0 e 2i t

2 0

=1 Re F(t;x) G (t;x) +F0 G0 e 2i!t

2

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Often in physics, we measure temporal averages (hi) of our physical observables. If so, we see that the average of the product of the two physical quantities represented by F and G can be expressed as

since the temporal average of the oscillating function expf 2i!tg vanishes.

Bibliography

[1]J. D. JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc., New York, NY . . . , 1999, ISBN 0-471-30932-X.

[2]W. K. H. PANOFSKY AND M. PHILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6.

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3

Electromagnetic Potentials

Instead of expressing the laws of electrodynamics in terms of electric and magnetic fields, it turns out that it is often more convenient to express the theory in terms of potentials. In this chapter we will introduce and study the properties of such potentials.

3.1 The Electrostatic Scalar Potential

As we saw in Equation (1.8) on page 5, the electrostatic field Estat(x) is irrotational. Hence, it may be expressed in terms of the gradient of a scalar field. If we denote this scalar field by stat(x), we get

Taking the divergence of this and using Equation (1.7) on page 5, we obtain

Poisson's equation

A comparison with the definition of Estat, namely Equation (1.5) on page 4, shows that this equation has the solution

where the integration is taken over all source points x0 at which the charge density (x0) is non-zero and is an arbitrary quantity which has a vanishing gradient. An example of such a quantity is a scalar constant. The scalar function stat(x) in Equation (3.3) is called the electrostatic scalar potential.

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3.2 The Magnetostatic Vector Potential

Consider the equations of magnetostatics (1.22) on page 9. From Equation (M.95) on page 187 we know that any 3D vector a has the property that r (r a) 0 and in the derivation of Equation (1.17) on page 8 in magnetostatics we found that r Bstat(x) = 0. We therefore realise that we can always write

where Astat(x) is called the magnetostatic vector potential.

We saw above that the electrostatic potential (as any scalar potential) is not unique: we may, without changing the physics, add to it a quantity whose spatial gradient vanishes. A similar arbitrariness is true also for the magnetostatic vector potential.

In the magnetostatic case, we may start from Biot-Savart's law as expressed by Equation (1.15) on page 8. Identifying this expression with Equation (3.4) allows us to define the static vector potential as

where a(x) is an arbitrary vector field whose curl vanishes. From Equation (M.91) on page 186 we know that such a vector can always be written as the gradient of a scalar field.

3.3 The Electrodynamic Potentials

Let us now generalise the static analysis above to the electrodynamic case, i.e., the case with temporal and spatial dependent sources (t;x) and j(t;x), and corresponding fields E(t;x) and B(t;x), as described by Maxwell's equations (1.45) on page 15. In other words, let us study the electrodynamic potentials

(t;x) and A(t;x).

From Equation (1.45c) on page 15 we note that also in electrodynamics the homogeneous equation r B(t;x) = 0 remains valid. Because of this divergence-free nature of the timeand space-dependent magnetic field, we can express it as the curl of an electromagnetic vector potential:

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Inserting this expression into the other homogeneous Maxwell equation, Equation (1.32) on page 13, we obtain

or, rearranging the terms,

@

r E(t;x) + @t A(t;x) = 0 (3.8)

As before we utilise the vanishing curl of a vector expression to write this vector expression as the gradient of a scalar function. If, in analogy with the electrostatic case, we introduce the electromagnetic scalar potential function(t;x), Equation (3.8) becomes equivalent to

This means that in electrodynamics, E(t;x) can be calculated from the formula

and B(t;x) from Equation (3.6) on the preceding page. Hence, it is a matter of taste whether we want to express the laws of electrodynamics in terms of the potentials (t;x) and A(t;x), or in terms of the fields E(t;x) and B(t;x). However, there exists an important difference between the two approaches: in classical electrodynamics the only directly observable quantities are the fields themselves (and quantities derived from them) and not the potentials. On the other hand, the treatment becomes significantly simpler if we use the potentials in our calculations and then, at the final stage, use Equation (3.6) on the facing page and Equation (3.10) above to calculate the fields or physical quantities expressed in the fields.

Inserting (3.10) and (3.6) on the facing page into Maxwell's equations (1.45) on page 15 we obtain, after some simple algebra and the use of Equation (1.11) on page 6, the general inhomogeneous wave equations

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which can be rewritten in the following, more symmetric, form

These two second order, coupled, partial differential equations, representing in all four scalar equations (one for and one each for the three components Ai;i = 1;2;3 of A) are completely equivalent to the formulation of electrodynamics in terms of Maxwell's equations, which represent eight scalar firstorder, coupled, partial differential equations.

3.3.1 Electrodynamic gauges

As they stand, Equations (3.11) on the preceding page and Equations (3.12) above look complicated and may seem to be of limited use. However, if we write Equation (3.6) on page 38 in the form r A(t;x) =B(t;x) we can consider this as a specification of r A. But we know from Helmholtz' theorem that in order to determine the (spatial) behaviour of A completely, we must also specify r A. Since this divergence does not enter the derivation above, we are free to choose r A in whatever way we like and still obtain the same physical results!

Lorentz equations for the electrodynamic potentials

because this condition simplifies the system of coupled equations Equation (3.12) above into the following set of uncoupled partial differential equa-

1In fact, the Dutch physicist Hendrik Antoon Lorentz, who in 1903 demonstrated the covariance of Maxwell's equations, was not the original discoverer of this condition. It had been discovered by the Danish physicist Ludwig Lorenz already in 1867 [4].

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tions which we call the Lorentz inhomogeneous wave equations:

where 2 is the d'Alembert operator discussed in Example M.6 on page 183. We shall call (3.14) the Lorentz potential equations for the electrodynamic potentials.

Gauge transformations

We saw in Section 3.1 on page 37 and in Section 3.2 on page 38 that in electrostatics and magnetostatics we have a certain mathematical degree of freedom, up to terms of vanishing gradients and curls, to pick suitable forms for the potentials and still get the same physical result. In fact, the way the electromagnetic scalar potential (t;x) and the vector potential A(t;x) are related to the physically observables gives leeway for similar “manipulation” of them also in electrodynamics. If we transform (t;x) and A(t;x) simultaneously into new ones 0(t;x) and A0(t;x) according to the mapping scheme

where (t;x) is an arbitrary, differentiable scalar function called the gauge function, and insert the transformed potentials into Equation (3.10) on page 39 for the electric field and into Equation (3.6) on page 38 for the magnetic field, we obtain the transformed fields

where, once again Equation (M.91) on page 186 was used. Comparing these expressions with (3.10) and (3.6) we see that the fields are unaffected by the gauge transformation (3.15). A transformation of the potentials and A which leaves the fields, and hence Maxwell's equations, invariant is called a gauge transformation. A physical law which does not change under a gauge transformation is said to be gauge invariant. By definition, the fields themselves are, of course, gauge invariant.

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The potentials (t;x) and A(t;x) calculated from (3.11) on page 39, with an arbitrary choice of r A, can be further gauge transformed according to (3.15) on the previous page. If, in particular, we choose r A according to the Lorentz condition, Equation (3.13) on page 40, and apply the gauge transformation (3.15) on the resulting Lorentz potential equations (3.14) on the previous page, these equations will be transformed into

We notice that if we require that the gauge function (t;x) itself be restricted to fulfil the wave equation

these transformed Lorentz equations will keep their original form. The set of potentials which have been gauge transformed according to Equation (3.15) on the preceding page with a gauge function (t;x) which is restricted to fulfil Equation (3.18) above, i.e., those gauge transformed potentials for which the Lorentz equations (3.14) are invariant, comprises the Lorentz gauge.

Other useful gauges are

•The radiation gauge, also known as the transverse gauge, defined by r A = 0.

•The Coulomb gauge, defined by = 0, r A = 0.

•The temporal gauge, also known as the Hamilton gauge, defined by

= 0.

•The axial gauge, defined by A3 = 0.

The process of choosing a particular gauge condition is referred to as gauge fixing.

3.3.2Solution of the Lorentz equations for the electromagnetic potentials

As we see, the Lorentz equations (3.14) on the preceding page for (t;x) and

A(t;x) represent a set of uncoupled equations involving four scalar equations

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