electrodynamics / Electromagnetic Field Theory - Bo Thide

4

The

Electromagnetic

Fields

While, in principle, the electric and magnetic fields can be calculated from the Maxwell equations in Chapter 1, or even from the wave equations in Chapter 2, it is often physically more lucid to calculate them from the electromagnetic potentials derived in Chapter 3. In this Chapter we will derive the electric and magnetic fields from the potentials.

We recall that in order to find the solution (3.35) for the generic inhomogeneous wave equation (3.19) on page 38 we presupposed the existence of a Fourier transform pair (3.20a) on page 38 for the generic source term

That such transform pairs exists is true for most physical variables which are not strictly monotonically increasing and decreasing with time. For charges and currents varying in time we can therefore, without loss of generality, work with individual Fourier components. Strictly speaking, the existence of a single Fourier component assumes a monochromatic source (i.e., a source containing only one single frequency component), which in turn requires that the electric and magnetic fields exist for infinitely long times. However, by taking the proper limits, we can still use this approach even for sources and fields of finite duration.

This is the method we shall utilise in this Chapter in order to derive the electric and magnetic fields in vacuum from arbitrary given charge densities(t;x) and current densities j(t;x), defined by the temporal Fourier transform

43

pairs

under the assumption that only retarded potentials produce physically acceptable solutions.1

The temporal Fourier transform pair for the retarded vector potential can then be written

where in the last step, we made use of the explicit expression for the temporal Fourier transform of the generic potential component !(x), Equation (3.32) on page 40. Similarly, the following Fourier transform pair for the vector potential must exist:

in order that all physical quantities be real. Similar transform pairs and requirements of real-valuedness exist for the fields themselves.

1In fact, John A. Wheeler and Richard P. Feynman derived in 1945 a fully self-consistent electrodynamics using both the retarded and the advanced potentials [6]; See also [1].

In the limit that the sources can be considered monochromatic containing only one single frequency !0, we have the much simpler expressions

where again the real-valuedness of all these quantities is implied. As discussed above, we can safely assume that all formulae derived for a general temporal Fourier representation of the source (general distribution of frequencies in the source) are valid for these simple limiting cases. We note that in this context, we can make the formal identification ! = 0 (! !0), j! =j0 (! !0) etc., and that we therefore, without any loss of stringence, let 0 mean the same as the Fourier amplitude ! and so on.

4.1 The magnetic field

Let us now compute the magnetic field from the vector potential, defined by Equation (4.5a) and Equation (4.5b) on the preceding page, and formula (3.7) on page 35:

The calculations are much simplified if we work in ! space and, at the final stage, Fourier transform back to ordinary t space. We are working in the Lorentz gauge and note that in ! space the Lorentz condition, Equation (3.13) on page 36, takes the form

which provides a relation between (the Fourier transforms of) the vector and scalar potentials.

Using the Fourier transformed version of Equation (4.8) and Equation (4.5b) on the preceding page, we obtain

The first term, the induction field, dominates near the current source but falls off rapidly with distance from it, is the electrodynamic version of the BiotSavart law in electrostatics, formula (1.13) on page 6. The second term, the radiation field or the far field, dominates at large distances and represents energy that is transported out to infinity. Note how the spatial derivatives (r) gave rise to a time derivative (˙)!

4.2 The electric field

In order to calculate the electric field, we use the temporally Fourier transformed version of formula (3.11) on page 35, inserting Equations (4.4b) and (4.5b) as the explicit expressions for the Fourier transforms of and A:

E!(x) = r !(x) +i!A!(x)

Doing so in the last term of Equation (4.14), and also using the fact that k = !=c, we can rewrite this Equation as

The last integral can be further rewritten in the following way:

where, according to Gauss's theorem, the last term vanishes if j! is assumed to be limited and tends to zero at large distances. Further evaluation of the derivative in the first term makes it possible to write

Using the triple product “bac-cab” formula (F.54) on page 155 backwards, and inserting the resulting expression for I! into Equation (4.17) on the preceding page, we arrive at the following final expression for the Fourier transform of the total E-field:

Taking the inverse Fourier transform of Equation (4.22) above, once again using the vacuum relation ! = kc, we find, at last, the expression in time do-

main for the total electric field:

1

E(t;x) = E!(x)e i!td!