electrodynamics / Electromagnetic Field Theory - Bo Thide

It is convenient to introduce the following quantities:

where Ue is the electric field energy, Um is the magnetic field energy, both measured in J, and S is the Poynting vector (power flux), measured in W/m2.

7.3.2 The momentum theorem in Maxwell's theory

Let us now investigate the momentum balance (force actions) in the case that a field interacts with matter in a non-relativistic way. For this purpose we consider the force density given by the Lorentz force per unit volume E+j B. Using Maxwell's equations (7.23) and symmetrising, we obtain

=E(r D) B (r H)

@ (D B) D (r E) +H(*+r -B.,)

@t

=0

= [E(r D) D (r E)] +[H(r B) B (r H)]

@@t (D B)

One verifies easily that the ith vector components of the two terms in square brackets in the right hand member of (7.32) can be expressed as

and

respectively.

Using these two expressions in the ith component of Equation (7.32) on the previous page and re-shuffling terms, we get

where S is the Poynting vector defined in Equation (7.29) on page 93. Integration over the entire volume V yields

which expresses the balance between the force on the matter, the rate of change of the electromagnetic field momentum and the Maxwell stress. This equation is called the momentum theorem in Maxwell's theory.

In vacuum (7.42) becomes

Bibliography

[1]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.

8

Electromagnetic

Radiation

In this chapter we will develop the theory of electromagnetic radiation, and therefore study electric and magnetic fields which are capable of carrying energy and momentum over large distances. In Chapter 3 we were able to derive general expressions for the scalar and vector potentials from which we then, in Chapter 4 calculated the total electric and magnetic fields from arbitrary distributions of charge and current sources. The only limitation in the calculation of the fields was that the advanced potentials were discarded.

We shall now study these fields further under the assumption that the observer is located in the far zone, i.e., very far away from the source region(s). We therefore study the radiation fields which are dominating in this zone.

8.1 The radiation fields

From Equation (4.12) on page 46 and Equation (4.23) on page 49, which give the total electric and magnetic fields, we obtain the radiation fields

Instead of studying the fields in the time domain, we can often make a spectrum analysis into the frequency domain and study each Fourier compon-

97

O

FIGURE 8.1: Relation between the surface normal and the k vector for radiation generated at source points x0 near the point x0 in the source volume V. At distances much larger than the extent of V, the unit vector nˆ, normal to the surface S which has its centre at x0, and the unit vector

ˆ 0

k of the radiation k vector from x are nearly coincident.

ent separately. A superposition of all these components and a transformation back to the time domain will then yield the complete solution.

The Fourier representation of the radiation fields (8.1a) (8.1b) were included in Equation (4.11) on page 46 and Equation (4.22) on page 48, respectively and are explicitly given by

If the source is located inside a volume V near x0 and has such a limited spatial extent that max jx0 x0j jx x0j, and the integration surface S , centred on x0, has a large enough radius jx x0j max jx0 x0j, we see from Figure 8.1

on the preceding page that we can approximate

k x x0 k (x x0) k (x x0) k (x0 x0)

(8.4)

k jx x0j k (x0 x0)

Recalling from Formula (F.48) and formula (F.49) on page 154 that

dS = jx x0j2 d= jx x0j2 sin d d'

ˆ

and noting from Figure 8.1 on the preceding page that k and nˆ are nearly parallel, we see that we can approximate.

Both these approximations will be used in the following.

Within approximation (8.4) the expressions (8.3a) and (8.3b) for the radiation fields can be approximated as

I.e., if max jx0 x0j jx x0j, then the fields can be approximated as spherical waves multiplied by dimensional and angular factors, with integrals over points in the source volume only.

8.2 Radiated energy

Let us consider the energy that carried in the radiation fields Brad, Equation (8.1a), and Erad, Equation (8.1b) on page 97. We have to treat signals with limited lifetime and hence finite frequency bandwidth differently from monochromatic signals.

Equation (8.10) on the preceding page can be written [cf. Parseval's identity]

11

S(t)dt = 2 (E! H !)d!

where the last step follows from the real-valuedness of E! and B!. We insert the Fourier transforms of the field components which dominate at large distances, i.e., the radiation fields (8.3a) and (8.3b).

The result, after integration over the area S of a large sphere which encloses the source, is

Inserting the approximations (8.4) and (8.5) into Equation (8.13) and also introducing

and recalling the definition (2.18) on page 26 for the vacuum resistance R0 we obtain

which, at large distances, is a good approximation to the energy that is radiated per unit solid angle d in a frequency band d!. It is important to notice that Formula (8.15) includes only source coordinates. This means that the amount of energy that is being radiated is independent on the distance to the source (as long as it is large).

8.3 Radiation from extended sources

As shown above, one can, at least in principle, calculate the radiated fields, Poynting flux and energy for an arbitrary current density Fourier component. However, in practice, it is often difficult to evaluate the integrals unless the current has a simple distribution in space. In the general case, one has to resort to approximations. We shall consider both these situations.

Certain radiation systems have a geometry which is one-dimensional, symmetric or in any other way simple enough that a direct calculation of the radiated fields and energy is possible. This is for instance the case when the current flows in one direction in space only and is limited in extent. An example of this is a linear antenna.

8.3.1 Linear antenna

Let us apply Equation (8.9) on page 100 for calculating the power from a linear, transmitting antenna, fed across a small gap at its centre with a monochromatic source. The antenna is a straight, thin conductor of length L which carries a one-dimensional time-varying current so that it produces electromagnetic radiation.

We assume that the conductor resistance and the energy loss due to the electromagnetic radiation are negligible. Since we can assume that the antenna wire is infinitely thin, the current must vanish at the end points L=2 and L=2. The current therefore forms a standing wave with wave number k = !=c and can be written

where the amplitude I0 is constant. In order to evaluate formula (8.9) on page 100 with the explicit monochromatic current (8.16) inserted, we need