15 真空中的电磁波

15 Electromagnetic Waves in Vacuum

Starting from Maxwell's equations, we want to treat electromagnetic fields that are rapidly alternating in time and space. At first, we will restrict ourselves to fields in vacuum. In this case, =D E and =B H ; furthermore, 0=j and 0ρ=. So, we have Maxwell's equations in the form

1c t

∂∇⨯=∂E B (15.1) 1c t

∂∇⨯=-∂B E (15.2) 0∇⋅=E (15.3) 0∇⋅=B . (15.4) We may eliminate the vectors E and B from the system of equations. Taking the curl of the equation (15.1):

1()()c t

∂∇⨯∇⨯=∇⨯∂B E and substituting the equation (15.2), then

2221()c t

∂∇⨯∇⨯=-∂B B . (15.5) Using the identity ()()∇⨯∇⨯=∇∇⋅-∆B B B due to 0∇⋅=B we obtain from equa- tion (15.5)

2221c t

∂∆=∂B B . (15.6a) Correspondingly, we may eliminate B and obtain

2221c t

∂∆=∂E E . (15.6b) Hence, both vectors E and B fulfill the same wave equation. We have encounter- ed wave equations already in mechanics in treating oscillation processes. There are various types of solutions for these equations. At first, we notice that the form of the wave equation

2221(,)0u t c t ⎛⎫∂∆-= ⎪∂⎝

⎭r (15.7) is relativistically invariant because the operator

2222242222222111c t x y z c t x x μμμ

=∂∂∂∂∂∂∂∆-=++-=∂∂∂∂∂∂∂∑ is equal to the scalar product of the four-gradient

1234,,,,,,()x x x x x x y z ict μ⎛⎫⎛⎫∂∂∂∂∂∂∂∂∂== ⎪ ⎪∂∂∂∂∂∂∂∂∂⎝⎭

⎝⎭ with itself. An equation of the form (15.7) is solved by any function

11223344(,)()()u t u k x u k x k x k x k x μμ==+++r (15.8)

if 222212

340k k k k k k μμ≡+++=, that is, k μ is a four-null-vector (light vector). Here, we use Einstein's sum convention. This means to sum up automatically over identical indices. We verify easily that

222221(,)()()()0u t u k x k u k x k k u k x c t x x ννμννμμννμμ

⎛⎫∂∂∂'''∆-==== ⎪∂∂∂⎝⎭r .

Writing 123(,,,)(,)k k k k i c i c μωω==k , the dispersion relation is 0k k μμ=, and thus k c ω=. Then, the solutions of the wave equations are:

(,)()()()u t u k x u t u kct μμω==⋅-=⋅-r k r k r .

This may be understood in the framework of a one-dimensional wave equation, 222221()0u x ct x c t ⎛⎫∂∂-±= ⎪∂∂⎝⎭

. Here, ()u x ct ± are again arbitrary functions possessing always the special combina- tion 14(,)(1,)(,)(,)x ct x ict i x ict k k ±=⋅=⋅ as its argument (k x μμ in the general case). In these cases 22120k k +=; hence, these functions are of the type (15.8). Some of them are illustrated in Figure 15.1. At time 0t =, the function ()u x ct ± describes the "mountain" ()u x . In the course of time this "mountain" does not change its shape but merely shifts to the position 0x ct ±=.

More precisely, ()u x ct + travels to the left to the position x ct =-, and ()u x ct - travels to the right to the position x ct =. Obviously, the propagation of the wave train proceeds with the velocity ||||v x c ==.

Now, we investigate plane waves representing space-time functions for which the geometric position of a certain phase of oscillation at a certain moment t is a plane. The equation of the surface of constant phase (Figure 15.2) is

const k x t μμω=⋅-=k r or const t ω⋅=+k r . (15.9) Obviously it is the representation of a plane in the so-called Hessian normal form.

The wave vector k is given by

grad()t ω=⋅-k k r .

It is normal to the plane of the wave. From (15.9),

0const t r k k

ω+⋅=≡k r . The surfaces of constant phase of the solutions ()u k x μμ of the wave equation turn out to be planes. Therefore we talk about plane waves. Since 0r is the distance of the plane from the origin we may conclude from (15.9) that this distance increases according to t k ω. Then, the velocity of propagation of the wave front is

0dr v dt k ω==. As already mentioned, the vector k is perpendicular to the surface of constant phase and points in the direction of propagation of the wave. Hence, we may conclude

k k

ω=k v .