DERIVATION OF THE WAVE EQUATION
Here is an alternative derivation of Eq. (33-9) for the speed of electromagnetic waves. It is more mathematical than our other treatment, but it includes a derivation of the wave equation for electromagnetic waves. This part of the section can be omitted without loss of continuity in the chapter.
During our discussion of mechanical waves in Section 19-4, we showed that a function y(x, t) that represents the displacement of any point in a mechanical wave traveling along the x-axis must satisfy a differential equation, Eq. (19-12):
This equation is called the wave equation, and u is the speed of propagation of the wave. To derive the corresponding equation for an electromagnetic wave, we again consider a plane wave. That is, we assume that at each instant, E, and B, are uniform over any plane perpendicular to the x-axis, the direction of propagation. But now we let E, and B, vary continuously as we go along the x-axis; then each is a function of x and t. We consider the values of E, and B, on two planes perpendicular to the x-axis, one at x and one at x + ~x .
Following the same procedure as previously, we apply Faraday’s law to a rectangle lying parallel to the xy-plane, as in Fig. 33-6. This figure is similar to Fig. 33-4. Let the left end gh of the rectangle be at position z, and let the right end ef be at position (x + ~x). At time t, the values of E, on these two sides are E,(x, t) and E,(x + ~x, t), respectively. When we apply Faraday’s law to this rectangle, we find that instead of
~E. di = -Eaas before, we have To find the magnetic flux IlIB through this rectangle, we assume that ~x is small enough that B, is nearly uniform over the rectangle. In that case, IlIB = Bz(x, t)A ;:
Bz(x, t)a ~x, and
This equation shows that if there is a time-varying component B, of magnetic field, there must also a component E, of electric field that varies with x, and conversely. We put this relation on the shelf for now; we’ll return to it soon. Next we apply Ampere’s law to the rectangle shown in Fig. 33-7. The line integral Now comes the final st ep. We take the partial derivatives with respect to x of bOthsides of Eq. (33-12), and we take the partial derivatives with respect to t of both sides ofEq. (33-14). The results are expression has the same form as the general wave equation, Eq. (33-10). Because the electric field E, must satisfy this equation, it behaves as a wave with a pattern that travels through space with a definite speed. Furthermore, comparison of Eqs. (33-15) and (33-10) shows that the wave speed v is given by agrees with Eq. (33-9) for the speed c of electromagnetic waves.
We can show that B, also must satisfy the same wave equation as E,. Eq. (33-15). To prove this, we take the partial derivative ofEq. (33-12) with respect to t and the partial derivative of Eq. (33-14) with respect to x and combine the results. We leave this derivation as a problem (see Problem 33-31).