31.5.4 The Wave Equation

In 《 SECTION 16.4, during our study of traveling waves, we derived the wave equation:

\frac{\partial^{2} D}{\partial t^{2}} = v^{2} \frac{\partial^{2} D}{\partial x^{2}}

(31.31)

There we learned that any physical system that obeys this equation for some type of displacement D can have traveling waves that propagate along the x-axis with speed v.

If we start with Equation 31.27, the Faraday’s law requirement for any electromagnetic wave, we can take the second derivative with respect to x to find

\frac{\partial^{2} E_{y}}{\partial x^{2}} = - \frac{\partial^{2} B_{z}}{\partial x \partial t}

(31.32)

You’ve learned in calculus that the order of differentiation doesn’t matter, so $\partial^{2} B_{z} / \partial x \partial t = \partial^{2} B_{z} / \partial t \partial x$ . And from Equation 31.30,

\frac{\partial^{2} B_{z}}{\partial t \partial x} = - ϵ_{0} μ_{0} \frac{\partial^{2} E_{y}}{\partial t^{2}}

(31.33)

Substituting Equation 31.33 into Equation 31.32 and taking the constants to the other side, we have

\frac{\partial^{2} E_{y}}{\partial t^{2}} = \frac{1}{ϵ_{0} μ_{0}} \frac{\partial^{2} E_{y}}{\partial x^{2}} (the wave equation for electromagnetic waves)

(31.34)

Equation 31.34 is the wave equation! And it’s easy to show, by taking second derivatives of $B_{z}$ rather than $E_{y}$ , that the magnetic field $B_{z}$ obeys exactly the same wave equation.

As we anticipated, Maxwell’s equations have led to a prediction of electromagnetic waves. Referring to the general wave equation, Equation 31.31, we see that an electromagnetic wave must travel (in vacuum) with speed

v_{em} = \frac{1}{\sqrt{ϵ_{0} μ_{0}}}

(31.35)

The constants $ϵ_{0}$ and $μ_{0}$ are known from electrostatics and magnetostatics, where they determined the size of $\vec{E}$ and $\vec{B}$ due to point charges. Thus we can calculate

v_{em} = \frac{1}{\sqrt{ϵ_{0} μ_{0}}} = 3.00 \times 10^{8} m/s = c

(31.36)

This is a remarkable conclusion. Coulomb’s law and the Biot-Savart law, in which $ϵ_{0}$ and $μ_{0}$ first appeared, have nothing to do with waves. Yet Maxwell’s theory of electromagnetism ends up predicting that electric and magnetic fields can form a self-sustaining electromagnetic wave if that wave travels with speed $v_{em} = 1 / \sqrt{ϵ_{0} μ_{0}}$ . No other speed will satisfy Maxwell’s equations.

Laboratory measurements had already determined that light travels at $3.0 \times 10^{8} m / s$ , so Maxwell was entirely justified in concluding that light is an electromagnetic wave. Furthermore, we’ve made no assumption about the frequency of the wave, so apparently electromagnetic waves of any frequency, from radio waves to x rays, travel (in vacuum) with speed c, the speed of light.