Chapter 32, Problem 56E

(a)

To determine

To verify: The solution to the wave equation is $E_y\left(x,t\right)=E_{\max }{e}^{-k_{C}x}\cos \left(k_{C}x-\omega t\right)$ by substituting $E_y\left(x,t\right)$ in $\frac{{\partial }^2{E}_y\left(x,t\right)}{\partial x^2}=\frac{\mu }{\rho }\frac{\partial E_y\left(x,t\right)}{\partial t}$

(b)

To determine

To explain: Why the electric field decreases in amplitude as wave propagates.

(c)

To determine

To show: The electric field amplitude decreases by a factor of $\frac{1}{e}$ in a distance $\frac{1}{k_C}=\sqrt{\frac{2\rho }{\omega \mu }}$ and to calculate that distance for a radio wave with frequency $f=1.0\text{MHz}$ in copper.