The Traveling Electromagnetic Wave, Quantitatively — Question 4

Question 4

A lossless dielectric changes the electric-to-magnetic field ratio

A homogeneous nondispersive dielectric has $\epsilon=4\epsilon_0$ and $\mu=\mu_0$. A plane wave inside it has electric amplitude E0=12.0 V/m and frequency 500 MHz. Use $v=1/\sqrt{\mu\epsilon}$, $E_0=vB_0$, $H_0=B_0/\mu$, and $c=3.00\times10^8$ m/s.

1. Find speed, wavelength, and magnetic induction amplitude.

2. Express the wave impedance E0/H0 in terms of vacuum impedance $Z_0=\mu_0c$, and explain why E0/B0 is not c.

Question 4 - Solution

A lossless dielectric changes the electric-to-magnetic field ratio

(a) Method and calculation

$$v=c/2=1.50\times10^8\,\mathrm{m/s},\quad\boxed{\lambda=v/f=0.300\,\mathrm m}.$$$$\boxed{B_0=E_0/v=8.00\times10^{-8}\,\mathrm T}.$$

(b) Interpretation and further calculation

$$Z=E_0/H_0=\mu_0E_0/B_0=\mu_0v\Rightarrow\boxed{Z=Z_0/2}.$$$$\boxed{E_0/B_0=v=c/2};\quad\epsilon\ne\epsilon_0.$$

Physical check

The wave frequency is specified inside the medium; its wavelength follows the medium speed rather than the vacuum speed.