Maxwell's Equations — Question 3

Question 3

Both curl equations determine vacuum propagation speed and field ratio

Consider source-free fields $\mathbf E=E_0\cos(kx-\omega t)\hat y$ and $\mathbf B=B_0\cos(kx-\omega t)\hat z$, with positive k and omega.

1. Apply Faraday law to relate amplitudes.

2. Apply the Ampere-Maxwell law to find phase speed and the amplitude ratio.

Question 3 - Solution

Both curl equations determine vacuum propagation speed and field ratio

(a) Method and calculation

$$(\nabla\times\mathbf E)_z=-kE_0\sin(kx-\omega t)=-\partial_tB_z.$$$$\boxed{kE_0=\omega B_0}.$$

(b) Interpretation and further calculation

$$(\nabla\times\mathbf B)_y=kB_0\sin(kx-\omega t)=\mu_0\epsilon_0\omega E_0\sin(kx-\omega t).$$$$\boxed{\omega/k=c=1/\sqrt{\mu_0\epsilon_0},\quad E_0=cB_0}.$$

Physical check

Each curl equation supplies one relation; using only one does not fix the propagation speed.