Maxwell's Equations — Question 7

Question 7

A vacuum standing wave separates electric and magnetic antinodes

A source-free standing electric field is $\mathbf E=E_0\cos kx\cos\omega t\,\hat y$ with $\omega=ck$. Seek Bz with no static offset.

1. Find Bz from Faraday law and verify its amplitude.

2. Locate electric and magnetic spatial nodes and compare the peak energy densities.

Question 7 - Solution

A vacuum standing wave separates electric and magnetic antinodes

(a) Method and calculation

$$\partial_tB_z=-\partial_xE_y=kE_0\sin kx\cos\omega t.$$$$\boxed{B_z=(E_0/c)\sin kx\sin\omega t}.$$

(b) Interpretation and further calculation

$$E\text{ nodes}:kx=(n+1/2)\pi;\quad B\text{ nodes}:kx=n\pi.$$$$\boxed{u_{E,\max}=\epsilon_0E_0^2/2=u_{B,\max}}.$$

Physical check

Equal peak energy scales do not mean equal electric and magnetic energy at every place and time.