Maxwell's Equations — Question 4

Question 4

A longitudinal electric traveling pattern requires sources

A proposed field is $\mathbf E=E_0\cos(kx-\omega t)\hat x$ and $\mathbf B=0$. Treat it as a local prescribed pattern in vacuum.

1. Find the charge and current densities required by Maxwell equations.

2. Check continuity and decide whether the pattern can be a source-free vacuum wave.

Question 4 - Solution

A longitudinal electric traveling pattern requires sources

(a) Method and calculation

$$\rho=\epsilon_0\nabla\cdot\mathbf E=-\epsilon_0kE_0\sin(kx-\omega t).$$$$\boxed{J_x=-\epsilon_0\partial_tE_x=-\epsilon_0\omega E_0\sin(kx-\omega t)}.$$

(b) Interpretation and further calculation

$$\partial_t\rho=\epsilon_0k\omega E_0\cos(kx-\omega t),\quad\partial_xJ_x=-\epsilon_0k\omega E_0\cos(kx-\omega t).$$$$\boxed{\dot\rho+\nabla\cdot\mathbf J=0;\quad\rho,\mathbf J\ne0\ \text{in general}}.$$

Physical check

Mathematical time dependence alone does not make a field a source-free electromagnetic wave.