Maxwell's Equations — Question 10

Question 10

The curl equations preserve initially satisfied Gauss constraints

Assume Faraday and Ampere-Maxwell equations hold in vacuum, and charge is conserved. Define $g_B=\nabla\cdot\mathbf B$ and $g_E=\nabla\cdot\mathbf E-\rho/\epsilon_0$.

1. Find the time derivative of each constraint.

2. If both constraints vanish initially, determine their later values and state the limitation if initial constraints are wrong.

Question 10 - Solution

The curl equations preserve initially satisfied Gauss constraints

(a) Method and calculation

$$\partial_tg_B=-\nabla\cdot(\nabla\times\mathbf E)=0.$$$$\partial_tg_E=c^2\nabla\cdot(\nabla\times\mathbf B)-\frac{\nabla\cdot\mathbf J+\dot\rho}{\epsilon_0}=0.$$

(b) Interpretation and further calculation

$$g_B(0)=g_E(0)=0\Rightarrow\boxed{g_B(t)=g_E(t)=0}.$$$$\boxed{\text{An initial constraint error persists; evolution alone does not remove it.}}$$

Physical check

Divergence of a curl is zero, so the field evolution propagates consistent initial constraints.