Maxwell's Equations — Question 2

Question 2

The displacement coefficient is fixed by charge conservation

Suppose an amended vacuum law is $\nabla\times\mathbf B=\mu_0\mathbf J+\lambda\partial_t\mathbf E$. Retain $\nabla\cdot\mathbf E=\rho/\epsilon_0$ and ordinary charge conservation.

1. Take a divergence to determine lambda for arbitrary time-dependent charge density.

2. Explain why setting lambda to zero is inconsistent for accumulating charge.

Question 2 - Solution

The displacement coefficient is fixed by charge conservation

(a) Method and calculation

$$0=\mu_0\nabla\cdot\mathbf J+\lambda\partial_t(\nabla\cdot\mathbf E)=(-\mu_0+\lambda/\epsilon_0)\dot\rho.$$$$\boxed{\lambda=\mu_0\epsilon_0}.$$

(b) Interpretation and further calculation

$$\lambda=0\Rightarrow\mu_0\nabla\cdot\mathbf J=0\Rightarrow\dot\rho=0.$$$$\boxed{\lambda=0\ \text{would exclude local charge accumulation}}.$$

Physical check

The added coefficient is constrained by conservation, rather than being an arbitrary adjustment.