Maxwell's Equations — Question 8

Question 8

Uniform static field additions leave the same local sources

A known pair of fields E and B satisfies vacuum Maxwell equations for rho and J. Add arbitrary constant vectors E0 and B0, independent of both position and time.

1. Check all four equations for the new field pair.

2. Explain what extra information is needed to select one physical solution among these possibilities.

Question 8 - Solution

Uniform static field additions leave the same local sources

(a) Method and calculation

$$\nabla\cdot\mathbf E_0=\nabla\cdot\mathbf B_0=0,\quad\nabla\times\mathbf E_0=\nabla\times\mathbf B_0=0.$$$$\boxed{\partial_t\mathbf E_0=\partial_t\mathbf B_0=0}.$$

(b) Interpretation and further calculation

$$(\mathbf E,\mathbf B)\to(\mathbf E+\mathbf E_0,\mathbf B+\mathbf B_0)\Rightarrow(\rho,\mathbf J)\ \text{unchanged}.$$$$\boxed{\text{Boundary and initial field data remove this ambiguity.}}$$

Physical check

These are physically different added fields, not merely a change of electromagnetic potentials.