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Electromagnetic Induction. Maxwells Equations

Problem 3.356

Demonstrate that the law of electric charge conservation, i.e. \(\boldsymbol{\nabla} \cdot \mathbf{j}=-\partial \rho / \partial t,\) follows from Maxwell's equations.

Reveal Answer
 3.356. Let us find the divergence of the two sides of the equation ∇×H=j+∂D/∂t . Since the divergence of a rotor is always equal  to zero, we get 0=∇⋅j+∂∂t(V⋅D) . It remains to take into  account that V⋅D=ρ\begin{aligned} &\text { 3.356. Let us find the divergence of the two sides of the equation }\\ &\boldsymbol{\nabla} \times \mathbf{H}=\mathbf{j}+\partial \mathbf{D} / \partial t \text { . Since the divergence of a rotor is always equal }\\ &\text { to zero, we get } 0=\mathbf{\nabla} \cdot \mathbf{j}+\frac{\partial}{\partial t}(\mathbf{V} \cdot \mathbf{D}) \text { . It remains to take into }\\ &\text { account that } \mathbf{V} \cdot \mathbf{D}=\rho \end{aligned}
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