All Problems Electromagnetic Induction. Maxwells Equations
Problem 3.357
Demonstrate that Maxwell's equations \(\nabla \times \mathbf{E}=-\partial \mathbf{B} / \partial t\) and \(\nabla \cdot B=0\) are compatible, i.e. the first one does not contradict the second one.
3.357. Let us consider the divergence of the two sides of the first equation. Since the divergence of a rotor is always equal to zero, \(\boldsymbol{\nabla} \cdot(\partial \mathbf{B} / \partial t)=0\) or \(\frac{\partial}{\partial t}(\mathbf{\nabla} \cdot \mathbf{B})=0 .\) Hence, \(\mathbf{\nabla} \cdot \mathbf{B}=\) const which does not contradict the second equation.