All Problems

Electromagnetic Induction. Maxwells Equations

Problem 3.357

Demonstrate that Maxwell's equations ∇×E=−∂B/∂t\nabla \times \mathbf{E}=-\partial \mathbf{B} / \partial t and ∇⋅B=0\nabla \cdot B=0 are compatible, i.e. the first one does not contradict the second one.

Reveal Answer
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, ∇⋅(∂B/∂t)=0\boldsymbol{\nabla} \cdot(\partial \mathbf{B} / \partial t)=0 or ∂∂t(∇⋅B)=0.\frac{\partial}{\partial t}(\mathbf{\nabla} \cdot \mathbf{B})=0 . Hence, ∇⋅B=\mathbf{\nabla} \cdot \mathbf{B}= const which does not contradict the second equation.
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