All Problems Electromagnetic Induction. Maxwells Equations
Problem 3.355
Using Maxwell's equations, show that (a) a time-dependent magnetic field cannot exist without an electric field; (b) a uniform electric field cannot exist in the presence of a timedependent magnetic field; (c) inside an empty cavity a uniform electric (or magnetic) field can be time-dependent.
3.355. (a) If \(\mathbf{B}(t),\) then \(\boldsymbol{\nabla} \times \mathbf{E}=-\partial \mathbf{B} / \partial t \neq 0\). The spatial derivatives of the field \(\mathbf{E},\) however, may not be equal to zero \((\mathbf{\nabla} \times \mathbf{E} \neq 0)\) only in the presence of an electric field. (b) If \(\mathbf{B}(t)\), then \(\boldsymbol{\nabla} \times \mathbf{E}=-\partial \mathbf{B} / \partial t \neq 0\). But in the uniform field \(\nabla \times \mathbf{E}=0\) (c) It is assumed that \(\mathbf{E}=\operatorname{af}(t),\) where a is a vector which is independent of the coordinates, \(f(t)\) is an arbitrary funetion of time. Then \(-\partial \mathbf{B} / \partial t=\boldsymbol{\nabla} \times \mathbf{E}=0,\) that is the field \(\mathbf{B}\) does not vary with time. Generally speaking, this contradicts the equation \(\boldsymbol{\nabla} \times \mathbf{H}=\) \(=\partial \mathbf{D} / \partial t\) for in this case its left-hand side does not depend on time whereas its right-hand side does. The only exception is the case when \(f(t)\) is a linear function. In this case the uniform field \(E\) can be time-dependent.