Question

An external magnetic field changes as a function of time such that: (1) it is momentarily zero at time $t=0$ , and (2) it produces an electric field $E=x\hat{y}-y\hat{z}$ that is constant in time. Determine the magnitude and direction of the magnetic field as a function of time

Answer 1

Given,

Electric field is given as $(E)= x\hat{y}-y\hat{z}$

We know that,

$\Rightarrow \nabla \times E=\frac{-dB}{dt}$

$\Rightarrow [\frac{\partial }{\partial x}\hat{i}+\frac{\partial }{\partial y}\hat{j}+\frac{\partial }{\partial x}\hat{z}]\times[x\hat{y}-y\hat{z}]=-\frac{dB}{dt}$

$\Rightarrow \hat{i}+\hat{k}=\frac{-dB}{dt}$

$B= -t(\hat{i}+\hat{k})$