Question

Imagine a uniform magnetic field, pointing in the $\mathbf{z}$direction and filling all space $\left(B=B_{0}z\right)$. A positive charge is at rest, at the origin. Now somebody turns off the magnetic field, thereby inducing an electric field. In what direction does the charge move?

Short answer

The direction of the charge is indetermined.

Step-by-step solution

Write the given data from the question.

The uniform magnetic field $\left(\mathrm{B}={\mathrm{B}}_0\mathrm{z}\right)$,

The initial magnetic field is$\mathrm{B}$

The positive charge is at rest.

Determine the direction of the charge move.

Let assume the charge is$\mathrm{q}$at the rest, and $\mathrm{E}$is the electric field.

The force experienced by the charges at the rest is given by,

$\mathbf{F}\mathbf{=}\mathbf{qE}$

The curl of the electric field is given by,

$\mathrm{\nabla }\times \mathrm{E}=-\frac{\mathrm{\partial }{\mathrm{B}}_{\widehat{\mathrm{C}}}}{\mathrm{\partial }\mathrm{t}}\mathrm{Z}$

Since the boundary condition is not mentioned therefore, it is difficult to move the electric field.

The electric field attain the zero value at infinity. Hence the direction of the charge is in determined due to no boundary condition.