Question

Show that there can be no net charge in a region in which the electric field is uniform at all points.

Answer 1

Hint: This case is clearly an application of Gauss’s law. We can approach this question by keeping the statement of the Gauss’s law in our mind. We can introduce a surface in the field and proceed to solve the given problem.

Complete step by step answer:
- According to Gauss’s law, the flux of an electric field has a value that is equal to the net charge that is enclosed by a region divided by $\epsilon_o$. We are given a region where the electric field is uniform at all the points. Let us consider a surface that is perpendicular to the given electric field.
- Now, when we take any point on the new surface, we find that the value of the potential is the same at all points. Hence it is an equipotential surface.
- The absence of potential difference implies the absence of a current, which further implies the absence of a charge. Hence, we have proved the existence of a no net charge region where the electric field is uniform at all the points of that particular region.

Note:
The electric flux in a given region can be found by taking the integral of the product of the electric field and the surface area. We are given that the electric field is constant. This implies that the integral value is also a constant. Thus the value of the net charge is again zero. The question can be addressed in this alternative way.