Question

If$E = 0$, at all points of a closed surface.
A. The electric flux through the surface is zero
B. The total charge enclosed by the surface is zero.
C. No charge resides on the surface
D. All of the above.

Answer 1

If the closed surface is situated in a uniform electric field, then the electric flux through the surface is zero. The net electric flux through the surface depends upon the nature as well as the quantity of charges enclosed by that surface. Also by Gauss Law, the electric field, the net charge and the electric flux are directly related to each other.

Formula Used: The electric field strength is: $E = \dfrac{1}{{4\pi {\varepsilon _0}}}\dfrac{q}{{{r^2}}}$

The Gauss Law is: $\phi = \dfrac{q}{{{\varepsilon _0}}}$

Complete step by step solution: The space around a charge in which its electric force can be experienced is called the electric field. The electric field strength due to a point source charge $q$ at a distance $r$from the source charge is given by

$E = \dfrac{1}{{4\pi {\varepsilon _0}}}\dfrac{q}{{{r^2}}}$ $\to (1)$

where $\dfrac{1}{{4\pi {\varepsilon _0}}}$is a constant. Therefore, if $E = 0$, then from equation (1),
$q = 0$. Also, no charge resides on the surface.

The total number of lines of force that pass through a closed surface in an electric field is called electric flux. (The lines of force are the lines drawn for visualizing the electric field). The electric flux is denoted by $\phi$. According to Gauss Law, the net electric flux through any closed surface is equal to $\dfrac{1}{{{\varepsilon _0}}}$times the total electric charge enclosed by the surface. Therefore,
$\phi = \dfrac{q}{{{\varepsilon _0}}}$. If $q = 0$, then $\phi = 0$. Thus the electric flux is also zero.

Hence, option (D) is the correct answer.

Note: The lines of electric force are the lines drawn for visualizing the electric field. The electric flux depends upon these lines. Also, these lines are defined by the electric field.

Thus, if the electric field is zero, then there will be no lines of electric force that pass through a closed surface and therefore, there will be no electric flux through the surface.