Question

In finding the electric field using Gauss law, the formula $|\overrightarrow{E}|=\frac{q_{enc}}{{\epsilon }_0|A|}$ is applicable. In the formula ${\epsilon }_0$ is permittivity of free space, $A$ is the area of Gaussian surface and $q_{enc}$ is charge enclosed by the Gaussian surface. This equation can be used in which of the following situation ?

• A. Only when the Gaussian surface is an equipotential surface.
• B. Only when a Gaussian surface is an equipotential surface and $|\overrightarrow{E}|$ is constant on the surface.
• C. Only when $|\overrightarrow{E}|=$ constant on the surface.
• D. For any choice of Gaussian surface.

Answer 1

The correct option is B Only when a Gaussian surface is an equipotential surface and $|\overrightarrow{E}|$ is constant on the surface.

The integral form of Gauss law is given by,

$\int \overrightarrow{E}\cdot \overrightarrow{dA}=\frac{q_{enc}}{{\epsilon }_0}$

Given that,
$|\overrightarrow{E}|=\frac{q_{enc}}{{\epsilon }_0|A|}\Rightarrow \frac{q_{enc}}{{\epsilon }_0}=|\overrightarrow{E}||\overrightarrow{A}|$

From these two equations, we get

$\int \overrightarrow{E}\cdot \overrightarrow{dA}=|\overrightarrow{E}||\overrightarrow{A}|$

The above expression is valid only when $|\overrightarrow{E}|$ is constant on the surface.

And also, the angle between $\overrightarrow{E}$ and $\overrightarrow{dA}$ is zero.

Therefore, $\overrightarrow{E}$ should be parallel to $\overrightarrow{dA}$, and which implies $\overrightarrow{E}$ should be perpendicular to the plane of the area of the surface.

$\overrightarrow{E}$ will be perpendicular to the plane of the area of the surface when the surface is an equipotential surface.

Hence, option $\left(B\right)$ is correct.