Question

If $\mathbf{V}\mathbf{=}\mathbf{0}$ at a point in space, must $\overrightarrow{\mathbf{E}}\mathbf{=}\mathbf{0}$ ? If $\overrightarrow{\mathbf{E}}\mathbf{=}\mathbf{0}$ at some point, must $\mathbf{V}\mathbf{=}\mathbf{0}$ at that point? Explain. Give examples for each.

Short answer

When$V=0$at a point in space, the electric field must be zero,$\overrightarrow{\mathrm{E}}=0$.

No, it is not mandatory for $V=0$ when $\overrightarrow{\mathrm{E}}=0$.

Step-by-step solution

Understanding the electric field

The electric field vector is the negative gradient of the electric potential at any point.

The electric field at any point is

$\overrightarrow{\mathrm{E}}=-\overrightarrow{\mathrm{\nabla }}V$.

Evaluation of electric field and electric potential

The electric field is zero. Then from equation (i),

$\begin{array}{rl}c\overrightarrow{\mathrm{\nabla }}V& =0\\ V=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\end{array}$

The electric potential is zero or any constant value; then, the electric field is zero at that point.

So when$V=0$at a point in space, the electric field must be zero,$\overrightarrow{\mathrm{E}}=0$.

When$\overrightarrow{\mathrm{E}}=0$, the value of electric potential is either zero or constant.

Thus, it is not mandatory for $V=0$ when $\overrightarrow{\mathrm{E}}=0$.