Question

If uniform electric field $\overrightarrow{E}=E_0\hat{i}+2E_0\hat{j}$ where $E_0$ is a constant, exists in a region of space and at (0, 0) the electric potential V is zero, then the potential at $(x_0,0)$ will be

• A. zero
• B. $-E_0{x}_0$
• C. $-2E_0{x}_0$
• D. $-\sqrt{5}{E}_0{x}_0$

Answer 1

The correct option is B $-E_0{x}_0$

Given :

$\overrightarrow{E}=E_0\hat{i}+2E_0\hat{j}$

$V(0,0)=0$

To find :

$V(x_0,0)$

Solution :

We know the relation between E and V as

$\nabla \overrightarrow{V}=-\int \overrightarrow{E}\cdot \overrightarrow{dl}$

$V(x_0,0)-V(0,0)=-{\int }_{(0,0)}^{(x_0,0)}\overrightarrow{E}\cdot \overrightarrow{dl}$

$V(x_0,0)-0=-{\int }_0^{x_0}(E_0\hat{i}+2E_0\hat{j})\cdot \left(dx\hat{i}\right)$ $(\because \text{change is only in x direction})$

$V(x_0,0)=-{\int }_0^{x_0}{E}_{0}dx$

$V(x_0,0)=-E_0{x}_0$

Correct Opt : {B}