Question

For a given $\overrightarrow{E}=2x\hat{i}+3y\hat{j}$. Find the potential $V$ at $(2,2)$ , when $V$ at origin is $5\text{volts}$.

• A. $1\text{V}$
• B. $8\text{V}$
• C. $-4\text{V}$
• D. $-5\text{V}$

Answer 1

The correct option is D $-5\text{V}$

Given,

$\overrightarrow{E}=2x\hat{i}+3y\hat{j}$ and

$V$ at $(0,0)$ is $5\text{volts}$.

Relation between the potential gradient $\frac{dV}{dr}$ and electric field is given by,$$E=-\frac{dV}{dr}$$$\Rightarrow dV=-E.dr$

Let the potential at point $(2,2)$ is $V$. Now, integrating above equation between $V_i=5$ to $V_f=V$ we have,

${\int }_5^{V}dV=-\int \overrightarrow{E}.\overrightarrow{dr}$

${\int }_5^{V}dV=-\int (2x\hat{i}+3y\hat{j}).(dx\hat{i}+dy\hat{j})$

$\Rightarrow {\int }_5^{V}dV=-{\int }_0^{2}2xdx-{\int }_0^{2}3ydy$

$\Rightarrow V-5=-\frac{2(2{)}^2}{2}-\frac{3(2{)}^2}{2}$

$\Rightarrow V=-4-6+5$

$\Rightarrow V=-5\text{V}$

Hence, option (d) is the correct answer.