Question

The electric field vector in a region is given by $\overrightarrow{\mathrm{E}}=(3\overrightarrow{\mathrm{i}}+4\mathrm{y}\overrightarrow{\mathrm{j}})\mathrm{V}/\mathrm{m}$ The potential at the origin is zero. Then the potential at a point $(2,1)$ m is

• A. $7\mathrm{V}$
• B. $8\mathrm{V}$
• C. $-8\mathrm{V}$
• D. $-7\mathrm{V}$

Answer 1

The correct option is C

$-8\mathrm{V}$

Step 1: Given information

Electric field vector, $\overrightarrow{\mathrm{E}}=(3\overrightarrow{\mathrm{i}}+4\mathrm{y}\overrightarrow{\mathrm{j}})\mathrm{V}/\mathrm{m}$

Then,

$$\begin{gathered} {\mathrm{E}}_{\mathrm{x}}=3 \\ {\mathrm{E}}_{\mathrm{y}}=4\mathrm{y} \end{gathered}$$

We have to find the potential at a point $(2,1)$ m.

Step 2: Calculation

The potential will be:

$\mathrm{V}=-{\int }_0^2{\mathrm{E}}_{\mathrm{x}}\mathrm{dx}-{\int }_0^1{\mathrm{E}}_{\mathrm{y}}\mathrm{dy}$

By substituting the given values,

$$\begin{gathered} \mathrm{V}=-{\int }_0^{2}3\mathrm{dx}-{\int }_0^{1}4\mathrm{ydy} \\ =-3{\left[\mathrm{x}\right]}_0^2-2{\left[{\mathrm{y}}^2\right]}_0^1 \\ =-3\times 2-2\times 1 \\ =-8\mathrm{V} \end{gathered}$$

Therefore, the potential is $-8\mathrm{V}$.