Question

The electric potential at any point $(x,y,z)$ is given as $V=\left(2y^3\right)\text{volt}$ . The electric field at point $(2,1,0)$ is
(($x,y,z)$ are in metre)

• A. $6\hat{j}{\text{Vm}}^{-1}$
• B. $-6\hat{j}{\text{Vm}}^{-1}$
• C. $2\hat{i}{\text{Vm}}^{-1}$
• D. $-2\hat{i}{\text{Vm}}^{-1}$

Answer 1

The correct option is B $-6\hat{j}{\text{Vm}}^{-1}$

From the relation between electric field and potential
$\overrightarrow{E}=-\frac{\partial V}{\partial x}\hat{i}-\frac{\partial V}{\partial y}\hat{j}-\frac{\partial V}{\partial z}\hat{k}$

$\overrightarrow{E}=-\frac{\partial V}{\partial y}\hat{j}[\text{As}\frac{\partial V}{\partial x}=\frac{\partial V}{\partial z}=0]$

$\Rightarrow \overrightarrow{E}=-\frac{\partial }{\partial y}\left(2y^3\right)\hat{j}$

$\Rightarrow \overrightarrow{E}=-6y^2\hat{j}$

At $y=1\text{m}$
$\Rightarrow \overrightarrow{E}=-6\hat{j}{\text{Vm}}^{-1}$