Question

Potential difference is given as $V(x,y)=-x^{2}y$ volt. Find electic field at a point $(1,2)?$

• A. $\hat{i}+4\hat{j}$ V/m
• B. $-4\hat{i}+\hat{j}$ V/m
• C. $4\hat{i}+\hat{j}$ V/m
• D. $4\hat{i}-\hat{j}$ V/m

Answer 1

Answer: (C)

As we know that,
$\vec{E}=E_x\hat{i}+E_y\hat{j}+E_z\hat{k}$
where, $E_x=-\frac{\partial V}{\partial x},E_y=-\frac{\partial V}{\partial y}$
and $E_z=-\frac{\partial V}{\partial z}$
Given, $V\left(x,y\right)=-x^{2}y$
So, $E_x=\frac{-\partial V}{\partial x}=\frac{-\partial \left(-x^{2}y\right)}{\partial x}=2xy$
$E_y=-\frac{\partial V}{\partial y}=\frac{-\partial }{\partial y}\left(-x^{2}y\right)=x^2$
so, $\vec{E}=E_x\hat{i}+E_y\hat{j}$
$\vec{E}=2xy\hat{i}+x^2\hat{j}$
Electric field at point (1, 2)
$\vec{E}=2\left(1\right)\left(2\right)\hat{i}+{\left(1\right)}^2\hat{j}$
$\vec{E}=4\hat{i}+\hat{j}V/m$