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

JIPMERJIPMER 2019

Solution:

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)
$\vec{E} = 2\left(1\right)\left(2\right)\hat{i}+\left(1\right)^{2}\hat{j}$
$\vec{E}=4\hat{i}+\hat{j} V/m$