Question

If potential in a region of space is given by $V=x^{2}y$, what is the electric field in the region?

• A. $\hat{i}+\hat{j}+\hat{k}$
• B. $-2xy\hat{i}-x^2\hat{j}$
• C. $-x^2\hat{i}-2xy\hat{j}$
• D. None of these

Answer 1

The correct option is B $-2xy\hat{i}-x^2\hat{j}$

We know that the electric field is related to potential difference as $\overrightarrow{E}=-[\frac{\partial V}{\partial x}\hat{i}+\frac{\partial V}{\partial y}\hat{j}+\frac{\partial V}{\partial z}\hat{k}]$
here the symbol $\partial$means partial derivative (remember that partial derivative of with respect to one variable, treating the other variables as constant)

$\frac{\partial V}{\partial x}=\frac{\partial \left(x^{2}y\right)}{\partial x}=2xy$
$\frac{\partial V}{\partial y}=\frac{\partial \left(x^{2}y\right)}{\partial y}=x^2$
$\frac{\partial V}{\partial z}=\frac{\partial \left(x^{2}y\right)}{\partial z}=0$
$so,\overrightarrow{E}=-2xy\hat{i}-x^2\hat{j}$