Question

The electric potential in a given region in free space is V = 3xy2 z. i) Obtain an expression for the electric field E

Answer 1

Given:

The electric potential in a given region in free space is V = 3xy²z.

To find:

Electrostatic field ?

Calculation:

The electrostatic field intensity can be calculated using partial differentiation as follows:

$\rm \therefore\vec{E} = - \bigg( \dfrac{ \partial V}{ \partial x} \hat{i} + \dfrac{ \partial V}{ \partial y} \hat{j} + \dfrac{ \partial V}{ \partial z} \hat{k} \bigg)$

$\rm \implies \: \vec{E} = - \bigg \{\dfrac{ \partial (3x {y}^{2} z)}{ \partial x} \hat{i} + \dfrac{ \partial (3x {y}^{2} z)}{ \partial y} \hat{j} + \dfrac{ \partial (3x {y}^{2}z) }{ \partial z} \hat{k} \bigg \}$

$\rm \implies \: \vec{E} = - \bigg\{ {y}^{2} z \hat{i}+ 6xyz \hat{j}+ 3x {y}^{2}\hat{k} \bigg \}$

So, final answer is :

$\boxed{ \bf \: \vec{E} = - \bigg\{ {y}^{2} z\hat{i} + 6xyz \hat{j}+ 3x {y}^{2}\hat{k} \bigg \}}$