Question

Q1.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

Step-by-step explanation:

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:

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

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

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

So, final answer is :

$\boxed{ \bf \: \vec{E} = - \bigg\{ {y}^{2} z + 6xyz + 3x {y}^{2} \bigg \}} </p><p>$