Question

Determine the electric field strength vector if the potential of this field depends on $x,y$ coordinates as $V=10axy$.

• A. $-10a(y\hat{i}+x\hat{j})$
• B. $10a(y\hat{i}+x\hat{j})$
• C. $-10a(x\hat{j}+y\hat{k})$
• D. $-a(y\hat{i}+x\hat{j})$

Answer 1

The correct option is A $-10a(y\hat{i}+x\hat{j})$

Given,
Electric potential , $V=10axy$

Relation between electric field vector and electric potential at any point is given by,

$\overrightarrow{E}=-[\frac{\partial V}{\partial x}\hat{i}+\frac{\partial V}{\partial y}\hat{j}+\frac{\partial V}{\partial z}\hat{k}]....\left(1\right)$

Now,

$\frac{\partial V}{\partial x}=\frac{\partial \left(10axy\right)}{\partial x}=10ay$

$\frac{\partial V}{\partial y}=\frac{\partial \left(10axy\right)}{\partial y}=10ax$

$\frac{\partial V}{\partial z}=\frac{\partial \left(10axy\right)}{\partial z}=0$

Substituting the values in equation $\left(1\right)$ we get,
$\overrightarrow{E}=-(10ay\hat{i}+10ax\hat{j}+0\hat{k})$
$\therefore \overrightarrow{E}=-10a(y\hat{i}+x\hat{j}+0\hat{k})$

Hence, option (b) is the correct answer.