Question

In a region, the potential is represented by $V(x,y,z)=6x-8xy+6yz$, where $V$ is in volts and $x$, $y$ and $z$ are in metres. The electric force experienced by a charge of $2$ Coulomb situated at point $(1,1,1)$ is

• A. $6\sqrt{5}N$
• B. $30N$
• C. $24N$
• D. $4\sqrt{35}N$

Answer 1

The correct option is D $4\sqrt{35}N$

Given that $V=6x-8xy+6yz$
Partially differentiating this with respect $x,y$ and $z$ separately
we get
${\overrightarrow{E}}_x=\frac{-\partial V}{\partial x}=-(6-8y)\hat{i}$
${\overrightarrow{E}}_y=\frac{-\partial V}{\partial y}=-(-8x+6z)\hat{j}$
${\overrightarrow{E}}_z=\frac{-\partial V}{\partial z}=-6y\hat{k}$
$\overrightarrow{E}=-(6-8y)\hat{i}+-(-8x+6z)\hat{j}+-6y\hat{k}........\left(i\right)$
Now at given point put $x=1,y=1$ and $z=1$
We get
$\overrightarrow{E}=2\hat{i}+10\hat{j}-6\hat{k}$
$\left|\overrightarrow{F}\right|=q\times \left|\overrightarrow{E}\right|$
$\left|\overrightarrow{F}\right|=2\times \sqrt{2^2+{10}^2+(-6{)}^2}$
$\left|\overrightarrow{F}\right|=4\sqrt{35}$