In a region, the potential is represented by $V(x, y, z)=$ $6 x-8 x y-8 y+6 y z$, where $V$ is in vo

Question

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

Professor PC Thomas Classes · Answer

Here, $V(x, y, z)=6 x-8 x y-8 y+6 y z$

The $x, y$ and $z$ components of electric field are
$E_{x}=-\frac{\partial V}{\partial x}=-\frac{\partial}{\partial x}(6 x-8 x y-8 y+6 y z)$
$=-(6-8 y)=-6+8 y$
$E_{y}=-\frac{\partial V}{\partial y}=-\frac{\partial}{\partial y}(6 x-8 x y-8 y+6 y z)$
$=-(-8 x-8+6 z)=8 x+8-6 z$
$E_{z}=-\frac{\partial V}{\partial z}=-\frac{\partial}{\partial z}(6 x-8 x y-8 y+6 y z)=-6 y$
$\vec{E}=E_{x} \hat{i}+E_{y} \hat{j}+E_{z} \hat{k}$
$=(-6+8 y) \hat{i}+(8 x+8-6 z) \hat{j}-6 y \hat{k}$
At point ( $1,1,1$ )
$\vec{E}=(-6+8) \hat{i}+(8+8-6) \hat{j}-6 \hat{k}=2 \hat{i}+10 \hat{j}-6 \hat{k}$
The magnitude of electric field $\vec{E}$ is
$\vec{E}=\sqrt{E_{x}^{2}+E_{y}^{2}+E_{z}^{2}}=\sqrt{(2)^{2}+(10)^{2}+(-6)^{2}}$
$=\sqrt{140}=2 \sqrt{35} \mathrm{~N} \mathrm{C}^{-1}$
Electric force experienced by the charge
$F=q E=2 \mathrm{C} 	imes 2 \sqrt{35} \mathrm{~N} \mathrm{C}^{-1}=4 \sqrt{35} \mathrm{~N}$