Question

In a region, the potential is represented by $V(x,y,z)=2xy–4y–yz+3zx$, where $V$ is in $\text{Volts}$ and $x,y$ and $z$ are in meters. The electric force experienced by a charge of $5$ coulomb at the point $(1,2,1)$ is $5\sqrt{n}$. Find $n$

Answer 1

Given:
$V(x,y,z)=2xy–4y–yz+3zx$
$q=5\text{C}$
Point $=(1,2,1)$
As we know that,

$E_{x(1,2,1)}=-\frac{\partial V}{\partial x}$

$E_{x(1,2,1)}=-\frac{\partial }{\partial x}(2xy–4y–yz+3zx)$

$E_{x(1,2,1)}=-(2y+3z)=-(2\times 2+3\times 1)$

$E_{x(1,2,1)}=-7\text{N/C}$

$E_{y(1,2,1)}=-\frac{\partial V}{\partial y}$

$E_{y(1,2,1)}=-\frac{\partial }{\partial y}(2xy–4y–yz+3zx)$

$E_{y(1,2,1)}=-(2x-4-z)=-(2\times 1-4-1)$

$E_{x(1,2,1)}=3\text{N/C}$

$E_{z(1,2,1)}=-\frac{\partial V}{\partial z}$

$E_{z(1,2,1)}=-\frac{\partial }{\partial z}(2xy–4y–yz+3zx)$

$E_{z(1,2,1)}=-(-y+3x)=-(-2+3\times 1)$

$E_{x(1,2,1)}=-1\text{N/C}$

$E=\sqrt{(E_x^2+E_y^2+E_z^2)}=\sqrt{(-7^2+3^2+(-1{)}^2)}$

$E=\sqrt{59}\text{N/C}$

Now $F=qE=5\times \sqrt{59}=5\sqrt{59}\text{N}$