# Electric Potential ?

Nov 30, 2017

We know that

$\vec{E} = - \frac{\partial V}{\partial x} \hat{i} - \frac{\partial V}{\partial y} \hat{j} - \frac{\partial V}{\partial z} \hat{k}$

$\therefore$

$V = - \int {E}_{x} \mathrm{dx} - \int {E}_{y} \mathrm{dy} - \int {E}_{z} \mathrm{dz}$

We are given that ${E}_{x} = 2 {x}^{3}$, ${E}_{y} = - 3 {x}^{2} y$, and ${E}_{z} = 2 y {z}^{2}$:

$V = - \int 2 {x}^{3} \mathrm{dx} + \int 3 {x}^{2} y \mathrm{dy} - \int 2 y {z}^{2} \mathrm{dz}$

$V = - \frac{1}{2} {x}^{4} + \frac{3}{2} {x}^{2} {y}^{2} - \frac{2}{3} y {z}^{3}$

Evaluate at the point #(1,3,-1)

$V = - \frac{1}{2} {\left(1\right)}^{4} + \frac{3}{2} {\left(1\right)}^{2} {\left(3\right)}^{2} - \frac{2}{3} \left(3\right) {\left(- 1\right)}^{3}$

$V = 15 \text{ V}$