## Electric Potential Sample Problem

Given electric potential equation $ V = x^3yz+2y^2z+xz^4 $, find:

a. The corresponding electric field equation for this potential

Using the identity $ E = - \nabla V $, we know that we need to compute the gradient of $ V $. We get:

$ \nabla V = \Bigg[\frac{\partial}{\partial x}(x^3yz+2y^2z+xz^4),\frac{\partial}{\partial y}(x^3yz+2y^2z+xz^4),\frac{\partial}{\partial z}(x^3yz+2y^2z+xz^4)\Bigg]\\ \nabla V = \Big[3x^2yz + z^4,x^3z + 4yz,x^3y + 2y^2 + 4xz^3\Big] $

b. The charge density of the field at $ (2,-3) $

$ E = - \nabla V $

$ \nabla \cdot E = \frac{\rho}{\epsilon_0} $

$ \Delta V = -\Large\frac{\rho}{\epsilon_0} $