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:

$\nabla V = \left[\begin{array}{l} \frac{\partial}{\partial x}(x^3yz+2y^2z+xz^4) \\ \frac{\partial}{\partial y}(x^3yz+2y^2z+xz^4) \\ \frac{\partial}{\partial z}(x^3yz+2y^2z+xz^4) \end{array}\right] \\ \nabla V = \left[\begin{array}{l} 3x^2yz + z^4 \\ x^3z + 4yz \\ x^3y + 2y^2 + 4xz^3 \end{array}\right]$

Applying the identity, we get:

$E = \left[\begin{array}{l} -3x^2yz - z^4 \\ -x^3z - 4yz \\ -x^3y - 2y^2 - 4xz^3 \end{array}\right]$

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

$E = - \nabla V$

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

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

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