Question

N a certain region, the electric potential due to a charge distribution is given by the equation v(x,y,z)=3(x^2)(y^2) +y(z^3) - 2x(z^3), where x,y, and z are measured in meters and v is in volts. calculate the magnitude of the electric field vector at the position (x,y,z)=(1.0,1.0,1.0).

Answer 1

The electric field $\mathbf E$ is the negative of the gradient of the electric potential:

$\mathbf E=-\nabla V(x,y,z)=-\left(6xy^2-2z^3,6x^2y+z^3,3yz^2-6xz^2\right)$

At the point (1.0, 1.0, 1.0), the electric field has magnitude

$\|\mathbf E\|=\|(4.0,7.0,-3.0)\|=8.6\dfrac{\rm V}{\rm m}$