Question

The electric potential at any point is given
by ¢= xy(z²— 3xy). Calculate the electric field E at that point.

Answer 1

$$\mathbf{E} = -\nabla \phi = - (\partial_x \phi, \partial_y \phi, \partial_z \phi)$$

$$\partial_x \phi = \partial_x (xyz^2 - 3x^2 y^2) = yz^2 -6xy^2$$ $$\partial_y\phi =xz^2 - 6x^2 y$$ $$\partial_z \phi = 2xyz$$

$$\mathbf{E} = (6xy^2 - yz^2, 6x^2 y - xz^2, -2xyz)$$