Question

The electric potential in a volume of space is given by V(x, y, z) = 2x2 - 3y2 + 5z.

Answer 1

The electric field at (x,y,z) its equal to the negative of the gradient if the electric potential.

We have

V(x, y, z) = 2x² - 3y² + 5z

so

E(x, y, z) = -grad(V) = -(dV/dx i + dV/dy j + dV/dz k)

where d/d(variable) is meant to be a partial derivative with respect to that variable. The partial derivatives are

dV/dx = 4x

dV/dy = -6y

dV/dz = 5

and so the electric field at any point is

E(x, y, z) = -4x i + 6y j - 5k

and at (3, 4, 5) it is

E(3, 4, 5) = -12i + 24j - 5k