Over a certain region of space the electric potential is giv

Over a certain region of space, the electric potential is given by V - 5x - 3x^y + 2yz^2. Find the expressions for the x, y, and z components of the electric field over this region. What is the magnitude of the field at the point P, which has coordinates (in meters) (1,0, - 2)?

Solution

E_x = - dV/dx = 5 - 6xy = 5 - 6 (1) (0) = - 5

E_y = - dV/dy = 0 - 6x² + 2z² = - 6 (1)² + 2 (-2)² = - 2

E_z = - dV/dz = 4yz = 4 (0) (-2) = 0

magnitude of electric field = E = sqrt(E_x² + E_y² + E_z²) = sqrt((-5)² + (-2)² + (0)²) = 5.4