Over a certain region of space, the electric potential is V = 4x - 7x2y + 7yz2. Find the expressions for the x, y, z components of the electric field over this region. (Use any variable or symbol stated above as necessary.)

Ex = - 4 + 14xy

Ey = 7x² - 7z²

Ez = - 14yz

Step-by-step explanation:

The relationship between Electric field Er (x, y, z) and Electric potential, V, is a differential relationship:

Er (x, y, z) = - dV/dr (x, y, z)

Where r (x, y, z) = distance in x, y and z components.

The x component of the electric field is:

Ex = - dV/dx

Given that:

V = 4x - 7x²y + 7yz²

Ex = - dV/dx

Ex = - (4 - 14xy)

Ex = - 4 + 14xy

The y component of the electric field is:

Ey = - dV/dy

Ey = - (-7x² + 7yz²)

Ey = 7x² - 7z²

The z component of the electric field is:

Ez = - dV/dz

Ez = - (14yz)

Ez = - 14yz