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

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for