Over a certain region of space, the electric potential is V = 5x - 5x2y + 2yz2.

Question

Over a certain region of space, the electric potential is V = 5x - 5x2y + 2yz2. (Use any variable or symbol stated above as necessary.)

Find the expression for the x component of the electric field over this region.
Ex =

Find the expression for the y component of the electric field over this region.
Ey =

Find the expression for the z component of the electric field over this region.
Ez =

What is the magnitude of the field at the point P, which has coordinates (5, 0, -5) m?
N/C

Explanation / Answer

E = - gradient V = - ? V ? V = i ?V/?x + j ?V/?y + k ?V/?z E = - [i ?V/?x + j ?V/?y + k ?V/?z] = Ex (i) + Ey (j) + Ez (k) ----------------------- comparing the coefficients of unit vectors Ex = - ?V/?x >>>>> so on derivatives are PARTIAL >> meaning when x varies > y, z are constants ----------------- NOTE V is not dependent on (z)??? ========== V(x,y,z) = Axy - Bx^2 + Cy ?V/?x = Ay - 2 Bx ?V/?y = Ax +C ?V/?z = 0 [Note above] ============= Ex = - Ay + 2 Bx Ey = - [Ax +C] Ez =0 ================ d) E = Ex (i) + Ey (j) + Ez (k) E = {- Ay + 2 Bx} (i) - [Ax +C] (j) field will be zero when> both Ex = Ey =0 Ex=0 > y = 2Bx/A Ey=0 > x = - C/A >>> y = -2BC/A^2

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