The electric potential V in a region of space is given by V ( x , y , z )= A ( x

Question

The electric potential V in a region of space is given by V(x,y,z)=A(x23y2+z2) where A is a constant.

Part A) Derive an expression for the electric field E  at any point in this region.

Enter your answers separated by commas. Express your answer in terms of the variables x, y, z and A.

Part B) The work done by the field when a 1.50- Ctest charge moves from the point (x,y,z)=(0,0,0.250m)to the origin is measured to be 6.00×105J. Determine A.

Part C) Determine the electric field at the point (0, 0, 0.250 m).

Enter your answers numerically separated by commas.

Explanation / Answer

(A)

The electric field E is minus the gradient of the potential :

E = - grad V
Ex = - dV/dx = - 2Ax
Ey = - dV/dy = 6Ay
Ez = - dV/dz = - 2Az
The vector E = - 2Ax i , 6Ay j, 2Az k
b) The potential energy of q = 1.5x10-6 C ,

is U = q V ;
the work done by the field when q moves from P to O is :
W = - [U(O) - U(P)] = U(P) - U(O)

= q[V(P) - V(O)]

= qA 0.252.
Then

A = 6x10-5/(1.5x10-6 x0.252)

= 640 V/m2

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