You are given a potential function: V (x, y, z) = x + 4xy + 2z^2 + 1 (Units are

Question

You are given a potential function: V (x, y, z) = x + 4xy + 2z^2 + 1 (Units are volts and metres). (a) Find the electric field E(x, y,z) at the origin (0,0, 0) AND at point (2, 2,2). You may leave the answers in (i,j, k) notation without finding magnitude and direction (10 marks) (b) Find the potential difference going from (2, 2, 2) to the origin (0, 0,0) (4 marks) (c) How much work has to be done to move a charge -e (an electron) from (2, 2, 2) to the origin? Express you answer in both electron volts (eV) and in Joules. (4 marks) Hint: use the potentials, not the electric field, to make it easy! (d) Does the charge move spontaneously from (2,2,2) to the origin, or is an external force required to push it? Explain why. (2 marks)

Explanation / Answer

A.

Here d is partial differentiation.

E= - dV/dX i - dV/dy j- dV/dz k ---------1

Substitute potential function in 1

E= (-x-4y) i - (4x) j - (4z) k ---------2

At (0,0,0) E=0 i+0 j+ 0k --------3

At (2,2,2) E=-12 i - 8 j - 8 k ------4

b.

V2-V1= (2+4*2*2+2*2^2+1) -(0+0+0+1)

V2-V1=26 V

c.

W= - eV = -26 eV = - 41.6*10^-19 J

d.

Electric force has to act on charge then only it can move.

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