# Tough electric flux problem

1. Feb 19, 2009

### Eric_meyers

1. The problem statement, all variables and given/known data
A very small electric dipole is a distance z from the center of a loop of radius a, with its dipole moment vector p directed along the axis of the loop. The component of the electric field of the dipole perpendicular to the loop at any point P is a distance r from the dipole given by

E(perpendicular) = p/(4 pi (epsilon) r^3) (3cos^2(theta) - 1)

-Where (theta) is the angle between the axis of the loop and the line from the dipole to P. Show that the electric flux through the loop is given by.

flux = [p/(2(epsilon))] a^2/(z^2 + a^2)^3/2

2. Relevant equations
flux = integral E(perpendicular) dA

3. The attempt at a solution

Ok, I've been at it for an hour and a half, and really from what I can surmise is this.

The loop is a circle whose center is at the origin, the dipole above it - a distance z, has a point a distance r from it. The cos factor in the electric field given to me corrects the field propagating from the dipole to only include the perpendicular components to my loop.

cos(theta) = z/r where r can't = 0

my area of my loop = pi * a^2

I'm having trouble thinking if this is an integral problem or if I can apply Guass's law and simply just multiple my electric field by my area to get the flux.

if I multiply the area with the corrected cos factor I get. 3cos^2(theta) - 1 = 3* z^2/r^2 - 1

3 * z^2 * p * pi * a^2/ (4 * pi * epsilon * r^3) [[[a being the radius of loop]]]

Which would then simplify to

3/4 * z^2 * p * a^2/(epsilon * r) and r would be (a^2 + z^2)^1/2

But clearly this is not right.