How Does Electric Flux Change with Dipole Orientation and Distance?

[Eric_meyers]

Homework Statement

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

Homework Equations

flux = integral E(perpendicular) dA

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.

 

[jbsteele]

So I assume it's an integral.I'm having trouble understanding how to set up the integral, so any help would be greatly appreciated. I assume my limits of integration, right? r= 0 to a theta = 0 to 2piI'm just not sure how to write this.

 

[nlightner]

Dear student,

Thank you for your question. It seems like you have made some progress in understanding the problem, but you are still unsure about how to approach it. Let me try to help you.

First, let's review the problem statement. We have a small electric dipole located at a distance z from the center of a circular loop with radius a. The dipole moment vector is directed along the axis of the loop. We are interested in finding the electric flux through the loop, which is given by the integral of the perpendicular component of the electric field over the loop's area.

Now, you correctly identified the expression for the perpendicular component of the electric field at a point P located at a distance r from the dipole. However, you are missing the integral part of the problem. In order to find the flux through the loop, we need to integrate the electric field over the loop's area. This is given by the surface integral:

flux = ∫∫ E(perpendicular) dA

where dA is an infinitesimal element of the loop's area.

So, how do we perform this integral? One way is to use Gauss's law, which states that the electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space (ε0). In our case, the loop is a closed surface, and the total charge enclosed is equal to the dipole moment p. Therefore, we can write:

flux = p/ε0

But this is not the same expression that we were asked to show. So, let's see if we can derive the given expression from this result.

The key is to relate the distance r from the dipole to the distance z from the center of the loop. We can do this by using the Pythagorean theorem, which tells us that r^2 = z^2 + a^2. Substituting this into our expression for the perpendicular component of the electric field, we get:

E(perpendicular) = p/(4πεr^3) (3cos^2θ - 1)

= p/(4πε(z^2 + a^2)^3/2) (3(z^2/(z^2 + a^2)) - 1)

= p/(4πε(z^2 + a^2)^3/2) (2z^2/(z^2