Physics question on calculation of electric flux.?

Question

A right angled isosceles triangle ABC of equal sides AB=BC=d and angle B = 90 is placed in a horizontal plane. A test charge "q" is placed vertically above vertex "A" at a distance "d". Calculate the flux of the electric field passing through the triangle.
Options are : q/6ε , q/18ε, q/24ε, q/48ε

Anonymous · Accepted Answer

Just so you know, people are much more ready to help you with a problem if you show them that you have at least tried (even if answer is wrong). I started helping people in Yahoo Answers about a week ago, then stopped because some of them were clearly using the Internet to do their homework (which is a bit silly, but anyhow).

This is Gaussian Integral problem as you know.

There are two ways to do this problem. The teacher clearly wants you to use one way over the other.

FIRST WAY (Do not use).

1. Calculate the intensity of electric field caused by q that penetrates the triangle at the (x, y) position in the triangle.
2. Use your intuition or other means to find the direction of this field.
3. Now you have your E vector at (x, y).
4. Because triangle is lying flat in plane, your E-dot-dA will result in only the Z component of your E vector being applicable. The X and Y components are unable to contribute to the penetration of the triangle because they are parallel to the triangle.
5. Now prepare to double-integrate with independent variable x from (0, d). 
6. Your limit for y on double integral will be dependent on your x. This part is very easy.
7. Calculate integral (x = 0, x = d)(y = 0, y = limit(y, x)) * z-component of E(x, y, z) dxdy.

SECOND WAY:

You already know what Gauss's Law says about amount of charge inside based upon field that penetrates outward through the surface of its container. Therefore, all you have to do is imagine that that the charge is at the center of a cube. Since it is at the center of cube, flux through all 6 faces will be equal since a cube is totally symmetric. Taking a single face into consideration, you have a reduction by factor of 6. Now that you know the flux through one face, you look at that face, and realize that your triangle is in one quadrant of the face, so now you have a reduction by factor of 4. In that quadrant, your triangle takes up only half, so now you have a reduction by 2.

Reduction of some quantity by 6, then 4, then 2, will give you your answer.

EDIT: OK. :) In that case, just-in-case, the answer is q/48ε.

Physics question on calculation of electric flux.?

1 Answer