Homework Help: Electric flux through cone

1. Apr 21, 2015

Tanya Sharma

1. The problem statement, all variables and given/known data

2. Relevant equations

Flux = ∫E.ds

3. The attempt at a solution

I need to get the projection of cone on a plane perpendicular to the electric field . The area thus obtained when multiplied by electric field would give the flux .

I am not able to imagine the projected area .Is there a systematic way to calculate the projected area ?

I would be grateful if somebody could help me with the problem .

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2. Apr 21, 2015

mooncrater

Hint: You can convert the electric field into components parallel and perpendicular to the base of the cone.

3. Apr 21, 2015

TSny

Is the flux you're looking for related to the flux through the blue and yellow shaded regions shown below?

[EDIT: Sorry, ignore this post! I didn't visualize it carefully enough. Attached diagram was deleted to avoid being misleading .]

Last edited: Apr 21, 2015
4. Apr 21, 2015

TSny

Consider a patch of area as shown. Can you express its unit normal vector $\hat{n}$ in terms of the angles $\theta$ and $\phi$ and the $\hat{i}, \hat{j}, \hat{k}$ unit vectors?

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5. Apr 21, 2015

Tanya Sharma

Very nice ! Thank you .

6. Apr 21, 2015

Tanya Sharma

Sorry . Could you please elaborate a little .

7. Apr 21, 2015

TSny

The unit normal vector is parallel to the area vector of the patch. So it can be used to help express the flux through the patch.

I'm not sure I'm interpreting the original question properly. "Find the magnitude of the flux that only enters the cone's curved surface. Do not count the outgoing flux."

I interpret that to mean that for a patch of area on the curved surface where the flux is outward rather than inward, then we do not count that flux. Is this the way you also interpret it?

8. Apr 21, 2015

Tanya Sharma

Yes .

9. Apr 21, 2015

TSny

OK, good. If you had an expression for $\hat{n}$, how could you use it to determine if a patch of area has inward flux or outward flux?

10. Apr 21, 2015

Tanya Sharma

Sign of $\vec{E} \cdot ds\hat{n}$ determines whether flux is positive or negative . $ds$ is the area element of the differential element .

Last edited: Apr 21, 2015
11. Apr 21, 2015

TSny

Yes. That's why I think it's a good idea to find vector expressions for $\vec{E}$ and $\hat{n}$.

12. Apr 21, 2015

Tanya Sharma

But how do I calculate $\hat{n}$ ?

I haven't done something like this before . Does it involve spherical coordinates and double integrals ? Could you give some relevant web link ?

13. Apr 21, 2015

TSny

It's similar to working in spherical coordinates. If you projected $\hat{n}$ onto the xy plane, how long would this projected vector be in the xy plane (expressed in terms of $\theta$)? How can you express the x and y components of this projected vector in terms of $\theta$ and $\phi$?