# Finding flux through a part of infinite disc

1. Dec 10, 2013

### Saitama

1. The problem statement, all variables and given/known data
A charge $(5\sqrt{2}+2\sqrt{5})$ coulomb is placed on the axis of an infinite disc at distance $a$ from the centre of disc. The flux of this charge on the part of the disc having inner and outer radius of $a$ and $2a$ will be

A)$\dfrac{3}{2\epsilon_0}$

B)$\dfrac{1}{2\epsilon_0}$

C)$\dfrac{2(\sqrt{5}+\sqrt{2})}{\epsilon_0}$

D)$\dfrac{2\sqrt{5}+5\sqrt{2}}{2\epsilon_0}$

2. Relevant equations

3. The attempt at a solution
I have solved the given problem through integration but since this is an exam problem, I am wondering if there is a smarter way to solve this problem. Since the disc is infinite, I think we can use this fact to find a shorter method. Or is integration the only way to solve the problem?

Any help is appreciated. Thanks!

2. Dec 10, 2013

### TSny

Did you integrate by breaking the annular region of the disk into nested rings and integrating the normal component of E over the rings?

If so, there is a trick you can do to set up a much simpler integral that will yield the result. (You can avoid actually doing this integration if you happen to know the formula for the surface area of a spherical cap (hint). But that's something I would have to look up )

3. Dec 10, 2013

### Saitama

Yes, that's exactly how I did it. :)

I think I understand your trick but I don't seem to able to apply it. Don't we have two spherical caps here? One of radius $\sqrt{2a^2}$ and the other of $\sqrt{5a^2}$.

4. Dec 10, 2013

### TSny

I was thinking of working with a single sphere which you could take to be a sphere of unit radius. But, yes, there will be two spherical caps on this sphere to consider, if you want to avoid integration. Or, you can set up a simple integral on this sphere and not worry about looking up a formula for the area of a spherical cap.

5. Dec 10, 2013

### Saitama

Umm...I don't get it, I have attached a figure, do I have to find the flux through the red spherical cap? I am thinking that I will have to deal with solid angle.

#### Attached Files:

• ###### spherical caps.png
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6. Dec 10, 2013

### TSny

Draw a small sphere around the charge with a radius considerably smaller than $a$. You can think of it as having unit radius.

EDIT: Here's a pic

#### Attached Files:

• ###### Annular flux.png
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Last edited: Dec 10, 2013
7. Dec 11, 2013

### Saitama

Brilliant! That is such a nice method, thank you very much TSny! :)

How did you think of the sphere? And what's wrong with my spherical caps?

8. Dec 11, 2013

### TSny

I'm not sure how the thought came to me. Probably I've seen similar examples before. (I'm old)

Your spherical caps are just fine! It's a nice way and it will require about the same work as using a single sphere.

9. Dec 13, 2013

### Saitama

Can you please tell me where you saw these kind of examples? :)

Yes, I reached the answer from my spherical caps too, thank you for the awesome solution TSny!

10. Dec 13, 2013

### sankalpmittal

This question was from FIITJEE AITS part test II. Are you giving this exam ? Same here. I was not able to solve this problem during exam. It was only at home that I solved it by manual integration.

But your method is awesome shortcut TSny !! Thanks !! :)

11. Dec 13, 2013

### TSny

I really can't say. You'll just pick up this sort of thing with practice.

Good work!