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Finding flux through a part of infinite disc

  1. Dec 10, 2013 #1
    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}##

    Answer: A


    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. jcsd
  3. Dec 10, 2013 #2

    TSny

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    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 :blushing:)
     
  4. Dec 10, 2013 #3
    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}##.
     
  5. Dec 10, 2013 #4

    TSny

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    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.
     
  6. Dec 10, 2013 #5
    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:

  7. Dec 10, 2013 #6

    TSny

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    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:

    Last edited: Dec 10, 2013
  8. Dec 11, 2013 #7
    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?
     
  9. Dec 11, 2013 #8

    TSny

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    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.
     
  10. Dec 13, 2013 #9
    Very sorry for the delay in reply, I did not have access to internet.

    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! :smile:
     
  11. Dec 13, 2013 #10
    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 !! :)
     
  12. Dec 13, 2013 #11

    TSny

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

    Good work!
     
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