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Finding electric field

  1. Nov 19, 2012 #1
    1. The problem statement, all variables and given/known data
    I have to find the electric field at the center of a hemisphere of radius R with charge density σ.


    2. Relevant equations



    3. The attempt at a solution
    The easiest way would be to select a ring element of small thickness, find the electric field due to it and integrate.
    I am trying to do it by an another method. Let's select a small element of area dA subtending a solid angle dΩ at the center, then
    [tex]dA=(dΩ)R^2[/tex]
    Charge of this small element is σdA=σ(dΩ)R^2.
    Therefore, electric field due to this element is
    [tex]dE=\frac{kσ(dΩ)R^2}{R^2}[/tex]
    The last job is to resolve the electric field into two components (see attachement) and integrate them separately but i am stuck here. I can't proceed after this because of the cosθ and sinθ.
     

    Attached Files:

  2. jcsd
  3. Nov 19, 2012 #2

    gneill

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    Staff: Mentor

    Your solid angle differential element dΩ would need to be swept over the surface of the hemisphere. But this is not something that's inherent in the formalism for the solid angle element; you would need to impose "direction" of the element via some other mathematical mechanism, and you won't be integrating over dΩ...

    Why not make your differential area out of a square patch ##(R d\theta)\times(R d\phi)## where ##\theta## and ##\phi## vary to cover the hemisphere? Use symmetry arguments to simplify the calculation (certain components of the field elements dE will cancel over the integration so they can be dropped).
     
  4. Nov 19, 2012 #3
    Do you have a link (except wiki) which explains this in a simple way? I am not too much familiar with solid angles. :smile:

    Assuming ##\theta## for the lateral surface of hemisphere and ##\phi## for semicircle, how do you get ##(R d\theta)\times(R d\phi)##. What i get is an expression like this: ##(R\sin\phi d\theta)\times(R d\phi)##. (see attachment)
     

    Attached Files:

    • dA.png
      dA.png
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    Last edited: Nov 19, 2012
  5. Nov 19, 2012 #4

    gneill

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    Sorry, I can't think of a web site offhand. Maybe the Wolfram Mathworld site will have something.
    Yeah, you're right. The latitude influences the length of the arc of longitude for spherical coordinates. So the area element looks like

    ##dA = R^2 sin(\phi) d\theta d\phi##
     
  6. Nov 19, 2012 #5
    I am myself confused now. :confused:
    It should be ##R\cos\phi## instead of ##R\sin\phi##
     
  7. Nov 19, 2012 #6

    gneill

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    Ah, well for spherical coordinates the usual practice is to take the angle ##\phi## between the z-axis and the radial vector, rather than between the x-y plane and the vector.
     
  8. Nov 19, 2012 #7
    I am not aware of spherical coordinate system.
    But is my expression for differential area correct?
     
  9. Nov 19, 2012 #8

    gneill

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    It's fine for the appropriate choice of angles. Set up the integration of the resulting field elements. Again, take advantage of symmetry to simplify the calculation.
     
  10. Nov 19, 2012 #9
    Here's my attempt:
    [tex]dA=R^2\cos(\phi)d\theta d\phi[/tex]
    [tex]dq=σR^2\cos(\phi)d\theta d\phi[/tex]
    [tex]dE=\frac{kdq}{R^2}[/tex]
    [tex]dE=kσR^2\cos(\phi)d\theta d\phi[/tex]
    We need to integrate only the sin component because the cos component cancels due to symmetry of the situation.
    [tex]dE\sin(\phi)=\frac{1}{2}kσ\sin(2\phi)d\theta d\phi[/tex]
    Integrating this, i get E=0. :confused:
     
  11. Nov 19, 2012 #10

    gneill

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    What were the limits of your integration?
     
  12. Nov 19, 2012 #11
    ##\phi## from 0 to ##\pi## and ##\theta## from 0 to ##2\pi##.
     
  13. Nov 19, 2012 #12

    gneill

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    Ah. You've made your ##\phi## cover the area twice, once from the x-y plane up to the zenith, and then from the zenith back down to the plane. Remember that the ##\theta## will swing the vector all the way around (since it's going from 0 to ##2\pi##), so at any given ##\phi## elevation it will already be swept around the whole surface. Make ##\phi## go from 0 to ##\pi/2##.
     
  14. Nov 19, 2012 #13
    Oh yes, i get it. I have got the right answer. Thanks a lot gneill. :smile:

    This method is way easier than selecting a small ring element.
     
  15. Nov 19, 2012 #14

    gneill

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    Glad it worked out. Cheers.
     
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