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Homework Help: Nonuniformly charged spherical surface

  1. Aug 27, 2012 #1
    A sphere of radius a in free space is nonuniformly charged over its surface such that the charge density is given by ρs(θ) = ρs0 sin 2θ, where ρs0 is a constant and 0≤θ≤∏. Compute the total charge of the sphere.

    So I know
    ρs = dQ/dS

    Integrating the surface charge density function will give me the charge Q. My question is how would you set up this integral?

    ∫ρs0 sin 2θ dS
    integrating 0 to ∏

    Or would this involve much more than that such as a triple integral?

    Any help getting this set up would be great! Thanks!
  2. jcsd
  3. Aug 27, 2012 #2
    This is not a triple integral. This is a surface integral.
  4. Aug 27, 2012 #3
    Alright then given the information above how would you set up the surface integral?
  5. Aug 27, 2012 #4
    You have already done that. Now you just need to integrate it. You need to convert to to a double integral via an appropriate coordinate system. What would it be?
  6. Aug 27, 2012 #5
    I haven't had calculus in a long time so I have forgotten most of it.

    ∫ ∫ ρs0 sin 2θ dA = ps0 sin 2θ * 8∏r dr dθ
    0 -> ∏ 0 ->a

    A = 4∏r2
    dA = 8∏r
  7. Aug 27, 2012 #6
    It can't be true, it is a sphere, so r = a = const. What coordinate system do you think would be most appropriate here?
  8. Aug 27, 2012 #7
    Well my first guess was a spherical coordinate system. Since it's a double integral it must be polar coordinates?

    What would the limits then be?
  9. Aug 27, 2012 #8
    You cant have "spherical" and "polar" at the same time.

    The domain of integration is the entire sphere as follows from the problem.
  10. Aug 27, 2012 #9
    Sorry I think you misunderstood me because that wasn't worded clearly. I originally thought that spherical coordinates were going to be used (before I posted this problem, using triple integrals) but since you stated it can be done using a double integral my thought was to use polar coordinates. You are given the limits for θ. Typically in polar you would use r,θ? So then all I would need to do is find the limits for r?
  11. Aug 27, 2012 #10
    You have a sphere. You can't describe a sphere in polar coordinates. You have a surface integral, where the surface is a sphere. You could use spherical coordinates, with which the surface integral over a sphere becomes a double integral. That's all there is to it, you just need to recall how you convert a surface integral to a double integral in a particular coordinate system. This is a very simple calculus problem, and I think you have studied the appropriate part of calculus.
  12. Aug 27, 2012 #11
    You are right and trying to pull it all back together is the hardest part. Does that involve parameterization described in terms of θ and Phi?
  13. Aug 27, 2012 #12
    Since r is fixed (it's a sphere), then you have only the angles that are variable.
  14. Aug 27, 2012 #13
    So then Phi from 0 to 2∏ and θ from 0 to ∏? Is this what you are saying?
  15. Aug 27, 2012 #14
    That's what you are saying, and that seems correct. But you need to express the surface element properly. That's part of the conversion from surface to double integral.
  16. Aug 27, 2012 #15
    Since r is constant then can I just say dS = r2 sin θ dθ dPhi?
  17. Aug 27, 2012 #16
    This is correct.
  18. Aug 27, 2012 #17
    Alright that is good to know. So then should this be the integral to evaluate?

    ρs0 * r2∫∫ sin(2θ) * sin(θ) dθ dPhi


    After integrating that I get 2∏*ρs0*r2.
  19. Aug 27, 2012 #18
    No, the does not seem correct. How do you integrate with respect to θ?
  20. Aug 27, 2012 #19
    For sin(2θ) * sin (θ) with respect to θ I got (2/3) sin(θ)^3.

    Evaluated from 0 to ∏ I got 0.

    Edit: TI89 got that answer which I have found isn't always reliable.
  21. Aug 27, 2012 #20
    So if the integral over θ is zero, what is the end result?
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