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Electrically charged sphere

  1. Oct 25, 2016 #1
    1. The problem statement, all variables and given/known data
    A sphere of radius ##a## is non-uniformly charged on its surface with a charge whose surface density is ##ρ_s(φ)=ρ_{so}(cosφ)^2## where ##φ## is the angle measures from the z axis, (0≤φ≤π) and ##ρ_{s0}## is a constant. Determine the expression for the total charge distributed on the sphere.
    2. Relevant equations
    ##dQ=ρ_sdS##
    3. The attempt at a solution
    I know im supposed to find the small surface element on which to integrate but the surface charge density is given by the angle and how am i supposed to make the surface element be in angle form. I tried thinking like this: In a circle the element ##dL## that is the small part of the circumference is ##rdφ## but dont know how to use that on the sphere..
    The solution should be ##Q=\frac{4π}{3}ρ_{s0}a^2##
    The problem i have now is how to start. I have to find the surface element and i dont know how, can you help?
     
  2. jcsd
  3. Oct 25, 2016 #2

    TSny

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  4. Oct 26, 2016 #3
    Hi, its great to be here :D
    I have solved the problem, i figured that the part of the sphere that is under the fixed angle can be integrated,
    ##dA=2rπdl## where the circumference at some radius ##r## that is equal to ##r=asinφ## multiplied by the ##dl## element equaling to ##adφ## gives out the area and the integral becomes ##Q=∫2πa^2ρ_{so}(cosφ)^2sinφ## integrated on the interval ##[0,π]## but just out of curiosity how would i use the area you provided?
    The surface element is ##dA=a^2sinφdφdθ## and the integral becomes ##Q=∫ρ_{so}(cosφ)^2a^2sinφdφdθ##? There are two differentials now, how to use this?
     
  5. Oct 26, 2016 #4

    haruspex

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    The integral with respect to θ is easy, so do that first.
     
  6. Oct 26, 2016 #5
    Well the limits are ##[0, π]## so it should be ##π## right? But then Im mising a factor of ##2## so it should be ##2π## somehow..
     
  7. Oct 26, 2016 #6

    haruspex

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    In polar, to cover the sphere, one angle goes 0 to π and the other from 0 to 2π.
     
  8. Oct 26, 2016 #7
    So i integrate one angle from ##[0, π]## and the other ##[0, 2π]##? How would i put the limits for the general expression?
     
  9. Oct 26, 2016 #8

    haruspex

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    You are asking about the notation? ##\int^{\pi}_{\phi=0}\int^{2\pi}_{\theta=0}##.
     
  10. Oct 26, 2016 #9
    oh yeah it would be a double integral, thanks :D!
     
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