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Finding flux through ellipsoid in Cylindrical Coordinates

  1. Jan 22, 2015 #1
    1. The problem statement, all variables and given/known data
    Using Cylindrical coordinates, find the total flux through the surface of the ellipsoid defined by x2 + y2 + ¼z2 = 1 due to an electric field E = xx + yy + zz (bold denoting vectors | x,y,z being the unit vectors)

    Calculate ∇⋅E and then confirm the Gauss's Law
    2. Relevant equations
    Cylindrical Coordinates being used: (s,φ,z)
    Conversion to Cylindrical Coordinates:
    x = scosφ
    y = ssinφ
    z= z

    Surface Element of a Cylinder:
    da = sdφdz

    3. The attempt at a solution
    I converted the ellipsoids equation into cylindrical, so it looks like:
    s2cos2φ + s2sin2φ + ¼z2 = 1

    solving for s looks like s = √(1-¼z2)

    I solved for both the volume and surface area of the ellipse through integration. Surface Area was as Follows:
    ∫ (from -2 to 2) ∫ (from 0 to 2π) sdφdz = ∫∫ √(1-¼z2) dφdz = 2π2

    How to use this to find Flux, I am unsure of.
    I know to convert E to cylindrical so E = scosφx + ssinφy + zz, but don't know what to do about the unit vectors.

    Is the flux just ∫E⋅da ? and if so, how do I take the dot product and what do I do about the unit vectors?
     
  2. jcsd
  3. Jan 22, 2015 #2
    Are you allowed to use the divergence theorem?

    Chet
     
  4. Jan 23, 2015 #3
    I think that's expected.
    The two are related right?
     
  5. Jan 23, 2015 #4
    Yes. Please state the divergence theorem in terms of E. In your problem, the divergence of E is a constant. What is that constant?

    Chet
     
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