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Gauss Law and Flux for variable charge density in a sphere

  1. Jan 14, 2012 #1
    Gauss Law and Flux for variable charge density in a sphere

    1. The problem statement, all variables and given/known data

    The charge density within a sphere varies as a constant, a, times its radius, r. Find an expression for the direction and magnitude of the electric flux, D, within the sphere


    2. Relevant equations

    Gauss' Law
    sphere volume = [itex]\frac{4}{3}[/itex] [itex]\pi[/itex] r[itex]^{3}[/itex]


    3. The attempt at a solution

    I have some difficulty understanding the question, but this is what I think I should do:

    Charge Density = ar

    Charge in volume moving from r to r+δr , δQ, is: ar x volume
    =(4/3) [itex]\pi[/itex] ar [(r+δr)^3 - r^3)]
    =(4/3) [itex]\pi[/itex] ar (3 r^2 δr) (ignoring double differentials as negligible)
    =4 [itex]\pi[/itex] a r^3 δr

    integrating from 0 to r(an arbitrary distance from centre) gives charge within that volume, Q:
    Q = [itex]\pi[/itex] a r^4

    Gauss Law says flux through an area is equal to charge enclosed so;
    Magnitude of the flux ,D, = Q = [itex]\pi[/itex] a r^4
    Direction is radially outward (assuming charge is positive). (not sure on how to argue this...)


    Notes: I have no idea if this is correct, or if I'm completely of the plot!! any help greatly appreciated.
    I thought D was the flux density, though the question calls it flux, so do I need to divide by area?

    Thanks Everyone!...
     
  2. jcsd
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