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Calculus to calculate charge

  1. Sep 19, 2013 #1
    1. The problem statement, all variables and given/known data

    A nonuniform, but spherically symmetric, distribution of charge has a charge density ρ(r) given as follows:
    ρ(r) = ρ0(1-r/R) for r≤R
    ρ(r) = 0 for r≥R

    where ρ0 = 3Q/∏R^3 is a positive constant.

    (a.) Show that the total charge contained in the charge distribution is Q.
    (b.) Show that the electric field in the region r≥R is identical to that produced by a point charge Q at r = 0.
    (c.) Obtain an expression for the electric field in the region r≤R.
    (d) Graph the electric field magnitude E as a function of r.
    (e.) Find the value of r at which the electric field is maximum, and find the value of that maximum field.

    2. Relevant equations

    Gauss' Law : Flux = ∫E dot dA = Qencl/ε 0
    Electric Field of a point charge: E = k*(q/r^2)

    3. The attempt at a solution

    I am really just having trouble setting up the integral. You would be integrating the electric field over a sphere, so you would need to choose dA (is this a slice of the area of the sphere?) and a formula to find electric field for a given r. In order to find the latter you would need to use Gauss' Law in which case a give r would give the electric field

    ρ(r) = q /A1 (is this the area or the volume of the sphere in question?)
    q = ρ(r)*A1

    E*A2 (what area is this?)= q/ε0
    E = ρ(r)*A1 / A2*ε0

    There are a few things I am confused about in here so please feel free to ask me to clarify anything. I need to know if I am on the right track here.

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

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 19, 2013 #2


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    For part (a), you want to use ##dq = \rho\,dv## and integrate over the entire sphere to find the total charge. You don't use Gauss's law here.
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