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Use Gauss' Law to find the magnitude of E in a sphere

  1. Sep 25, 2007 #1
    Problem
    Problem 27.56

    A sphere of radius R_0 has total charge Q. The volume charge density (C\m^3) within the sphere is [tex]\rho[/tex](r) = C/(r^2), where C is a constant to be determined.

    Part A
    The charge within a small volume dV is dq = [tex]\rho[/tex]dV. The integral of [tex]\rho[/tex]dV over the entire volume of the sphere is the total charge Q. Use this fact to determine the constant C in terms of Q and R_0.
    Express your answer in terms of pi, Q, and R_0.

    Part B
    Use Gauss's law to find an expression for the magnitude of the electric field E inside the sphere, r [tex]\leq[/tex] R_0.
    Express your answer in terms of Q, R_0, r and appropriate constants.

    Relevant equations and whatnot :-)
    Gauss' Law for a sphere: 4pi(r^2)E = Q_in/[tex]\epsilon[/tex]_0
    Q = [tex]\rho[/tex] * V

    Solution attempt(s) (and the miserable failure(s))
    I've already found Part A, through [tex]\rho[/tex] = Q/v; the answer is C = Q/(4*pi*R_0).

    For Part B, I've tried the following:

    Use Gauss' Law to set up:

    E = Q_in/(A*[tex]\epsilon[/tex]_0)

    Define Q_in as:

    Q_in / Q = (4/3)*pi*r^3 / (4/3)*pi*((R_0)^3) ==> Q_in = Q*r^3/((R_0)^3)

    Define A as:

    A = 4*pi*r^2

    and plug everything in to get:

    E = Qr / ( (R_0)^3) * 4 * pi * [tex]\epsilon[/tex]_0 )

    Which is apparently wrong :-(

    So, I figured that they probably were intending to make the students use the solution to Part A, in which we found the C constant of [tex]\rho[/tex](r) = C/(r^2).

    So, using the same setup and the same A and the same [tex]\epsilon[/tex]_0 (like that's going to change :-), I defined Q_in as:

    Q_in = [tex]\rho[/tex](r) * V(r) = (C/r^2) * ((4/3)*pi*r^3) = (Qr/3*(R_0))

    Again plugging in all the variables, I got stuck with:

    E = Q / ( 3 * 4 * pi * r * (R_0) * [tex]\epsilon[/tex]_0 )

    And that too is also wrong, apparently (unless there's a huge conspiracy against me involving Dr. Randall Knight and quite possibly the US government whom are intent on sabotaging my efforts). And now, I'm lost and have no idea what to do. There's a question in the book that asks for the electric field inside a uniformly charged sphere, and the answer is E = Qr / ( (R_0)^3) * 4 * pi * [tex]\epsilon[/tex]_0 ), which is what I got at my first try. Am I missing something here? Helpy please! :-D
     
  2. jcsd
  3. Sep 25, 2007 #2

    Doc Al

    User Avatar

    Staff: Mentor

    The problem is that this is not a uniformly charged sphere. To find the charge enclosed within a Gaussian sphere of radius r, set up the integral of [itex]\rho dV[/itex]. (The integral is easy to solve.)
     
  4. Mar 13, 2009 #3
    Whats the answer for the second part then?
     
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