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Gauss's law problem

  1. Mar 23, 2007 #1
    1. The problem statement, all variables and given/known data
    A uniformly charged ball of radius a and a total charge -Q is at the center of a hollow metal shell with inner radius b and outer radius c. The hollow sphere has a net charge +2Q. Find the magnitude of the electric field in the regions: [tex]r_1 < a[/tex],[tex] a < r_2 < b[/tex],[tex] b < r_3 < c[/tex], and [tex]r_4 > c.[/tex]

    2. Relevant equations

    [tex]V = \frac{4}{3} \pi R^3[/tex]
    [tex]S = 4 \pi R^2[/tex]
    [tex]\oint E(x)dA = \frac{q_{in}}{\epsilon_o}[/tex]

    3. The attempt at a solution

    For E(r1 < a):
    [tex]\rho = \frac{Q_{tot}}{\epsilon_o}[/tex]
    [tex]Q_{in,tot} = \rho*\frac{4}{3} \pi r_1^3[/tex]
    [tex]\oint_0^rE(x)dA = \frac{q_{in}}{\epsilon_o}[/tex]

    [tex]E(r_1) = \frac{\rho\frac{4}{3} \pi r_1^3}{\epsilon*4 \pi r_1^2}[/tex]

    [tex]E(r_1) = \frac{\rho*r_1}{3\epsilon_o}[/tex]

    This is actually where I am stuck, I got everything else. Am I supposed to get rid of that volume charge density, [tex]\rho[/tex]?
    Last edited: Mar 23, 2007
  2. jcsd
  3. Mar 23, 2007 #2


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    Gold Member

    Yes; in textbook problems, they expect you to express the answer in terms of the variables given in the statement of the problem.

    The density is uniform and you know the total charge in the little ball. So just divide that by its volume to find its density.
  4. Mar 23, 2007 #3
    The following may help.



    http://islam.moved.in/tmp/c.jpg [Broken]
    Last edited by a moderator: May 2, 2017
  5. Mar 24, 2007 #4
    There is actually a second and third part to this question:

    b) Find potentials at points in the regions: [tex]r_1 < a[/tex], [tex]a < r_2 < b[/tex], [tex]b < r_3 < c[/tex], and [tex] r_4 >c[/tex]

    For r1 < a,
    I use the formula

    [tex]V_o - V_{r_1} = \int_0^r E(r_1)dr[/tex]

    [tex]E(r_1) = \frac{\rho*r_1}{3\epsilon_o}[/tex]

    [tex]V_o - V_{r_1} = \frac{\rho}{3\epsilon_o} \int_0^r r dr[/tex]

    [tex]V_o - V_{r_1} = \frac{\rho*r_1^2}{6\epsilon_o}[/tex]

    Solving the same way, I got::

    [tex]V_o - V_{r_2} = \frac{Q}{4 \pi \epsilon_o r_2}[/tex]

    [tex]V_o - V_{r_3} = 0[/tex]

    [tex]V_o - V_{r_4} = \frac{-Q}{4 \pi \epsilon_o r_4}[/tex]

    Did anyone get the same answer as I did?

    Also, I am supposed to sketch the graphs of how Ex depends on a distance(r) from the center of the sphere, and how V depends on a distance(r) from the center of the sphere.

    I drew my graphs, but I don't know how to post it on. I don't have a scanner handy, so if anyone can help, please let me know what your graphs looks like.
    Thank you very much, you are all so helpful.
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