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Gauss' law

  1. Feb 25, 2014 #1
    1. The problem statement, all variables and given/known data

    A positive charge q is placed at the center of a hollow electrically neutral conducting sphere (inner radius R1 9cm, outer radius R2 10 cm.

    Using Gauss' law determine the electric field of every point in space, as a function of r (the distance from the center of the sphere). Only the algebraic expression is required.

    2. Relevant equations


    3. The attempt at a solution

    I'm unsure of how to go about this problem, all the examples I have seen discuss thin spheres.

    A positive charge at the center of the sphere would induce a positive charge on the surface of the sphere, whilst a negative one would be induced on the inner surface.

    I think it is safe to assume the charge at the center is a point and therefore the charge will be evenly distributed creating a symmetrical electric field.

    [itex]\Phi _{0} = ( \Sigma \cos \phi) \Delta A[/itex]

    I know I need to 'construct' a gaussian surface with radius r (r>R2) concentric with the shell, but I don't know how to use the information about the two radii I was given - Are they important here?

    Since the electric field is everywhere perpendicular to the gaussian surface, [itex]\phi = 0^{o}[/itex] and [itex] \cos \phi = 1[/itex].

    The electric charge has the same value all over the surface, so we can say that;

    [itex]\Phi _{0} = E( \Sigma \Delta A) = E(4 \pi r^{2}) [/itex]

    Setting [itex]\Phi _{0} = \frac{q}{\epsilon _{0}} [/itex]

    We can say that [itex] E = \frac{q}{4 \pi \epsilon _{0} r^{2}}[/itex]

    For r > R2 since Gauss' law also shows us that there is no net charge inside the sphere.

    The above makes sense to me, but at no point did I use the radii given to me in the question...

    What am I doing wrong?

    Thanks!

    BOAS
     
  2. jcsd
  3. Feb 25, 2014 #2

    Doc Al

    User Avatar

    Staff: Mentor

    They are important because they mark off three distinct regions: r < R1; R1 < r < R2; r > R2.

    You only dealt with the last region. What about the others?
     
  4. Feb 25, 2014 #3
    Ok, that was surprisingly obvious.

    I'll have three distinct expressions, does that satisfy the question of finding the electric field of every point in space?

    I suppose that it does, but on first reading of the question I was expecting a single expression.

    Thanks for your help, i'm confident I can do this now.
     
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