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Applying Gauss's Law Several Times

  1. Feb 12, 2013 #1
    1. The problem statement, all variables and given/known data
    I am working on a problem that is similar to this one:

    https://www.physicsforums.com/showthread.php?t=598914



    2. Relevant equations



    3. The attempt at a solution
    When r < a, why doesn't the electric fields generated by the concentric conducting, hollow sphere affect the Gaussian surface at that point??

    Why is the electric field at a point in the concentric conducting, hollow sphere zero? After all, it doesn't specify that it is in electrostatic equilibrium.

    Also, what is the difference between the entire system being in electrostatic equilibrium and just the concentric conducting, hollow sphere being in electrostatic equilibrium? Can an insulator be in electrostatic equilibrium?
     
  2. jcsd
  3. Feb 12, 2013 #2

    rude man

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    You will probably never have to worry about electrostatic disequilibrium. You were given that the insulating sphere had a uniform charge density. Therefore it's in equilibrium.
     
  4. Feb 13, 2013 #3
    So, you are saying that if I was to generate a Gaussian surface inside of the conductor (say, a sphere), then the flux through this surface is zero, because the electric field through this surface is zero? How can this be? I know that the charge isn't distributed inside of the conductor, the charge resides on the surface, but the Gaussian surface would still enclose the insulator, which has a charge.

    EDIT: Something just came to my mind: does the insulator induce the electro-static equilibrium? And if the insulator had a positive charge, it would cause the electrons in the conductor to move to the surface closest to the insulator? Leaving the other surface "less" negative?
     
  5. Feb 13, 2013 #4

    Doc Al

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    Yes.

    Charge can rearrange itself on the inner surface of the conducting shell to cancel any charge placed within the hollow of the sphere.

    Yes. Any charge placed inside the conducting sphere will induce a charge on the inner surface that cancels its field within the conductor (and for all points beyond).
     
  6. Feb 13, 2013 #5
    So, using the diagram given in the link, if I was to place a Gaussian sphere directly between the insulator and conductor, the electric field lines emanating from the insulator will "hit" the sphere from the inside, and the electric field lines emanating from the conductor "strike" the very surface of the Gaussian sphere; and so, the electric fields combine to cancel out?

    And is this the reason why we don't have to consider the electric field of the conductor when generating a Gaussian surface that meets the condition of r < a, because the electric fields lines never even make it that far?

    EDIT: Also, is there a way of quantifying the amount of induction that the insulator affects? Would it just be equal and opposite to the charge of the insulator? And the remaining charge would reside on the outside?
     
    Last edited: Feb 13, 2013
  7. Feb 13, 2013 #6
    For example, say the insulator has a charge of Q1 = 3.60 C, and the conductor had a total charge of -2.00 C. Would the inner surface of the conductor would have a charge of Q2 = -Q1 ---> Q2 = -3.60 C; and would this then imply that Q3, the outer surface of the conductor, would have a charge of Q2 + Q3 = -2.00 C ----> Q3 = 1.6 C?
     
  8. Feb 13, 2013 #7

    Doc Al

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    That is correct.
     
  9. Feb 13, 2013 #8
    Why is it correct, if you don't mind me asking?
     
  10. Feb 13, 2013 #9
    Is there some law or principle of induction?

    Is it possible for someone to answer these questions I have from an earlier post?
     
    Last edited: Feb 13, 2013
  11. Feb 13, 2013 #10

    SammyS

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    The reason that we know the charge on the inner surface has equal magnitude and opposite sign of the charge on the inner charged sphere, is that if we use a Gaussian surface that lies within the conducting material, such as a sphere with radius between b & c, the E field is zero everywhere on the Gaussian surface so the net charge within the Gaussian surface is zero.
     
  12. Feb 13, 2013 #11
    So, would electric field at a radius of a < r < b be zero?

    EDIT: And would it be zero because the number of electric field lines radiating out of the Gaussian surface equals the number of electric field lines coming in?
     
    Last edited: Feb 13, 2013
  13. Feb 13, 2013 #12

    rude man

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    No. Put a Gaussian spherical surface anywhere on a < r < b and what is the charge inside that surface? And so what does Gauss say about the E flux thru that surface?
     
  14. Feb 13, 2013 #13

    Doc Al

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    No. The field there would equal that from the charged insulator. The induced charge on the inner surface of the conducing shell will have no field for r < b.
     
  15. Feb 13, 2013 #14
    So, the flux through a Gaussian only depends on the charge enclosed, further implying that it only depends on the electric field emanating from that enclosed charge?
     
  16. Feb 13, 2013 #15

    SammyS

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    Whoa !

    E field lines originate from Positive and terminate on Negative charge. The negative charge on the inner surface of the conductor does not interfere with the electric field at a < r < b .

    So the field at a radius of a < r < b is not zero. Gauss's Law says the flux though a sphere of such radius is not zero.
     
  17. Feb 13, 2013 #16
    Why won't it have a field there?
     
  18. Feb 13, 2013 #17

    Doc Al

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    The total flux calculated over the entire closed Gaussian surface depends only on the net charge enclosed within that surface. (That's what Gauss's law says.) Note that the field at any given point on the surface may well depend on other charges.
     
  19. Feb 13, 2013 #18

    Doc Al

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    The field produced by a uniform shell of charge is zero at all points within the shell. A result derivable from calculus.
     
  20. Feb 13, 2013 #19

    rude man

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    Just put Gaussian spherical surfaces around a < r < b, at b < r < c, and at r > c.
    Inside a < r < b you have charge Q due to the inner sphere. Inside b < r < c you have zero total charge inside that surface since the E field all over that surface has to be zero. Look at my equation below. Inside r > c you have nothing but Q since the conductor is uncharged.

    In all cases, total free charge inside a surface = 4πr^2 ε0E.
     
  21. Feb 13, 2013 #20
    I'm sorry, I am still having difficulty with this. If I wanted to calculate the electric field at a point in between the insulated sphere and the conducting sphere--that is, I create a Gaussian sphere around the insulated sphere (a < r < b)--I would have the consider the electric field contribution of both the conducting sphere AND the insulated sphere, or would I just figure out the electric field at that point due to the enclosed charge in the Gaussian sphere, which would be the insulated sphere?
     
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