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Individual Spherical Capacitors

  1. May 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Two conducting spheres of radii R1 and R2 are kept widely separated from each other. What are their individual capacitance? If the spheres are connected by a metal wire, what will be the capacitance of the combination? Think in terms of series-parallel connections.


    2. Relevant equations



    3. The attempt at a solution
    Individual capacitance are simply [itex]4\pi\epsilon_oR_1[/itex] and [itex]4\pi\epsilon_oR_2[/itex]. I have no idea on what happens when they are connected. I can't understand the type of combination. Will it be series or parallel?

    Any help is appreciated.
     
  2. jcsd
  3. May 25, 2012 #2
    A single spherical shell is a capacitor only if you imagine that the other shell (ground) is at infinitity. Now, when you connect the two spheres together, you join a set of plates of each of the capacitors together, and hold them at a fixed potential? What happens to the "other plates"? Is this connection in series or parallel?
     
  4. May 25, 2012 #3
    Sorry but i have no idea, can you provide me some more explanation?
     
  5. May 25, 2012 #4
    What does it mean for capacitors to be connected in series? How about in parallel?
     
  6. May 26, 2012 #5

    ehild

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    Remember the definition of capacitance: C=Q/V. V is the potential with respect to infinity in this case. The shape does not count, any kind of a metal piece has capacitance.
    Charge the spheres to equal potential. Q1=C1V, Q2=C2V. Connect them with a wire. What happens? What is the potential of this structure? What is the net charge on it? So what is the capacitance?

    ehild
     
  7. May 26, 2012 #6
    If the capacitors are in series, the charge is same and if they are in parallel, the potential is same.

    Nothing should happen as they are at same potential.
     
  8. May 26, 2012 #7

    ehild

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    And what is the charge of the whole system?

    ehild
     
  9. May 26, 2012 #8
    The charge is (C1+C2)V.
    And capacitance is (C1+C2).
     
  10. May 26, 2012 #9

    ehild

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    Splendid! So it is like parallel capacitors.

    ehild
     
  11. May 26, 2012 #10
    Yes, but in the question its not mentioned that they are at same potential.
     
  12. May 26, 2012 #11

    ehild

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    Well, if they are connected with a wire, they become the same potential anyway.

    The capacitance does not depend on the charge or on the potential. But you can determine it by measuring charge and potential, and taking the ratio C=Q/V.

    It is the same as with the mass. m=F/a, mass equals to force over acceleration. The mass of the object is its attribute, but you can measure it by applying a known force and detecting the acceleration.

    If you know the charge of an object and its potential with respect to infinity you can attribute a capacitance to it as C=Q/V.

    If your spheres are at different potential initially, after connecting them, there will be some charge transfer from one sphere to the other, till both of them have he same potential V=kQ1/R1=kQ2/R2. The first sphere is charged to Q1=VR1/k, the second one is charged to Q2=VR2/k. The resultant capacitance is (Q1+Q2)/V.

    ehild
     
  13. May 26, 2012 #12
    Thank you very much for your explanation! :smile:

    This is just a question out of curiosity.
    What if i connected the whole system with the battery? Will they be in series or parallel?
     
  14. May 26, 2012 #13
    What do you mean? Connect one of the spheres with one lead, and the other sphere with the other lead of the battery?
     
  15. May 26, 2012 #14
    Oh yes, that's what i mean. :)
     
  16. May 26, 2012 #15
    I told you. Each spherical capacitor has the other of its plates at infinity (and all these plates may be thought of as connected to the ground). If you draw the relevant diagram, you will immediately conclude if they are connected in series or parallel.
     
  17. May 26, 2012 #16

    ehild

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    See attachment. The connected spheres as capacitors are analogous to two "normal" grounded capacitor with the free plates connected. They are in parallel between B and the ground. What happens if you connect a battery in the way you said: one terminal to one plate of the first capacitor and the other terminal to the connected plate of the other capacitor?????

    See the connected spheres. They are of metal. So you make a metallic contact between the terminals of the battery. It is called "short", fatal for a battery. It gets very hot and starts to smoke and gets black and it is not a battery any more... RIP...

    In case of a sphere as capacitor the other "plate" is at infinity. You need to connect the terminals of the battery to the wire connecting the spheres and to the other common point of the capacitors, which is at infinity.

    ehild
     

    Attached Files:

  18. May 27, 2012 #17
    Thank you ehild for the explanation, it was an interesting read. :smile:
     
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