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Homework answers / question archive / Consider the capacitor formed by a conducting sphere with radius R1 surrounded by a concentric conducting spherical shell with inside radius R2 and outside radius R3
Consider the capacitor formed by a conducting sphere with radius R1 surrounded by a concentric conducting spherical shell with inside radius R2 and outside radius R3.
(1) Calculate the potential of the outer shell when charge Q is put on it and the inner shell is grounded.
(2) Calculate the potential of the inner shell when charge Q is put on it and the outer shell is grounded.
(3) Calculate the charge on the outer sphere when it is grounded and the inner sphere is at potential V.
(4) Convert your calculations above into the coefficients of capacitance---a.k.a. the capacitance tensor.
(5) Convert your calculations above into the potential coefficients---a.k.a. the potential tensor.
There can be some small variety in the definitions of potential and capacitance tensors.
As you have not written the definitions you use, I had to choose some, and chose the potential tensor P to be defined by equation V_i = sum_j P_ij Q_j
and the capacitance tensor as C = inverse of P.
If your definitions are different you can either disregard all that follows and request refund, or tell me the definitions you want to use so that I translate the results into the definitions you ask.
Another point of ambiguity is in question (3) where "outer sphere" may mean either the outer shell or its outer surface. I have explained what would be the answer in both cases and you can pick your choice.
In fact, if you have to write down this assignment as a homework, I would suggest writing down both interpretation of the faulty language there.
In the calculation of the inverse matrix, it is natural to assume that you know how to do it given that fact that you received this assignment.
If however you need help in how to calculate inverse matrices, you can post a separate posting asking some OTA to teach you.
Just in case, I give you here a general formula for a 2x2 matrix
inverse( a, b ; c, d ) = [1/(ad-bc)] * ( d, -b ; -c, a)
