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Question

Suppose the space between the two inner shells of the previous problem is filled with a dielectric of dielectric constant K. Find the capacitance of the system between A and B.

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Solution

Since the space between the two inner shells is filled with a dielectric, capacitance CAB becomes CAB=4π0abK(b-a) and capacitance CBC becomes CBC=4π0bc(c-b).
Now, as the capacitors are in series, the equivalent capacitance is given by

1C=1CAB+1CBCC=CABCBCCAB+CBCC=(4π0)2ab2Kc(b-a)(c-b)4π0abk(b-a)+4π0bc(c-b)C=4π0Kab2c(b-a)(c-b)abk(c-b)+bc(b-a)(b-a)(c-b)C=4π0Kab2c[abk(c-b)+bc(b-a)]C=4π0Kabc[ak(c-b)+c(b-a)]

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