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Question
Suppose the space between the two inner shells 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 `C_(AB) = (4pi∈_0abK)/((b-a))` and capacitance CBC becomes `C_(BC) = (4pi∈_0bc)/((c-b))` . Now, as the capacitors are in series, the equivalent capacitance is given by
`1/C = 1/C_(AB) + 1/C_(BC)`
`⇒ C = (C_(AB)C_(BC))/(C_(AB)+C_(BC))`
`⇒ C = (((4pi∈_0)^2 ab^2 Kc)/((b-a)(c-b)))/((4pi∈_0abk)/((b-a))+(4pi∈_0bc)/((c-d)))`
`⇒ C = ((4pi∈_0Kab^2c)/((b-c)(c-b)))/(((abk(c-b)+bc(b-a))/((b-a)(c-b))))`
`⇒ C = (4pi∈_0Kab^2c)/[[abk(c-b)+bc(b-a)]]`
`⇒ C = (4pi∈_0Kabc)/[[ak(c-b)+c(b-a)]]`
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