#### 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.

#### Solution

Since the space between the two inner shells is filled with a dielectric, capacitance C_{AB } becomes `C_(AB) = (4pi∈_0abK)/((b-a))` and capacitance C_{BC}_{ }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)]]`

Is there an error in this question or solution?

Solution 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. Concept: Capacitors and Capacitance.