A sphercial capacitor is made of two conducting spherical shells of radii a and b. The space between the shells is filled with a dielectric of dielectric constant K up to a radius c as shown in figure . Calculate the capacitance.
We have two capacitors: one made by the shells a and c and the other made by the shells b and c.
The capacitance of the capacitor `C_(ac)` is given by
`C_(ac) = (4pi∈_0acK)/((c-a))`
The capacitance of the capacitor `C_(cb)` is given by
`C_(cb) = (4pi∈_0bcK)/(K(b-c)`
The two capacitors are in series; thus, the equivalent capacitance is given by
`1/C = 1/C_(ac) + 1/C_(cb)`
⇒ `1/C = ((c-a))/(4pi∈_0acK) + ((b-c))/(4pi∈_0cb)`
⇒ `1/C = (b(c-a)+Ka(b-c))/(K4pi∈_0abc)`
⇒ `C = (K4pi∈_0abc)/(b(c-a)+Ka(b-c))`