A spherical capacitor is formed by two conducting spherical shells...

The given spherical capacitor can be considered as a combination of two spherical capacitors connected in series:

1. A capacitor $C_1$ with a dielectric of constant $K$ between the radii $a and$c$.<br>$C_1 = \dfrac{4\pi\varepsilon_0 K a c}{c - a}$<br><br>2. A capacitor$C_2$with air between the radii$c$and$b$.<br>$C_2 = \dfrac{4\pi\varepsilon_0 b c}{b - c}$<br><br>Since the electric field lines pass radially through both regions, the two capacitors are in series. The equivalent capacitance$C$is given by:<br><br>$\dfrac{1}{C} = \dfrac{1}{C_1} + \dfrac{1}{C_2}$<br><br>$\dfrac{1}{C} = \dfrac{c - a}{4\pi\varepsilon_0 K a c} + \dfrac{b - c}{4\pi\varepsilon_0 b c}$<br><br>$\dfrac{1}{C} = \dfrac{1}{4\pi\varepsilon_0 c} \left( \dfrac{c - a}{K a} + \dfrac{b - c}{b} \right)<br><br>\dfrac{1}{C} = \dfrac{1}{4\pi\varepsilon_0 c} \left( \dfrac{b(c - a) + K a(b - c)}{K a b} \right)

$C = \dfrac{4\pi\varepsilon_0 K a b c}{K a(b - c) + b(c - a)}$