In a spherical capacitor made of two concentric spheres, one-half the annular space between the spheres is filled with one dielectric of relative permittivity $ε_{r 1}$ and the remaining half with another dielectric of relative permittivity $ε_{r 2},$ the dividing surface between the two dielectrics being a plane through the center of the spheres. The capacitance will be the same as though the dielectric in the whole annular space has a uniform permittivity $\in_{r}$ .Then $\in_{r} i s e q u a l t o$

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$\frac{ε_{r 2}}{2}$

$\frac{ε_{r 1}}{2}$

$\frac{ε_{r 1} + ε_{r 2}}{2}$

answer is C.

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Detailed Solution

In this case, the dielectric interface is parallel to the vectors $D$ and $E$ . Hence, the arrangement can be considered as two capacitors in parallel. Since each capacitor carries half the charge that would be carried by a complete spherical capacitor, the capacitance of the two parts will be

$\frac{2 π ε_{0} ε_{r 1}}{\frac{1}{a} - \frac{1}{b}} a n d \frac{2 π ε_{0} ε_{r 2}}{\frac{1}{a} - \frac{1}{b}}$