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 r1 and the remaining half with another dielectric of relative permittivity ε r2 , the dividing surface between the two dielectrics being a plane through the centre of the spheres. Show that the capacitance will be the same as though the dielectric in the whole annular space has a uniform average relative permittivity, i.e. ( ε r1 + ε r2 )/2.
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Sol. 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 capacitances of the two parts will be
and 
∴ The total capacitance, C = C 1 + C 2 = 
Thus, 
is the average relative permittivity.
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