The space between the plates of a parallel plate capacitor is filled with a 'dielectric' whose 'dielectric constant' varies with distance as per the relation: K ( x )= K _ o + x ( = a constant ) The capacitance C , of the capacitor, would be related to its vacuum capacitance C _ o for the relation :

Step-by-step solution

The value of dielectric constant is given as, aligned & K = K ₀+ x & And, V = ₀^ d Edr & V = ₀^ d K dx aligned aligned &= ₀^ d 1 ( ~K ₀+ x . dx &= [ ( K ₀+ d - K ₀ ] . &= (1+ d K ₀ ) aligned Now it is given that capacitance of vacuum = C ₀ . Thus, C = Q V aligned &= . s v ( Let surface area of plates = s ) &= . s (1+ d K ₀ ) aligned = s d d 1 (1+ d K ₀ ) ( . in vacuum . ₀=1 ) c = d (1+ d K ₀ ) C ₀ ( . here, . C ₀= s d )