A parallel plate capacitor is filled equally (half) with two dielectrics of dielectric constant ₁ and ₂ , as shown in figures. The distance between the plates is d and area of each plate is A. If capacitance in first configuration and second configuration are C₁ and C₂ respectively, then C₁ C₂ is :

Step-by-step solution

Area of plate is A . then aligned & C = ₂ ₀ ~A ~d / 2 = 2 ₂ ₀ ~A ~d & C ^ = ₁ ₀ ~A ~d / 2 = 2 ₁ ₀ ~A ~d aligned Let C ₀= ₀ ~A ~d aligned & C =2 ₂ C ₀ & C ^ =2 ₁ C ₀ aligned C & C ^ are in series C ₁= CC ^ C + C ^ = 4 ₂ ₁ C ₀^2 2 C ₀ ( ₂+ ₁ ) = 2 ₂ ₁ C ₀ ( ₂+ ₁ ) Here C = ₁ ₀ ~A 2 ~d = ₁ C ₀ 2 C ^ = ₂ C ₀ 2 C & C ^ are inparallel C ₂= C ^ + C = ( ₁+ ₂ ) C ₀ 2 Thus C ₁ C ₂ = 2 ₂ ₁ C ₀ ( ₂+ ₁ ) 2 ( ₁+ ₂ ) C ₀ = 4 ₂ ₁ ( ₂+ ₁ )^2