A parallel plate capacitor is completely filled by an insulating material 3 mm thick and of relative permittivity 4 . The distance between the plates is increased to allow the insertion of a second sh

Question

A parallel plate capacitor is completely filled by an insulating material 3 mm thick and of relative permittivity 4 . The distance between the plates is increased to allow the insertion of a second sheet 5 mm thick and of relative permittivity ε r ​ also. If the capacitance of the capacitor so formed is one-half of the original capacitance, then the value of ε r ​ -

TestHub · Accepted Answer

Let the original capacitance be C 1 ​ = d 1 ​ ϵ r ​ ϵ 0 ​ A ​ . Given d 1 ​ = 3 mm and ϵ r ​ = 4 , so C 1 ​ = 3 × 1 0 − 3 4 ϵ 0 ​ A ​ The new capacitor has two dielectrics in series. The total distance is d total ​ = 3 + 5 = 8 mm. The new capacitance C 2 ​ = ϵ r 1 ​ d 1 ​ ​ + ϵ r 2 ​ d 2 ​ ​ ϵ 0 ​ A ​ = 4 3 × 1 0 − 3 ​ + ϵ r ​ 5 × 1 0 − 3 ​ ϵ 0 ​ A ​ Given C 2 ​ = 2 1 ​ C 1 ​ ( 4 3 ​ + ϵ r ​ 5 ​ ) 1 0 − 3 ϵ 0 ​ A ​ = 2 1 ​ 3 × 1 0 − 3 4 ϵ 0 ​ A ​ 4 3 ​ + ϵ r ​ 5 ​ 1 ​ = 3 2 ​ 4 3 ​ + ϵ r ​ 5 ​ = 2 3 ​ ϵ r ​ 5 ​ = 2 3 ​ − 4 3 ​ = 4 6 − 3 ​ = 4 3 ​ ϵ r ​ = 3 20 ​

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