Three parallel plate capacitors C₁, C₂ and C₃ each of capacitance 6 F are connected such that C₁ and C₂ are in series, and this combination is in parallel with C₃ . The effective capacitance between the terminals of the circuit is found to be 10 F when the space between the parallel plates of C₂ is filled with a dielectric medium of unknown dielectric constant k . The value of k is :

Step-by-step solution

Let the initial capacitance of each capacitor be C = 6 F . When a dielectric of constant k is inserted into C₂ , its new capacitance becomes C₂' = kC = 6k F . The capacitors C₁ and C₂' are in series. Their equivalent capacitance C_s is given by: C_s = C₁ C₂' C₁ + C₂' = 6 6k 6 + 6k = 36k 6(1+k) = 6k 1+k F This series combination is in parallel with C₃ . The total equivalent capacitance C_ eq is: C_ eq = C_s + C₃ = 6k 1+k + 6 We are given that C_ eq = 10 F . So, 6k 1+k + 6 = 10 6k 1+k = 4 6k = 4(1+k) 6k = 4 + 4k 2k =