Two parallel plate capacitors of capacitances 2 F and 4 F are connected in parallel and charged to a potential difference of 12 V . The battery is then disconnected. The space between the plates of the 2 F capacitor is completely filled with a material of unknown dielectric constant K . If the final potential difference across the combination is 4 V , the value of K is _____.

Step-by-step solution

Initial equivalent capacitance of the parallel combination is C_ eq = C₁ + C₂ = 2 F + 4 F = 6 F . Initial total charge on the system is Q = C_ eq V_i = 6 F 12 V = 72 C . When the battery is disconnected, the total charge remains conserved. After inserting the dielectric of constant K into the 2 F capacitor, its new capacitance becomes C₁' = 2K F . The new equivalent capacitance of the parallel combination is C_ eq ' = C₁' + C₂ = (2K + 4) F . The final common potential difference is given as V_f = 4 V . Using Q = C_