A capacitor having capacitance C is given an initial charge Q . At time t = 0 , it is connected across an uncharged capacitor of the same capacitance via a resistance R . Determine the charge accumulated on the second capacitor as a function of time.

Step-by-step solution

Let q₁ and q₂ be the charges on the first and second capacitors at time t . By conservation of charge, q₁ + q₂ = Q . Applying Kirchhoff's voltage law in the circuit: q₁ C - iR - q₂ C = 0 Since i = dq₂ dt and q₁ = Q - q₂ , we have: Q - q₂ C - q₂ C = R dq₂ dt Q - 2q₂ C = R dq₂ dt dq₂ Q - 2q₂ = dt RC Integrating both sides with limits q₂ = 0 at t = 0 to q₂ at time t : ₀^ q₂ dq₂ Q - 2q₂ = ₀^ t dt RC - 1 2 ( Q - 2q₂ Q ) = t RC ( Q - 2q₂ Q ) = - 2t RC Q - 2q₂ = Q e^ -2t/RC q₂ = Q 2 (1 - e^ -2t/RC )