Now solve the differential equation V (t) = -CRdV (t) dt for the initial conditions given in the problem introduction to find the voltage as a function of time for any time t. Express your answer in terms of q0, C, R, and t.

V (t) = (q0/C) * e^ (-t/RC)

Explanation:

If there were a battery in the circuit with EMF E, the equation for V (t) would be V (t) = E - (RC) (dV (t) / dt). This differential equation is no longer homogeneous in V (t) (homogeneous means that if you multiply any solution by a constant it is still a solution). However, it can be solved simply by the substitution Vb (t) = V (t) - E. The effect of this substitution is to eliminate the E term and yield an equation for Vb (t) that is identical to the equation you solved for V (t). If a battery is added, the initial condition is usually that the capacitor has zero charge at time t=0. The solution under these conditions will look like V (t) = E (1-e-t / (RC)). This solution implies that the voltage across the capacitor is zero at time t=0 (since the capacitor was uncharged then) and rises asymptotically to E (with the result that current essentially stops flowing through the circuit).