A 200volt electromotive force is applied to an RCseries circ

A 200-volt electromotive force is applied to an RC-series circuit in which the resistance is 1000 ohm and the capacitance is 5 times 10^-6 farad. Write an initial-value problem for the charge q(t)on the capacitor at time t if initially the current in the circuit is 0.4 amperes.

Solution

R*(dq/dt) + (1/C) * q = E(t)

R = Resistance ; q = charge ; E = Electromotive force ; C = Capacitance

1000*dq/dt + 2*10^5*q = 200

dq/dt + 200 q = 1/5

This is a linear differential equation of first-order

The solution for this is calculated by integrating factor

q(t) = 0.001 + Ce^(-200t)

i(t) = q\'(t) = - 200*Ce^(-200t)

Since i(0) = 0.4 ===> - 200*C e^0 = 0.4 ==> C = - 0.002

So the charge q(t) = 0.001 - 0.002e^(-200t).

The current i(t) = - 200*(-0.002) e^(-200t) = 0.4e^(-200t)