Answer to Question #122487 in Differential Equations for jse

Question

A 100-volt electromotive force is applied to an RC-series circuit in which the resistance is 500 ohms and the capacitance is 10^−4 farad. Find the charge q(t) on the capacitor if q(0) = 0.

q(t) = _____

Find the current i(t).
i(t) = ____

Answer 1

The Differential equation for RC-series circuit is

$R \frac{dq}{dt} + \frac{1}{C} q = E(t)$ .

Now, $R = 500$ ohm, C = $10^{-4 }$ F, E(t) = 100 V.

So, $500 \frac{dq}{dt} + 10^4 q = 100 \implies \frac{dq}{dt} + 20 q = 0.2$

This is standard first-order differential equation.

So, Integrating factor = $e^{\int20dt} = e^{20t}$ .

Hence solution is $q e^{20t} = \int(0.2)e^{20t} dt + c$

$\implies qe^{20t} = (0.2) \frac{e^{20t} }{20}+ c =0.01 e^{20t} +c$

Now, $q(0)=0 \implies 0.01 + c = 0\implies c = -0.01$

$\implies qe^{20t} = 0.01(e^{20t}-1) \\ \implies q = \frac{0.01(e^{20t}-1)}{e^{20t}} = 0.01 (1-e^{-20t})$

And current $i(t) = \frac{dq}{dt} = 0.2 e^{-20t}$