Question #122487
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) = ____
1
Expert's answer
2020-06-24T18:22:14-0400

The Differential equation for RC-series circuit is

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

Now, R=500R = 500 ohm, C = 10410^{-4 } F, E(t) = 100 V.

So, 500dqdt+104q=100    dqdt+20q=0.2500 \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 = e20dt=e20te^{\int20dt} = e^{20t} .

Hence solution is qe20t=(0.2)e20tdt+cq e^{20t} = \int(0.2)e^{20t} dt + c

    qe20t=(0.2)e20t20+c=0.01e20t+c\implies qe^{20t} = (0.2) \frac{e^{20t} }{20}+ c =0.01 e^{20t} +c

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

    qe20t=0.01(e20t1)    q=0.01(e20t1)e20t=0.01(1e20t)\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)=dqdt=0.2e20ti(t) = \frac{dq}{dt} = 0.2 e^{-20t}


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Comments

Assignment Expert
16.07.21, 00:14

Dear Helen, thank you for correcting us.


Helen
07.07.21, 15:01

Hi, Thank you for the solution :) I think you made a slight mistake though. Isn't i(t) supposed to be i(t)=0.2e^-20t . You forgot the -ve sign.

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