Problem 2

An initially uncharged capacitor

C=100\ \mu\mathrm F

is connected at

t=0

in series with a resistor

R=10\ \mathrm{k\Omega}

and an ideal source of emf

E=12

V. Let

q(t)

be the capacitor charge and

i(t)=\dot q(t)

the current. (a) Find the time constant

\tau

and the long-time charge

q_\infty

. (b) Write expressions for

q(t)

and

i(t)

. (c) Find when the charge reaches

90\%

of its final value and the current at that instant.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

q(t)=q_\infty(1-e^{-t/\tau})=1.20\times10^{-3}\left(1-e^{-t}\right)\ \mathrm C

problems.proof.analysis

The general solution is the final charge $q_\infty$ plus an exponential term with a free constant. The condition $q(0)=0$ fixes that constant and gives the familiar $1-e^{-t/\tau}$ form.