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

i(t_{90})=\frac{E}{R}(0.10)=1.20\times10^{-4}\ \mathrm A;\qquad \tau=1.00\ \mathrm s,\ q_\infty=1.20\ \mathrm{mC},\ t_{90}=2.30\ \mathrm s

problems.proof.analysis

When $e^{-t/\tau}=0.10$ , the current is 10% of its initial value. Final results: $\tau=1.00$ s, $q_\infty=1.20$ mC, $i(t)=1.20e^{-t}$ mA, and $t_{90}=2.30$ s.