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

1-e^{-t_{90}}=0.90\Rightarrow t_{90}=\tau\ln10=2.30\ \mathrm s

problems.proof.analysis

Set the ratio $q/q_\infty=0.90$ and solve the exponent: $e^{-t/\tau}=0.10$ . Taking the natural logarithm gives the time; with $\tau=1$ s it is just $\ln 10$ .

problems.proof.pitfall. The 90% instant is not

0.9\tau

: exponential approach slows, so it takes

\tau\ln 10

, beyond one time constant but still short of asymptote.