Problem 2

An ideal constant source of emf

E=12.0\,\mathrm V

is connected at

t=0

in series with a resistor

R=200,Omega

and capacitor

C=50.0,\mu\mathrm F

. The capacitor is initially uncharged, and the wires and switch are ideal. Let

q(t)

be the charge on the plate connected toward the positive source terminal, and let

i(t)=dq/dt

be the charging current, for

tge0

. (a) Find the time constant and functions

q(t)

and

i(t)

. (b) Find when the capacitor voltage reaches

90%

of its final value and the current then. (c) Find the stored energy at that instant and the limiting charge and current as

t\toinfty

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\frac{dq}{dt}+\frac{q}{RC}=\frac ER,\quad \tau=RC=1.00\times10^{-2}\,\mathrm s

problems.proof.analysis

Substituting $i=dq/dt$ and dividing by $R$ gives a first-order linear equation. The product $RC$ has units of seconds and equals $0.0100\,\mathrm{s}$ , the charging timescale.