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

q(t)=CE(1-e^{-t/RC})

problems.proof.analysis

The first-order equation has equilibrium value $CE$ plus an exponentially decaying transient. The initial condition $q(0)=0$ fixes its coefficient and yields this expression, which tends to $CE$ .