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

t_{90}=-RC\ln(0.10)=2.303\times10^{-2}\,\mathrm s,\quad i(t_{90})=0.10E/R=6.00\,\mathrm{mA}

problems.proof.analysis

Since $V_C=q/C=E(1-e^{-t/RC})$ , setting $V_C=0.90E$ gives $e^{-t/RC}=0.10$ . The current is then $10%$ of its initial value, not $90%$ , because the resistor drop supplies the remaining source voltage.

problems.proof.pitfall. At 90% of the final capacitor voltage, the current is only 10% of its initial value.