Problem 2

A capacitor

C

initially carrying charge

Q_0

is connected at

t=0

to a resistor

R

. Find charge

Q(t)

, current

I(t)

, and total heat dissipated by time

t

, assuming no external source. Model each stage with the appropriate conservation law or equation of motion; quantities stated as constant remain fixed during that stage. (a) Establish the governing physical relation for each stage. (b) Determine the requested observables from the stated initial conditions and constraints. (c) Check signs, dimensions, and the physical limiting behavior of the results.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\frac{dQ}{dt}=-\frac{Q}{RC},\qquad Q(t)=Q_0e^{-t/(RC)}

problems.proof.analysis

Separate variables to obtain $dQ/Q=-dt/(RC)$ ; integrating with $Q(0)=Q_0$ gives the exponential decay. This follows from the chosen physical model and the stated initial conditions; the sign convention must remain consistent throughout. A dimensional check catches algebraic errors before the result is interpreted.

problems.proof.pitfall. Do not mix sign conventions or omit a system constraint; check the sign and units before interpreting the result.