188 Week 5: Resistance
Nowmultiply through bydt, divide through byQ−CV 0 :dQ
Q−CV 0=−
dt
RC(395)
and integrate both sides (indefinite integral on the right) to get:
ln(Q−CV 0 ) =−t
RC+A (396)
(whereAis the constant of integration). As before, to getQ, we exponentiate both sides:
Q(t)−CV 0 =eln(Q−CV^0 )=e−
RCt+A
=eAe−t/RC (397)Finally, we solve forQ(t):
Q(t) =CV 0 +eAe−t/RCCV 0 +Be−t/RC (398)and set the constant of integrationB=eA(the exponential of an unknown constant is still an
unknown constant^57 ) from the initial conditions, so thatQ(0) = 0. Our final answer is:
Q(t) =CV 0(
1 −e−t/RC)
(399)
It is left as an exercise to evaluate the same list of quantities that wedid for the discharging
capacitor: I(t), VC(t), VR(t), PC(t), PR(t). To this we addPV(t), the total power provided to the
circuit by the voltage, and suggest that you demonstrate that ast→ ∞the total energy provided
to the circuit by the voltage equals the total energy stored in the capacitor in the end plus the total
energy burned in the resistor. Note well that because our solutionwas based on Kirchhoff’s loop
rule, whichisthe constraint that work-energy be satisfied, it should come as nosurprise that in the
end energy conservation is precisely embodied in the full integratedsolution we obtain.
Yet to me, it always does. There is something amazing, almost magical,in the way that energy
conservation works out in the equations of electromagnetism, given the complexity, the structure,
thedetailwe see in the many different problems we work throughout the semester and beyond (as
electromagnetism is a major foundation of our understanding ofeverything, in both classical and
quantum physics). But it does.
We live in an enormously conservative Universe, where there are, quite rigorously,no free lunches,
where mass-energyneverwhimsically appears or disappears, where one can, with sufficient care, trace
out and balance every conserved quantity in any problem no matterhow many bodies are involved
or how complex the dynamics of the system.
This concludes our examination ofRCcircuits and our return to the world of dynamical equations
of motion with nontrivial solutions, in this case exponential solutions(although we have done our
best to keep our hand in with the occasional “discovered” oscillatoror constant acceleration problem
on the homework so far). RCcircuits are quite important and occur in nature as well as in most
electronic devices, where they are often used for timing purposesor whereRCexponential charging
or discharging behavior is an artifact of the circuit design that “softens” the edges of sudden square-
wave-like transitions in voltage as they propagate into a circuit leg with nonzero resistance and
capacitance.
The most important place that they occur in nature is probably insidethe brain. The nervous
system is decently modelled by neurons as tiny bioelectrical batteries that charge up capacitance
across a membrane with variable resistance, a resistance that goes from very high to very low
“suddenly” as the membrane depolarizes and channels open that permit the transport of e.g. sodium
(^57) At your convenience, meditate upon theunitsimplicit in this constant and figure out how they make it through
the process above, where certain things have to be dimensionless and others do not...