Week 5: Resistance 161
- RCcircuits are simple loops where a capacitor is charged or discharged through a resistance.
You should be able to derive the time-dependent discharge of a capacitor through a resistor as
the followingexponential decay:
VC(t) =V 0 e−t/RC (283)or:
Q(t) =Q 0 e−t/RC (284)whereQ 0 is the initial charge on the capacitor andV 0 =Q 0 /Cis the initial potential across
the capacitor. This result follows from applying Kirchhoff’s voltage lawaround a loop and
converting it into a first order, linear, ordinary differential equation of motion that can be
directly integrated.- The “exponential time constant” of this decay isτ=RC. Recall that the time constantτis
the fixed time interval in which the initial charge/potential decays to 1/eof its value at the
start of the interval. Exponential processes always gain/lose thesamefractionof their initial
value in any given interval of time. - A charging capacitor (initially uncharged) can similarly be shown to exponentially approach
an asymptotic charge or potential:
VC(t) =V 0(
(1−e−t/RC)
(285)
or:
Q(t) =Q 0(
1 −e−t/RC)
(286)
whereV 0 is the magnitude of the charging potential andQ 0 =CV 0 , in both cases thefinalval-
ues found on the capacitor after a verylongtime, specifically many exponential time constant
intervals.Note on notation:At one time the voltage produced by e.g. a battery or mechanical power
supply was called (by Allesandro Volta, one of the original discoverers of the chemical electrical
cell) anelectromotive force, and this usage was continued by later researchers such as Faraday. This
was a horrible misnomer – Volta’s model for the cause of the voltage (that “motivated” the choice)
was incorrect, and of course theunits of force, Newtons, are completely different from the
units of voltage, Joules per Coulomb. The SI unit of potential and potential difference, the
Volt, is named after Volta.
Unfortunately many physics textbooks perpetuate the traditionof referring to the voltage pro-
duced byanymeans as an electromotive force or use the acronym “EMF” to describe this voltage
without actually using the word force. In addition, the symbolEis often used in place of the symbol
Vto label the voltage of a cell or induced voltage (discussed in a few chapters) as anE-MF. Although
this is a calligraphic/script font version ofE, it is still remarkably easy to confuse with the electric
field and of course a voltage isn’t conceptually or dimensionally an electric field, either!
This book will (hopefully consistently) use the symbolVto describe the voltage sources or sinks
of a circuit element or the circuit itself, including electrical cells or induced voltages, and will eschew
the use of the symbolEor the descriptors EMF or (worse) “electromotive force” used todescribe
a potential or potential difference no matter what it results from. This should do no conceptual
harm to the general topic of electricity and magnetism; indeed it should simplify the treatment of
potential differences. Students should be aware of the more common usage, however, to the extent
that they use additional textbooks or references to supplementthis one as they study.