Week 5: Resistance 163
and water. For every mole of lead converted into lead sulphate, twomoles of electrons have to move
from cathode to anode. That is 1. 2 × 1024 / 1. 6 × 1019 = 0. 75 × 105 Coulombsof charge, enough to
drive an Ampere of current (one Coulomb/second) for around a day. A mole of lead is around 207
grams, which weighs around a half a pound. Allowing for the electrolyte and sulphuric acid, roughly
a pound of battery will drive a load of two watts (one ampere at two volts) for just under a day
(where we’ll work out energy relations below to justify this in a moment).
A second advantage of this particular battery is that it isrechargable. If one simply places a
voltage across the cell that exceeds its terminal voltage, chargeflowsthe other way, reversing the
reaction and turning lead sulphate back into lead or lead oxide. By careful design, one can charge
and discharge the battery many times before too much lead sulphate falls off of the electrodes or
crystalizes out across the space in between the terminals and shorts out the batter, at which time
the battery must be remanufactured (to avoid dumping toxic lead into the environment).
Vehicle batteries, of course, weight many pounds – as many as fiftyor sixty – and have six cells,
and therefore can drive bigger currents at higher voltages, currents that can easily be large enough
to be dangerous. In fact, a car battery^49 , and can easily kill you if you handle it carelessly by the
poles with e.g. wet hands or cuts on your fingers! I’ve gotten “hit” this way myself handling a car
battery by the poles in a rainstorm, and it hurts! This kind of battery can (multiplying out the
coulombs, volts, and seconds) do around 150,000 joules of work per pound in the ideal case, probably
less than half this in the real world case.
However, all batteries have afinite rateat which they can dowork, determined by the physical
limitations on the rate at which the chemical reaction can proceed. So even if one shorts out a
battery with aperfectconductor, one won’t get an infinite current at a constant voltage. As the
current goes up, the voltage goes down, until at some point all of the energy is released as the heat
of reaction in the electrolyte and none to the battery load. Some batteries are designed to provide
a fixed voltage and low current for a long time; others are designed to produce a fixed voltage and
alargecurrent for ashorttime. Car batteries in particular are usually pretty good at both.
The Symbol for a Battery
All of this is too complicated for intro physics, of course. We want tostart by idealizing a battery
and replacing it in all circuits we consider with a single simple symbol. The symbol we will use
is
V +
, whereV is the nominal potential difference maintained by the battery between its
terminals (its “terminal voltage”) and where the + sign (and longer plate) indicate theanode, the
side of the batteryfromwhich positive current flows (where we are suffering from Franklin’s Mistake,
because the actual motion of charge in the chemical reaction above is negative electrons flowing the
other way). Again, the battery behaves like an “inexhaustible capacitor” in an electrical circuit,
increasingthe potential byVas one moves from the cathode (small plate) to the anode (large plate)
in any circuit diagram containing this symbol.
Ouridealbattery never runs out of power, has no limitations on the amount of current it can
provide at its rated voltage, and its voltage is rigorously constant. None of these is going to be
true in practice for real batteries, and after we define resistance and work out Ohm’s Law, series
resistance addition rules, and Kirchoff’s rules below, we’ll revisit the battery and see how we can
compensatefor these features by assigning aninternal resistancerto the battery itself. This internal
resistance is not entirely a fiction – batteries and other power suppliesdohave some actual internal
resistance – but it often also represents the practical effect of other rate limiting physics, such as
the maximum rate that some given force can do work on a piece of generating apparatus.
(^49) http://www.darwinawards.com/darwin/darwin1999-50.html Not just a car battery. You can kill yourself with a
nine volt transistor radio battery, and one of my favorite Darwin awards went to a Navy officer who demonstrated
this the hard way after being warned about the danger.