182 Week 5: Resistance
If oneshort-circuitsa battery – connects a very low/zero resistance across its terminals –
then the battery will usually deliver its maximum power, and its maximumpossible current. A
very simple, but quite accurate, model for this limiting is indicated in the figure 58 above. In it,
a hypothetical chemical battery is represented as the two circuitelements inside the dotted box.
One is the actual chemical potential generated by the chemical reaction. This is called theinternal
voltageof the battery^56. As we shall see, this is also the voltage between the terminals of thebattery
when there isno load, if the chemical process has not exhausted the reactants (if youlike, the “fuel”
of the battery). In addition to this, the battery is considered to have aninternal resistancer
that limits the current the batter can deliver even when completely short circuited.
We are now (with both Kirchoff’s rules and/or series resistances in hand) well capable of under-
standing how all of this works. Kirchoff’s rule for the circuit loop is:
V 0 −Ir−IR=V 0 −I(r+R) = 0 (349)or:
I=V 0
r+R(350)
Theterminal voltageis defined to beVt=V 0 −Ir, the voltage between thephysical terminals
of the battery when it is delivering any given currentI. IfR=∞,I= 0 andVt=V 0 as indicated
above. IfR= 0 (the battery is “short circuited” when a zero resistance is connected across the
terminals) we find that:
Imax=V^0
r(351)
In practical terms, the internal voltage is usually known, fixed by the chemistry of the battery, and
one canmeasurethe internal resistance indirectly by short-circuiting the batterywhile measuring
the delivered current. As batteries are discharged (or as rechargable batteries age) this internal
resistance increases until their terminal voltage effectively dropsto zero if a load of any sort is
connected across the terminals.
Note well that we can easily compute the power delivered to the internal resistance (the battery
itself, generally heating up the battery with its internal Joule heating) versus its load resistanceR:
Pr=I^2 r=V 02r
(r+R)^2(352)
and
PR=I^2 R=V 02R
(r+R)^2(353)
The sum of theses add up to the total power provided to the circuitby the internal voltage/energy
source, as it must.
It is an instructive exercise to demonstrate that the power delivered to the load is amaximum
whenr=R, when the load resistance matches the internal resistance of thepower supply. This
is calledimpedance matching– impedance is a sort of generalized resistance that we will study in
more detail in the chapter on AC circuits, but in the case of DC circuits it is equal to ordinary
resistance. Impedance matching is an essential part of the engineering of things like earphones or
speakers, where one limits the power deliverable to the load by any given amplifier.
(^56) This was historically called the “electromotive force”, or“EMF” of the battery, and it is still often represented
asEin physics textbooks and called the EMF. I find it difficult to call or label something that is clearly avoltage
aforce, even by obscure inheritance. This is doubly so for chemistry, where the actual motivation is caused by the
discretequantumenergy changes between the reactants and the products and where it is a lot of work to even define
a good quantum analog of “force” at all. I therefore rebel in my own small way and just call a voltage a voltage and
differentiate only with modifiers.