Batteries
A battery (or power supply - PS) is a source of emf, acting like a charge pump to
maintain a voltage in a circuit.
The actual voltage supplied to a circuit is the terminal voltage, or voltage at the terminals
of the battery/PS. If the battery/PS is ideal, meaning that it has no internal resistance and so
there is no internal I2R power loss, then the terminal voltage is equal to the emf. {Note that emf
stands for electromotive force – a misnomer since it is not a force- but a potential – we use the
term emf for that particular voltage supplied by a power source such as a battery or PS with its
own source of energy, chemical or electric or otherwise.}
In general, a battery/PS has an internal resistance r and when the battery/PS is connected
in a circuit and has a current I flowing from the battery/PS will have a terminal voltage that is
less than the emf, ε, and given by
a
b
∆V = Vb – Va = ε – I r
If I = 0 then ∆V = ε , a situation known as an open circuit.
If the external resistance in the circuit is R, known as the
load resistance, then we have that
ε = IR + Ir
so that I = ε/(R + r). Often R is much larger than r and we can neglect r.
Remember that the power loss in a resistor is P = I ∆V = I2R, so if we multiply the above
equation by I we have
Iε = I2R + I2r,
which says that the power input by the battery/PS is equal to the power loss at both resistors from
heating.
Example: Show that the maximum power delivered to the load resistor R (for a fixed ε) occurs
when R = r.
P = I2R = ε2R/(R + r)2
But dP/dR = ε2 d [R/(R+r)2]/dR = 0 at maximum or
1/(R+r)2 + R(-2)[1/(R+r)3] = 0 or
(R+r) – 2R = 0 or R = r.
Then Pmax = ε2r/(2r)2 = ε2/4r at R = r.