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>> A pretty accurate model of a battery is a perfect voltage source in
>> series with an "internal" resistor. This then nicely explains how the
>> more current you try to draw the lower the apparent voltage will be at
>> the terminals of the battery.
>
> As far as I'm aware, resistors affect only current, not potential.
That statement doesn't even make sense. The voltage across the terminals of
a resistor is equal to the current going through it times the resistance,
that's all you need to know and then you can figure out any resistor circuit
with a bit of algebra.
By the way, notice how the lights dim on your car (and maybe your radio also
shuts off?) when you start the engine? That's because the starter motor is
drawing a huge current and causing the battery voltage to drop to way below
12V.
Your lights are connected directly to the battery supply, so they dim, and
the radio might only be specced to work down to 8V or something, in which
case it shuts itself off until the voltage comes back up again. Some radios
can survive very low voltages for short periods of time, so they usually
stay on while the engine is being started, but not on all cars.
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