A lead-acid battery’s original purpose was to replace the cranking handle, and start the auto engine automatically at the push of a button. However, it soon replaced acetylene lamps with electric headlights, and cavalry bugles co-drivers once honked by squeezing rubber bulbs. A reader wrote in, and asked why the engine still runs when the battery is too weak to crank the starter motor. What does a battery do with regard to this, they asked.
What Does a Battery Do With Reference to the Coil?

A starter battery could not possibly produce the thousands of volts necessary to cause a spark plug to flash. And thereby produce sufficient energy to cause the gaseous petroleum fumes to ignite.
Instead, it feeds twelve-volt electricity to the primary winding of the ignition coil, comprising a few turns of heavy wire. This transfers to a secondary winding of thousands of turns of thinner wire. This arrangement generates a magnetic field that jumps across the plug at the top of the piston stroke. So the high-voltage spark fires just as it begins to thrust down again.
A Short History of Magnetos and Coils and Batteries
Early gasoline vehicles did not have starter motors. Drivers had to use a crank handle to turn over the engine manually. However, their engines still had spark plugs. And these derived their electricity from magnetos instead.

Those magnetos produced high-tension current on a single rotation. However, they faded from automobile use owing to cheaper battery-coil technology. We still find magnetos on internal combustion engines that require a hand start such as lawnmowers.
Furthermore, piston-engine aircraft also use them, to ensure continuity of energy in the event of an electrical failure.
Thus, the short answer to what does a battery do after the auto engine starts, is it keeps feeding power to the coil. The longer answer is it also supplies smoothed lower voltage to secondary systems. So the navigation lights are consistently bright, and our entertainment sounds as good as ever as we cruise the highway.
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Preview Image: 1907 Cadillac Tourer at Autoworld Brussels