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Back when I was younger, I used to take apart the old disposable cameras I'd take to camp to have fun playing with the parts and doing a bit of exploring. One year I ended up with a camera with a flash, and got a bit of a surprise.

After having removed the cardboard I took a moment to look it over, with the thumb of one hand resting along the AA battery on the bottom. As I'm finishing my evaluation of the piece, my index finger from my other hand touched the top of the flash, which sent a current up one arm and back down the other, setting off the flash.

This caught me completely by surprise, because I'd always thought batteries only conducted if there was a complete circuit between the poles, and even holding a battery by both ends at once I've never noticed a charge before. What exactly happened here?

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You discharged the high voltage flash capacitor, not the battery. Most likely the side of the battery acted as the ground. You're lucky you're not dead. –  Samuel Jun 3 at 6:42
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With most single-cell batteries (e.g. AA), the case is connected to its negative pole.

A battery by itself does not provide enough voltage to feel by hand. However, the flash circuitry charges one end of the flash tube to a high voltage (between 250 and 5000 Volts, depending on the tube). This energy is discharged as soon as the trigger circuit is closed, which your hand did.

Make sure electrical systems are disconnected from power, and make sure high-voltage systems (such as flashes) are completely discharged before disassembling them!

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