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The electrons must go from the battery to the light-bulb, then come back again to the battery through the other connection. If there is only one connection, there will soon (within microseconds or so) be an accumulation of electrons that will repel any additional electrons - the current will stop flowing.

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Q: Why must there be two connections to a batter and to a light bulb?
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Related questions

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What switches must be closed in order for the light bulb to light up?

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