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When adding bulbs in parallel, the brightness is brighter than that of series. But does that mean adding bulbs in parallel will increase the brightness of the other bulbs?

My intuition is as follows: When adding a bulb in parallel the current doubles, but that current splits between the two branches such that both bulbs receive the same current and the same voltage, so brightness doesn't increase, but it is still brighter relative to adding bulbs in series. Is this correct?

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You are correct.

When you put them in parallel, each bulb is seeing the full supply voltage. Hence each bulb will get the same current as it did on its own. So, each bulb shines with the same brightness it would have if there was only one bulb. Of course this assumes the supply is able to provide twice the current.

When you put the bulbs in series, the total resistance in the circuit doubles, hence the current halves. This half current flows through both bulbs, so they shine at a reduced brightness.

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You just get the brightness of two ore more bulbs, every single bulb keeps its brightness, if they are parallel.

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You have it correct except that the parallel bulbs aren't in a branch and the current doesn't double. The power generator produces a constant voltage. Both are part of the same system so both have full voltage through that system and will always have full voltage.

The difference with the other bulbs comes because you add a branch to that series circuit. Adding bulbs to that branch means there is an additional path for the current to take. So, that voltage will make those bulbs appear less bright while the current from the bulbs in the parallel never changes.

It's like a river. If you plant a tree by a river, it will always get the same amount of water. If the river has a branch veering off, some water will go in that direction but it won't be at the same speed or amount as the larger river. A tree planted along the branch isn't going to get the same amount of water as the original tree since the total amount of water in the river remains the same.

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