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I am wondering how coiled wire creates so much heat when a current is passed through it. Is it just a larger surface area of heated substance is available or are interacting forces at work eg magnetic and electric fields interacting? Does adding more coils simply add more surface area or are there more interacting forces as well.

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I'm not sure I understand what you're asking exactly, but there are two effects you might be seeing:

  1. A coiled wire is longer than a straight wire that fits in the same space. Therefore it has higher resistance ($R=\rho\dfrac{l}{A}$) and so produces more heat from the same current flowing through it.

  2. A coiled wire can't cool itself as efficiently as a straight wire because radiation from one part of the wire can impinge on another part of the wire instead of out into space; and because air flowing over one part of the wire might then flow past another part of the wire instead of away from the wire.

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Coiling a light filament has several effects.

  • Because the bulb is not under vacuum, the gas inside cools the filament. The coiling reduces the cooling effect from the gas (substantially).

  • An uncoiled filament with the required surface area will be much longer than an coiled one. A longer filament needs more support (expense) in a commercial bulb.

  • A coiled wire intercepts some of the radiation. So you need somewhat more surface area radiating than you would with a straight wire.

Agrawal suggests that the resistance changes due to coiling amount to about 1%.

Lamptech document discussing coiling.

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I would add a third reason to The Photon's and Paul's answers.

The electrons in a current flow in a straight wire are bouncing permanently with the atoms and running zig zag. As The Photon pointed out, Th e wire has an ohmic resistance.

The electrons in a flow in a coil beside the bouncing have to run in circles and this is clearly an acceleration. Due to this accelerate the involved electrons emit electromagnetic radiation. Some part of this radiation produces the EM field of the coil and an other part get received from the wires atoms and get re-emitted with lower frequencies. This raises the temperature of the wire.

BTW the acceleration accompanied by the EM radiation is one reason for the additional heating with AC currents as Paul pointed out.

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