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Why does an increase in current increase the conductor temperature?

I wonder if I understand it well .If charged particles are at an electric potential V, measured in Volts, this means that each Coulomb of charge has this amount of electrical potential energy. So, 1 V = 1 J/C. If the charged particles can move and flow to a point of lower potential, then they will do so, transferring the extra energy into other forms, Each Coulomb of charge flowing around the external circuit, must lose an amount of energy equal to the pd, and this energy will be transferred as internal energy to the particles of the conductor, raising its temperature. An increase in the current will increase the rate at which the conductor absorbs energy, therefore heating it more strongly.

To sum up the increased current brings more energy and thus increases the temperature.

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Why does an increase in current increase the conductor temperature?

Because the "conductor" isn't perfect. Other than the special case of super-conductors, the wire you use has finite resistance. The power dissipated by a resistance is proportional to the square of the current thru it. In common units:

    W = A2 Ω

where A is the current thru a resistance in Amperes, Ω the resistance in Ohms, and W the power dissipated in Watts.

For example, let's say you are running 20 A thru a piece of wire with 100 mΩ resistance. The wire will dissipate (20 A)2(100 mΩ) = 40 W. Depending on the length of the wire, it could get anywhere from hot enough to melt, to slightly warm, to not even noticeably warm without careful measurement.

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