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The rate of heat production in a wire carrying an electric current is proportional to...

The answer is meant to be current with a power (e.g. the square root of current, just current, current squared).

I know that the rate of heat production in a wire carrying electric current is proportional to resistance; How can I deduce that the heat production is proportional to current squared?

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The heat production in a current carrying wire is equal to: $P = I^2 R$.

You can derive this formula as follows. First we assume that the material behaves Ohmic and therefore: $V = I \cdot R$. Further we know that $P = I \cdot V$. Substitute Ohms law into the previous equation to get the formula you are looking for.

Alternative derivation would be more formal using the Poynting vector. You imagine having a wire with a constant current through it. This will give rise to a magnetic field. (Use Ampere's law to find this field). Then also calculate the electric field from using the value of the current and the resistance of the material. (You take Ohms law to calculate the potential difference and from there the electric field). Then you calculate the Poynting vector: $S = E x B/\mu_0$. Integrate this vector field over a closed surface around the wire and you will find $P = I^2 R$.

Here is a link that works the problem out with the Poynting vector: http://web.mit.edu/8.02t/www/materials/InClass/_notes/IC_Sol_W15D1_1.pdf

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The relation can be established for ohmic conductors as follows: dH = VdQ => dH = V(dQ/dt)dt As we know dQ/dt =I and V=IR => dH= IIRdt Integrating on both sides we get H= I^2*R*t Hence the heat evolved is proportional to the square of the current through the conductor.

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