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Prove. (1) The effective value I_{eff} of an alternating current is given by

Proof. The rate of development of heat is the instantaneous power P given by

2) P = i^{2}R = I_{m}^{2} R sin^{2} ωt

and the heat H developed in a complete cycle of period T is given by

Now

and

5) T = 2π/ω

Utilizing 4) and 5), 3) evaluates to

The heat that would be developed by a direct current I_{eff} in the same time would be

Equating 6) and 7), we get

or

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