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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 = i²R = I_m² R sin² 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