Average value of EMF or current over a complete cycle is zero How to derive this mathematically - Physics - Alternating Current

Question

Average value of EMF or current over a complete cycle is zero How
to derive this mathematically ?

Sushant · Accepted Answer

hi Ebenez
Jacober ,

Please find below the solution to the asked query:
In an alternating current circuit:

When an alternating current passed through a moving coil galvanometer it shows no deflection ,this is because for one complete cycle mean value of alternating current is zero as AC flows in one direction during one half cycle and in opposite direction during another half cycle
let  i = iosinwt +∅iavg(T/2) = ∫0T/2idt∫0T/2dt = ∫0T/2iosinwt +∅dt∫0T/2dtiavg(T/2) = iowcoswt +∅0T/2T/2 =iow cosw(T/2) +∅ - cos∅T/2iavg(T/2) = iow cosπ+∅ - cos∅T/2 = 2ioπ as w = 2πTso iavg(T/2) = 2ioπ =  +0
636iosimilarly for the other half cycleiavg(T/2) = ∫T/2Tidt∫T/2Tdt = ∫T/2Tiosinwt +∅dt∫T/2Tdt = -2ioπ = -0
636iothus iavg(T)  =+0 636io-0
636io = 0or directly iavg(T) = ∫0Tidt∫0Tdt  = ∫0Tiosinwt +∅dt∫0Tdt = iowcoswt +∅0TTiavg(T) = iow coswT +∅ - cos∅T = iow cos2π +∅ - cos∅T iavg(T) = iow cos∅ - cos∅T = 0

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Average value of EMF or current over a complete cycle is zero. How to derive this mathematically ?