What is meant by mean or average value of alternating current ? Show that mean value of ac over a complete cycle is zero.

Explanation

Mean value of alternating emf:
It is that steady emf which would send the same amount of charge through a circuit during half the period of alternating emf as is done by the alternating emf through the same circuit during same interval of time.
For calculating its value, let an alternating emf be given by
`E= E_0 sinomega" "...(I)`
If R be the resistance of the circuit, then
`I = E/R = E_0/Rsinomegat` [Using eq. (i)]
If the current remains constant for a small interval of time dt, then small amount of charge passed through time dt is.
`dq =1 dt = E_0/Rsinomegatdt`
`:.` Total charge passes during first half-cycle is
`q=int_0^(T//2)E_0/Rsinomegatdt=(E_0)/Rint_0^(T//2)sinomegatdt`
`E_0/R[(-cosomegat)/omega]_0^(T//2)`
`=-(E_0)/(omegaR)[cosomegaT/2-cos0^@]`
`=-(E_0)/(omegaR)[cos""(2pi)/(T)T/2-cos0^@] " "( :. omega=(2pi)/T)`
`=(E_0)/(omegaR)[cospi-cos0^@]`
or `q=-E_0/(omega_0R)[-1-1]=(2E_0)/(omega_0R)" "...(ii)`
If `E_m` be the mean value of alternating emf over half the cycle, then
`q=E_m/RT/2" "...(iii)`
From eqns. (i) and (iii),
`E_m/R(T/2)=(2E_0)/(omegaR)=(2E_0)/((2pi//T)R)=(2E_0T)/(2piR)`
`E_m=(2E_0)/pi`