What is meant by average value of alternating current? Obtain an expression for it. Prove that the average value of alternating current over one complete cycle is zero.

Explanation

At any time, alternating current flowing through the circuit is given by
`I= l_0 sinomegat`
Let dq = Small amount of charge sent by a.c. in small time dt
Suppose current I remains constant for time dt
`:. dq = I dt`
`dq = I_0 sin omegat dt ` [Using.eq. (1)]
Charge sent by a.c. over a complete cycle is
`intdq=int_0^TI_0sinomegat dt`
`q=I_0|(-cosomegat)/(omega)|_0^T`
`q=I-_0/omega|cosomegaT-costheta|`
`q=I_0/omega|cos""(2pi)/T(T)-1|" "[ :. omega=(2pi)/T]`
`q=I_0/(omega)|1-1|=0`
Let `I_(av)` = .Mean value of a.c. over the complete cycle.
Then charge sent by it is given by
`q= _(av) (T) " "....(ii)`
From eqns. (i) and (ii)
`I_(av) (T)=0`
`I_(av) =0`
So mean value of a:c: over a complete cycle is zero.