`(i)` When an `AC` source is connected to an ideal inductor show that the average power supplied by the source ever a complete cycle is zero.
`(ii)` A lamp is connected in series with an inductor and an `AC` source. What happens to the brightness of the lamp when the key is plugged in and an iron rod is inserted the indcutor ? Explain.

Explanation

`(i)` In an ideal inductor current lags by `pi//2`
`V=V_(0)sin(wt-pi//2)=-I_(0)sin(pi//2-wt)=-I^(@)coswt`
Work done in small time `dt`
`dw=Pdt=-V_(0)I_(0)sinwt cos wt dt`
`=-(V_(0)I_(0))/(2)sin2wtdt`
Average power associated per cycle
`P_(av)=(W)/(T)=(I)/(T)int_(0)^(T)dw=-(V_(0)I_(0))/(2T)int_(0)^(T)sin2wt dt=(V_(0)I_(0))/(2T)[(cos2wt)/(2w)]_(0)^(T)`
`=(V_(0)I_(0))/(2T)[(cos4pi-cos0)/(2w)]=0`.
`(ii) L` will increase `mu_(r)` times, this will increase `X_(L)` and hence, current decrease. So brightness decreases.