(a) For a given a.c., i=imsinωt, show that the average power dissipated in a resistor R over a complete cycle is 1/2im2R (b) A light bulb is rated at 100 W for a 220 V a.c. supply. Calculate the resistance of the bulb.

(a) : Current is given as
i=imsin(wt)
Average power dissipated over a complete cycle

=∫0Ti2Rdt/T
∴

=∫0Tim2sin2(wt)Rdt/T
OR

=im2R/T∫0Tsin2(wt)dt=im2R/T∫0T(1-cos(2wt))/2dt
OR

=im2R/2T ×[t-sin(2wt)/2]0T
OR

=im2R/2T ×[T-0] =>

=im2R/2
(b) : Given
P=100W
V=220volts
∴Resistance of the bulb
R=(220)2/100=484Ω