Question

Show that the average value of ac over a complete cycle is zero

Answer 1

Answer:

instantaneous current= I=Imsinωt

average current=

I avg = 1/T∫Idt                 [ taking limit 0 to T]

           = 1/T∫Imsinωtdt

          =  Im/T∫sinωtdt

          =  Im/T[-cosωt/ω]

           =Im/T[-cosωT+cos0]

            =Im/T[-1+1]

            =im/T[0]

            = 0

Answer 2

Answer:

Step-by-step explanation:

The equation for AC current is $I=I_{0}sin\omega{t}$

Therefore the total current of a full cycle is

$\text{Total current}={\int_{0}^{T}I_{0}sin\omega{t}dT$

Average of the current of a full cycle is =

$\frac{\int_{0}^{T}I_{0}sin\omega{t}dT}{\int dT}\\\\\Rightarrow\frac{\int_{0}^{T}I_{0}sin\omega{t}dT}{T}\\\\\Rightarrow\frac{I_{0}}{T}\times(\int_{0}^{T}sin\omega{t}dT)\\\\\Rightarrow\frac{I_{0}}{T}\times(\int_{0}^{T}\frac{-cos\omega{t}}{\omega})\\\\\Rightarrow\frac{-I_{0}}{\omega{T}}\times(\int_{0}^{T}cos\omega{t})\\\\\Rightarrow\frac{-I_{0}}{\omega{T}}\times(cos0-cos2\pi)\\\\\Rightarrow\frac{-I_{0}}{\omega{T}}\times(1-1)\text{ }[cos0=cos2\pi=1]}\\\\\Rightarrow0\text{ (Proved)}$