Question

Physics class 12th Alternating Current question "Find out the average value of current from the equation in the first half cycle?".
Equation mentioned in the snap.
Thanks in advance.

Answer 1

we know according to math

if average value of any function f(x ) in interval x1 to x2 is
= f(x).dx/(x2 - x1) .

use this ,

i =I°sin( wt + ∅)

it half circle means 0 to π
hence, time taken to complete half circle is 0 to T/2
where T is time period .
[becoz period of sine function is 2π , hence half circle π ]

hence,

average value of i = < i > = i.dt/(T/2- 0)

< i > ={ 2 i°/T}sin( wt + ∅).dt

= { 2i°/T } [ -cos(wt + ∅)]/w

={ 2i°/wT} [ -cos( wt + ∅ ) ] put limit 0 ≤ t ≤ T/2

= {2i°/wT} [- cos( wT/2 + ∅) + cos( ∅) ]

we know
w = 2π/T put this ,

< i > = { 2i°/2π} [ cos∅ - cos( π + ∅) ]

< i > = {2i°/π } cos∅

∅ is phase constant .
assume initial phase of wave is 0
then cos∅ = 1

now ,
< i > = 2i°/π × 1

< i > = 2i°/π