In an AC circuit, the current has the form for constants The power is defined as for a constant . Find the average value of the power by integrating over the interval

Question

In an AC circuit, the current has the form  for constants  The power is defined as  for a constant  . Find the average value of the power by integrating over the interval

Quizplus · Accepted Answer

The average value of power over a period T is given by:

P_avg = (1/T) * ∫[0,T] p(t) dt

In this case, the given interval is [0, 2π]:

P_avg = (1/(2π)) * ∫[0,2π] ½V₀I₀cos(ωt)² dt

We know that cos²(ωt) = ½[1 + cos(2ωt)], so we can write:

P_avg = (1/(4π)) * ∫[0,2π] V₀I₀[1 + cos(2ωt)] dt

Using the fact that ∫cos(2ωt) dt = 0 and ∫1 dt = t, we get:

P_avg = (1/(4π)) * [V₀I₀(2π)]

Simplifying, we obtain:

P_avg = (1/2) V₀I₀

Therefore, the average power is D) 11eaa88b_9137_921f_a696_49967719ddd5_TB5869_11.

In an AC Circuit, the Current Has the Form