The average current of a sinusoidal luxury varying alternating current of peak value 5th with initial phase zero bw the instants t=T/8 to t=T/4 Pls answer soon

To find the average current of a sinusoidal alternating current (AC) over a specific time interval, we can use the properties of sinusoidal functions. In your case, you have a peak current (I₀) of 5 A and an initial phase of zero. The time interval you are interested in is from t = T/8 to t = T/4, where T is the period of the sinusoidal wave.

Understanding the Sinusoidal Function

The sinusoidal current can be expressed mathematically as:

I(t) = I₀ * sin(ωt + φ)

Here, I₀ is the peak current, ω is the angular frequency, t is time, and φ is the phase angle. Since the phase angle is zero in your case, the equation simplifies to:

I(t) = 5 * sin(ωt)

Determining the Angular Frequency

The angular frequency (ω) is related to the period (T) by the formula:

ω = 2π/T

For our calculations, we will keep ω in terms of T, as we will be integrating over a fraction of the period.

Calculating the Average Current

The average current over a time interval can be calculated using the integral of the current function over that interval, divided by the length of the interval:

I_avg = (1/Δt) * ∫ I(t) dt

In this case, Δt = T/4 - T/8 = T/8. Therefore, we need to evaluate the integral:

I_avg = (1/(T/8)) * ∫(from T/8 to T/4) 5 * sin(ωt) dt

Setting Up the Integral

Substituting ω = 2π/T into the integral gives:

I_avg = (8/T) * ∫(from T/8 to T/4) 5 * sin(2πt/T) dt

This simplifies to:

I_avg = (40/T) * ∫(from T/8 to T/4) sin(2πt/T) dt

Evaluating the Integral

The integral of sin(2πt/T) can be computed as follows:

∫ sin(2πt/T) dt = - (T/2π) * cos(2πt/T)

Now, we evaluate this from T/8 to T/4:

∫(from T/8 to T/4) sin(2πt/T) dt = - (T/2π) * [cos(2π(T/4)/T) - cos(2π(T/8)/T)]

This simplifies to:

∫(from T/8 to T/4) sin(2πt/T) dt = - (T/2π) * [cos(π/2) - cos(π/4)]

Since cos(π/2) = 0 and cos(π/4) = √2/2, we have:

∫(from T/8 to T/4) sin(2πt/T) dt = - (T/2π) * [0 - √2/2] = (T√2)/(4π)

Final Calculation of Average Current

Substituting this back into our average current formula gives:

I_avg = (40/T) * (T√2)/(4π) = (10√2)/(π)

Thus, the average current over the interval from T/8 to T/4 is:

I_avg ≈ 3.98 A (approximately, depending on the value of π).

This result shows how the average current can be derived from a sinusoidal function over a specified time interval, illustrating the relationship between peak current, phase, and the average value over time.