A alternating current at any instant is given by i=[6+√56 sin (100 π t+π / 3)] A. The r m s value of the current is ______ A.

Solution

Physics sir · 3 steps

Do not add the two rms values; square the current, average it over a cycle, then take the root.

1. Use the definition. The rms current is the square root of the mean of i² over one full cycle. Write i = 6 + √56 sin θ with θ = 100π t + π/3, then square it.

   i² = 36 + 12√56 sin θ + 56 sin² θ

2. Average each term. Over a full cycle, ⟨ sin θ ⟩ = 0 and ⟨ sin² θ ⟩ = 1/2. The phase π/3 only shifts the wave, so it does not change these averages. That gives ⟨ i² ⟩ = 36 + 0 + 56/2 = 36 + 28 = 64 A².
3. So. iᵣₘₛ = √64 = 8 A.

The common slip: 11.29. Adding the DC value to the AC rms value directly, 6 + √28, gives about 11.29, not 8. The mean squares add, not the rms values.

The idea to keep. For a DC part plus a sine, the squares add, not the currents: iᵣₘₛ = √(I_(dc)² + I₀²/2). The cross term always averages to zero over a cycle.

Answer from the NTA final answer key. Solution by PrepWiser.