An ac current is represented as
i=5 √2+10 cos (650 π t+π/6) A m p
The r.m.s value of the current is

Solution

Physics sir · 3 steps

Square the current, average it over a cycle, then take the root. The cross term averages to zero.

1. Two parts. The current is a steady (dc) part I₀ = 5√2 A plus an oscillating part with amplitude 10 A. The rms value is √(⟨ i² ⟩), the square root of the mean of i² over one cycle.
2. Average each term. Squaring gives three terms. The dc term averages to (5√2)² = 50. The cos² term averages to 10²/2 = 50. The cross term 2 × 5√2 × 10 cos(…) averages to zero, because cos is positive and negative for equal times.

   iᵣₘₛ² = I₀² + Iₘ²/2 = 50 + 50 = 100 A²

3. So. iᵣₘₛ = √100 = 10 A, which is option (a).

The idea to keep. For dc plus a sinusoid, add the squares, do not add the currents: iᵣₘₛ = √(I_(dc)² + Iₘ²/2). The product term always averages out over a full cycle.

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