An alternating current is given by the equation i = i₁ sin ωt + i₂ cos ωt. The rms current will be :

Solution

Physics sir · 4 steps

Square the current and average it over a cycle: the cross term sin ω t cos ω t averages to zero.

1. Start from the definition. The rms current is the square root of the average of i² over one full cycle. So first find ⟨ i² ⟩.

   i² = i₁²sin² ω t + i₂²cos² ω t + 2i₁ i₂sin ω t cos ω t

2. Average each term. Over a full cycle ⟨ sin² ω t ⟩ = ⟨ cos² ω t ⟩ = 1/2. The cross term is i₁ i₂ sin 2ω t, which averages to zero. So ⟨ i² ⟩ = 1/2(i₁² + i₂²).
3. Same result, another way. The two terms combine into a single sine wave i = I₀sin(ω t + φ) with amplitude I₀ = √(i₁² + i₂²), because sine and cosine are 90° apart. Then iᵣₘₛ = I₀/√2.
4. So. iᵣₘₛ = √(⟨ i² ⟩) = 1/√2(i₁² + i₂²)^(1/2), which is option (a).

The idea to keep. Sine and cosine of the same frequency are 90° apart, so their amplitudes add like perpendicular vectors, √(i₁² + i₂²). Then divide the combined peak by √2.

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