A resistance of 40 Ω is connected to a source of alternating current rated 220 V, 50 Hz. Find the time taken by the current to change from its maximum value to the rms value :

Solution

Physics sir · 4 steps

Ignore the 40 ohm and 220 V; only the frequency matters, because this is a phase question.

1. Start at the peak. In a pure resistor the current is i = I₀ cos ω t, so it is at its maximum I₀ at t = 0. The resistance and voltage only set the size of I₀, not the timing.
2. Find the phase. The rms value is I₀/√2. Set cos ω t = 1/√2, which first happens at ω t = π/4.

   t = (π/4)/ω = (π/4)/(2π f) = 1/8f

3. Put in numbers. With f = 50 Hz, t = 1/(8 × 50) = 1/400 s = 2.5 × 10⁻³ s. That is one-eighth of the period T = 20 ms.
4. So. The current takes 2.5 ms to fall from its maximum to its rms value, which is option (a).

The idea to keep. Peak to rms is a phase change of π/4, so it always takes T/8, whatever the resistance or voltage.

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