JEE Main 2025 Jan 24 Shift 1, Physics Q16: Introduction to Alternating Currents and Circuits

JEE Main2025Jan 24, Shift 1Physics

An alternating current is given by $I = I_{A} sin ω t + I_{B} cos ω t$ . The r.m.s. current will be :-

A
$I_{A}^{2} + I_{B}^{2}$
B
$\frac{I _{A}^{2} + I _{B}^{2}}{2}$
C
$\frac{I _{A}^{2} + I _{B}^{2}}{2}$
D
$\frac{∣ I _{A} + I _{B} ∣}{2}$

Solution

The sum of a sine and a cosine of the same frequency is a single sinusoid:

$I_{A} sin ω t + I_{B} cos ω t = I_{0} sin (ω t + ϕ)$ , where $I_{0} = I_{A}^{2} + I_{B}^{2}$ .

The r.m.s. value of a sinusoid is its peak value divided by $2$ :

$I_{r m s} = \frac{I _{0}}{2} = \frac{I _{A}^{2} + I _{B}^{2}}{2}$

Cross-check by direct averaging: $⟨ sin^{2} ω t ⟩ = ⟨ cos^{2} ω t ⟩ = \frac{1}{2}$ and $⟨ sin ω t cos ω t ⟩ = 0$ , so $⟨ I^{2} ⟩ = \frac{I _{A}^{2} + I _{B}^{2}}{2}$ .

$∴$ Option (3).

Concept behind this question

Introduction to Alternating Currents and CircuitsNotes, formulas and examples →

Practice more Physics

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →

JEE Main 2025 Jan 24 Shift 1, Physics Q16: Introduction to Alternating Currents and Circuits

Concept behind this question

More previous year questions on Introduction to Alternating Currents and Circuits

Practice more Physics