In an a.c. circuit, the instantaneous e.m.f. and current are given by \[ \begin{aligned} & e=100 \sin 30 t \\ & i=20 \sin \left(30 t-\frac{\pi}{4}\right) \end{aligned} \] In one cycle of a.c., the average power consumed by the circuit and the wattless current are, respectively :

(C) $\frac{1000}{\sqrt{2}}, 10$

JEE Main · 2018 · 08 Apr 2018 · Offline · Alternating Current

In an a.c. circuit, the instantaneous e.m.f. and current are given by

e = 100 sin 30 t i = 20 sin (30 t - \frac{π}{4})

In one cycle of a.c., the average power consumed by the circuit and the wattless current are, respectively :

Answer: (C)

From the given expressions the peak emf is

100

V, the peak current is

20

A and the current lags the voltage by

ϕ = \frac{π}{4}

. The average power over a cycle uses the peak values and the power factor,

⟨ P ⟩ = \frac{ε _{0} I _{0}}{2} cos ϕ = \frac{100 \times 20}{2} \times \frac{1}{2} = \frac{1000}{2}

W. The wattless current is the part of the rms current in quadrature with the voltage,

I_{r m s} sin ϕ = \frac{20}{2} \times \frac{1}{2} = 10

A, so the pair is

\frac{1000}{2}

and

10

JEE Main · 2018 · 08 Apr 2018 · Offline · Single correct option