A circuit connected to an ac source of emf $e=e_0 \sin (1000 t)$ with t in seconds, gives a phase difference of $\frac{\pi}{4}$ between the emf e and current i. Which of the following circuits will exhibit this?

(D) RC circuit with $R=1 k\Omega$ and $C=10 \mu F$

(A) RC circuit with $R=1 k\Omega$ and $C=1 \mu F$

(B) RL circuit with R $=1 k\Omega$ and L $=10 mH$

(C) RL circuit with R $=1 k\Omega$ and L $=1 mH$

JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Alternating Current

A circuit connected to an ac source of emf

e = e_{0} sin (1000 t)

with t in seconds, gives a phase difference of

\frac{π}{4}

between the emf e and current i. Which of the following circuits will exhibit this?

Answer: (D)

A phase difference of

\frac{π}{4}

between emf and current needs

tan \frac{π}{4} = 1

, which means the reactance must equal the resistance,

X = R

. The source gives

ω = 1000 rad/s

and every choice has

R = 1 k Ω

. Test the capacitive options through

X_{C} = \frac{1}{ω C}

: with

C = 10 μ F

we get

X_{C} = \frac{1}{1000 \times 10 \times 1 0 ^{- 6}} = \frac{1}{0.01} = 1000 Ω

, exactly

R

. The others miss, since

C = 1 μ F

gives

1 0^{4} Ω

and the inductive choices give

X_{L} = ω L

of only

10 Ω

1 Ω

. So the circuit that produces the required

\frac{π}{4}

is the RC combination with

R = 1 k Ω

and

C = 10 μ F

JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Single correct option