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Alternating Current question

2021 · 20 Jul · Shift 1 · Q69
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Alternating Current question

2021 · 20 Jul · Shift 1 · Q69

JEE MainPhysicsAlternating CurrentNumerical+4 / −1
In an LCR series circuit, an inductor 30 mH and a resistor 1 Ω\OmegaΩ are connected to an AC source of angular frequency 300 rad/s. The value of capacitance for which, the current leads the voltage by 45 ∘^\circ∘ is 1x×10−3{1 \over x} \times {10^{ - 3}}x1​×10−3 F. Then the value of x is ‾\underline{\hspace{2cm}}​.
Numerical answer
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Correct answer: 3

  1. Given data
  • Inductance: L=30 mH=30×10−3 HL = 30\,\text{mH} = 30 \times 10^{-3}\,\text{H}L=30mH=30×10−3H
  • Resistance: R=1 ΩR = 1\,\OmegaR=1Ω
  • Angular frequency: ω=300 rad/s\omega = 300\,\text{rad/s}ω=300rad/s

We need the capacitance CCC such that current leads voltage by 45∘45^\circ45∘ in a series LCR circuit.

  1. Phase angle condition in series LCR

For a series LCR circuit,

tan⁡ϕ=XL−XCR\tan \phi = \frac{X_L - X_C}{R}tanϕ=RXL​−XC​​

where ϕ\phiϕ is the phase angle by which voltage leads current.

If current leads voltage by 45∘45^\circ45∘, then voltage lags current by 45∘45^\circ45∘, so

ϕ=−45∘\phi = -45^\circϕ=−45∘

Hence,

tan⁡ϕ=tan⁡(−45∘)=−1\tan \phi = \tan(-45^\circ) = -1tanϕ=tan(−45∘)=−1

Therefore,

XL−XCR=−1\frac{X_L - X_C}{R} = -1RXL​−XC​​=−1

Since R=1 ΩR=1\,\OmegaR=1Ω,

XL−XC=−1X_L - X_C = -1XL​−XC​=−1

or

XC−XL=1X_C - X_L = 1XC​−XL​=1
  1. Compute inductive reactance
XL=ωL=300×30×10−3=9 ΩX_L = \omega L = 300 \times 30 \times 10^{-3} = 9\,\OmegaXL​=ωL=300×30×10−3=9Ω

So,

XC−9=1X_C - 9 = 1XC​−9=1 XC=10 ΩX_C = 10\,\OmegaXC​=10Ω
  1. Use capacitive reactance formula
XC=1ωCX_C = \frac{1}{\omega C}XC​=ωC1​

Thus,

10=1300C10 = \frac{1}{300C}10=300C1​ 300C=110300C = \frac{1}{10}300C=101​ C=13000 FC = \frac{1}{3000}\,\text{F}C=30001​F

Now,

13000=13×10−3\frac{1}{3000} = \frac{1}{3} \times 10^{-3}30001​=31​×10−3

So,

C=13×10−3 FC = \frac{1}{3} \times 10^{-3}\,\text{F}C=31​×10−3F

Comparing with

C=1x×10−3 FC = \frac{1}{x} \times 10^{-3}\,\text{F}C=x1​×10−3F

we get

x=3x=3x=3
  1. Final answer
3\boxed{3}3​

The derived answer matches the stored correct answer.

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