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Alternating Current question

2022 · 27 Jul · Shift 2 · Q52
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  5. /2022 · 27 Jul · Shift 2 · Q52

Alternating Current question

2022 · 27 Jul · Shift 2 · Q52

JEE MainPhysicsAlternating CurrentMCQ+4 / −1
A series LCR circuit has L=0.01 H,R=10 Ω\mathrm{L}=0.01\, \mathrm{H}, \mathrm{R}=10\, \OmegaL=0.01H,R=10Ω and C=1μF\mathrm{C}=1 \mu \mathrm{F}C=1μF and it is connected to ac voltage of amplitude (Vm)50 V\left(\mathrm{V}_{\mathrm{m}}\right) 50 \mathrm{~V}(Vm​)50 V. At frequency 60%60 \%60% lower than resonant frequency, the amplitude of current will be approximately :
  1. A
    466 mA
  2. B
    312 mA
  3. C
    238 mA
  4. D
    196 mA
View written solutionFree

Correct answer: C

  1. Given data

    L=0.01 H,R=10 Ω,C=1 μF=10−6 FL=0.01\,\text{H},\quad R=10\,\Omega,\quad C=1\,\mu\text{F}=10^{-6}\,\text{F}L=0.01H,R=10Ω,C=1μF=10−6F Vm=50 VV_m=50\,\text{V}Vm​=50V

    The circuit is driven at a frequency 60% lower than the resonant frequency.

  2. Resonant angular frequency

    For a series LCR circuit, ω0=1LC\omega_0=\frac{1}{\sqrt{LC}}ω0​=LC​1​

    Compute: LC=0.01×10−6=10−8LC=0.01\times 10^{-6}=10^{-8}LC=0.01×10−6=10−8 LC=10−4\sqrt{LC}=10^{-4}LC​=10−4 ω0=110−4=104 rad/s\omega_0=\frac{1}{10^{-4}}=10^4\,\text{rad/s}ω0​=10−41​=104rad/s

  3. Operating frequency

    “60% lower than resonant frequency” means ω=0.4 ω0=0.4×104=4000 rad/s\omega=0.4\,\omega_0=0.4\times 10^4=4000\,\text{rad/s}ω=0.4ω0​=0.4×104=4000rad/s

  4. Reactances at this frequency

    Inductive reactance: XL=ωL=4000×0.01=40 ΩX_L=\omega L=4000\times 0.01=40\,\OmegaXL​=ωL=4000×0.01=40Ω

    Capacitive reactance: XC=1ωC=14000×10−6=14×10−3=250 ΩX_C=\frac{1}{\omega C}=\frac{1}{4000\times 10^{-6}}=\frac{1}{4\times 10^{-3}}=250\,\OmegaXC​=ωC1​=4000×10−61​=4×10−31​=250Ω

    Net reactance: X=XL−XC=40−250=−210 ΩX=X_L-X_C=40-250=-210\,\OmegaX=XL​−XC​=40−250=−210Ω

  5. Impedance of the series circuit

    Z=R2+(XL−XC)2Z=\sqrt{R^2+(X_L-X_C)^2}Z=R2+(XL​−XC​)2​ Z=102+2102Z=\sqrt{10^2+210^2}Z=102+2102​ Z=100+44100Z=\sqrt{100+44100}Z=100+44100​ Z=44200≈210.24 ΩZ=\sqrt{44200}\approx 210.24\,\OmegaZ=44200​≈210.24Ω

  6. Current amplitude

    Im=VmZ=50210.24≈0.238 AI_m=\frac{V_m}{Z}=\frac{50}{210.24}\approx 0.238\,\text{A}Im​=ZVm​​=210.2450​≈0.238A

    Im≈238 mAI_m\approx 238\,\text{mA}Im​≈238mA

  7. Option check

    • A: 466 mA466\,\text{mA}466mA ❌
    • B: 312 mA312\,\text{mA}312mA ❌
    • C: 238 mA238\,\text{mA}238mA ✅
    • D: 196 mA196\,\text{mA}196mA ❌

Therefore, the correct option is C.

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