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NEET-XII-Physics

39: Alternating Current

with Solutions - page 5
Qstn# iv-16 Prvs-QstnNext-Qstn
  • #16
    An inductance of 2.0 H, a capacitance of 18μF and a resistance of 10 kΩ are connected to an AC source of 20 V with adjustable frequency. (a) What frequency should be chosen to maximise the current in the circuit? (b) What is the value of this maximum current? (a) What frequency should be chosen to maximise the current in the circuit? (b) What is the value of this maximum current? (a) What frequency should be chosen to maximise the current in the circuit? (b) What is the value of this maximum current?
    Ans : Given:
    Inductance of inductor, L = 2.0 H
    Capacitance of capacitor, C = 18 μF
    Resistance of resistor, R = 10 kΩ
    Voltage of AC source, E = 20 V (a) In an LCR circuit, current is maximum when reactance is minimum, which occurs at resonance, i.e. when capacitive reactance becomes equal to the inductive reactance,i.e.
    XL = XC
    `` \Rightarrow \omega L=\frac{1}{\omega C}``
    `` \Rightarrow {\omega }^{2}=\frac{1}{LC}=\frac{1}{2\times 18\times {10}^{-6}}``
    `` \Rightarrow {\omega }^{2}=\frac{{10}^{6}}{36}``
    `` \Rightarrow \omega =\frac{{10}^{3}}{6}``
    `` \Rightarrow 2\,\mathrm{\,\pi \,}f=\frac{{10}^{3}}{6}``
    `` \Rightarrow f=\frac{1000}{6\times 2\,\mathrm{\,\pi \,}}=26.539\,\mathrm{\,Hz\,}``
    `` =27\,\mathrm{\,Hz\,}`` (b) At resonance, reactance is minimum.
    Minimum Reactance, Z = R
    Maximum current (I) is given by,
    `` I=\frac{E}{R}``
    `` \Rightarrow I\mathit{=}\frac{20}{10\times {10}^{3}}``
    `` \Rightarrow I=\frac{2\,\mathrm{\,A\,}}{{10}^{3}}=2\,\mathrm{\,mA\,}``
    Page No 330: (a) In an LCR circuit, current is maximum when reactance is minimum, which occurs at resonance, i.e. when capacitive reactance becomes equal to the inductive reactance,i.e.
    XL = XC
    `` \Rightarrow \omega L=\frac{1}{\omega C}``
    `` \Rightarrow {\omega }^{2}=\frac{1}{LC}=\frac{1}{2\times 18\times {10}^{-6}}``
    `` \Rightarrow {\omega }^{2}=\frac{{10}^{6}}{36}``
    `` \Rightarrow \omega =\frac{{10}^{3}}{6}``
    `` \Rightarrow 2\,\mathrm{\,\pi \,}f=\frac{{10}^{3}}{6}``
    `` \Rightarrow f=\frac{1000}{6\times 2\,\mathrm{\,\pi \,}}=26.539\,\mathrm{\,Hz\,}``
    `` =27\,\mathrm{\,Hz\,}`` (b) At resonance, reactance is minimum.
    Minimum Reactance, Z = R
    Maximum current (I) is given by,
    `` I=\frac{E}{R}``
    `` \Rightarrow I\mathit{=}\frac{20}{10\times {10}^{3}}``
    `` \Rightarrow I=\frac{2\,\mathrm{\,A\,}}{{10}^{3}}=2\,\mathrm{\,mA\,}``
    Page No 330: (a) In an LCR circuit, current is maximum when reactance is minimum, which occurs at resonance, i.e. when capacitive reactance becomes equal to the inductive reactance,i.e.
    XL = XC
    `` \Rightarrow \omega L=\frac{1}{\omega C}``
    `` \Rightarrow {\omega }^{2}=\frac{1}{LC}=\frac{1}{2\times 18\times {10}^{-6}}``
    `` \Rightarrow {\omega }^{2}=\frac{{10}^{6}}{36}``
    `` \Rightarrow \omega =\frac{{10}^{3}}{6}``
    `` \Rightarrow 2\,\mathrm{\,\pi \,}f=\frac{{10}^{3}}{6}``
    `` \Rightarrow f=\frac{1000}{6\times 2\,\mathrm{\,\pi \,}}=26.539\,\mathrm{\,Hz\,}``
    `` =27\,\mathrm{\,Hz\,}`` (b) At resonance, reactance is minimum.
    Minimum Reactance, Z = R
    Maximum current (I) is given by,
    `` I=\frac{E}{R}``
    `` \Rightarrow I\mathit{=}\frac{20}{10\times {10}^{3}}``
    `` \Rightarrow I=\frac{2\,\mathrm{\,A\,}}{{10}^{3}}=2\,\mathrm{\,mA\,}``
    Page No 330:
  • #16-a
    What frequency should be chosen to maximise the current in the circuit?
    Ans : In an LCR circuit, current is maximum when reactance is minimum, which occurs at resonance, i.e. when capacitive reactance becomes equal to the inductive reactance,i.e.
    XL = XC
    `` \Rightarrow \omega L=\frac{1}{\omega C}``
    `` \Rightarrow {\omega }^{2}=\frac{1}{LC}=\frac{1}{2\times 18\times {10}^{-6}}``
    `` \Rightarrow {\omega }^{2}=\frac{{10}^{6}}{36}``
    `` \Rightarrow \omega =\frac{{10}^{3}}{6}``
    `` \Rightarrow 2\,\mathrm{\,\pi \,}f=\frac{{10}^{3}}{6}``
    `` \Rightarrow f=\frac{1000}{6\times 2\,\mathrm{\,\pi \,}}=26.539\,\mathrm{\,Hz\,}``
    `` =27\,\mathrm{\,Hz\,}``
  • #16-b
    What is the value of this maximum current?
    Ans : At resonance, reactance is minimum.
    Minimum Reactance, Z = R
    Maximum current (I) is given by,
    `` I=\frac{E}{R}``
    `` \Rightarrow I\mathit{=}\frac{20}{10\times {10}^{3}}``
    `` \Rightarrow I=\frac{2\,\mathrm{\,A\,}}{{10}^{3}}=2\,\mathrm{\,mA\,}``
    Page No 330: