Question

An LCR series circuit with $100\Omega $ resistance is connected to an AC source of $200V$ and angular frequency $300rad/\sec $. When only the capacitor is removed the current lags the voltage by ${60^ \circ }$. When only the inductor is removed the current leads the voltage by ${60^ \circ }$. The average power dissipated is:
$\left( A \right)6A,400W$
$\left( B \right)2A,800W$
$\left( C \right)2A,400W$
$\left( D \right)5A,400W$

Answer 1

Hint: LCR electronic circuit consists of resistor, capacitor and inductor and they are connected in series. We need to find the impedance of the capacitor and inductor. We need to find the impedance of the circuit and using it determines the current flowing through the circuit. Then we can determine the power dissipated.

Formula used:
$Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} $
$P = EI\cos \phi = EI\dfrac{R}{Z}$
$I = \dfrac{E}{Z} = \dfrac{{200}}{{100}} = 2A$
Here $Z$ is the impedance, $R$ is the resistance and ${X_L},{X_c}$ are the impedance of the inductor and capacitor.

Complete step by step answer:
The LCR electronic circuit consists of resistor, capacitor and inductor and they are connected in series. LCR circuits can act only as a resistor, inductor or as a capacitor. This circuit will also enhance the circuit. External voltage can be less than this voltage.
Resonance occurs in a circuit that is connected in series when the supply frequency causes the voltage across the inductor and capacitor to be equal. Q factor will be affected if there is resistive loss. Q factor is a unit less dimensionless quantity. Q factor can be defined as to how quickly the energy of the oscillating system decays.
When capacitor is removed,
$\Rightarrow$ $\tan 60 = \dfrac{{{X_L}}}{R}$
$\Rightarrow$ ${X_L} = \sqrt 3 R$
When inductor is removed,
$\Rightarrow$ $\tan 60 = \dfrac{{{X_C}}}{R}$
$\Rightarrow$ ${X_C} = \sqrt 3 R$
Hence ${X_c} = {X_L}$
Then the impedance is given by
$\Rightarrow$ $Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} = 100$
Then the current is
$\Rightarrow$ $I = \dfrac{E}{Z} = \dfrac{{200}}{{100}} = 2A$
Then the power is given by
$\Rightarrow$ $P = EI\cos \phi = EI\dfrac{R}{Z}$
$\Rightarrow$ $P = \left( {200} \right) \times 2 \times \dfrac{{100}}{{100}} = 400W$

Hence the correct option is $\left( C \right).$

Note: LCR circuit can act only as a resistor, inductor or as a capacitor. This circuit will also enhance the circuit. Q factor is the energy stored per unit cycle to energy dissipated per cycle. Higher the Q factor means more energy is stored. Quality factor controls the damping of oscillations. It is a dimensionless quantity.