Question

In \[R - L - C\]series circuit, the potential difference across each circuit is $20V$. Now the value of resistance alone is doubled, the P.D across $R$, $L$and $C$respectively
A) \[20V,10V,10V\]
B) \[20V,20V,20V\]
C) \[20V,40V,40V\]
D) \[10V,20V,20V\]

Answer 1

Hint-An AC circuit in which the resistor of resistance $R$, inductor of inductance $L$ and the capacitor of capacitance $C$are connected in series, and the $RLC$ series is connected across the voltage source the circuit is called as a $RLC$ series circuit. As resistance, inductance and the capacitances are connected in the circuit, the effective resistance is called Impedance. Impedance gives opposition to the flowing AC current by any of the three components like resistance, capacitance or inductance.

Formula used:

\[{V_R} = IR\]
\[{V_C} = I{X_C}\]
\[{V_L} = I{X_L}\]
Where,
\[{V_R},{V_C},{V_L}\]= The potential differences across resistance, capacitance and the inductance respectively,
\[{X_C},{X_{_L}}\]= Capacitance and inductance respectively
\[R\]= Resistance

Complete step by step answer:
(i) Considering the voltage across the resistance, capacitance and the inductance is 20V. Therefore, \[{V_R} = {V_C} = {V_L} = 20V\]
(ii) Now the value of resistance alone is doubled. If the value of resistance is doubled, the value of capacitance and the inductance will be halved. Hence
\[{V_L} = \dfrac{{I{X_L}}}{2}\]
\[{V_L} = \dfrac{{20}}{2} \Rightarrow 10V\]
\[{V_C} = \dfrac{{I{X_C}}}{2}\]
\[{V_C} = \dfrac{{20}}{2} \Rightarrow 10V\]
(iii) Therefore from the obtained solution, we can say the resistance, inductance and the capacitance are $20V$, $10V$and $10V$respectively.

Hence the correct option is A.

Additional information:
(i) $RLC$ circuits are used in filter circuits, oscillators, voltage multipliers and it is also very useful in tuning circuits of radio and TV circuits.
(ii) When the frequency of the applied ac source and the natural frequency is equal, the electrical resonance occurs. The resonance occurs in a certain frequency called resonant frequency.

Note:The resonance occurs only in the circuit having inductance and the capacitance. When the voltage across the capacitance and the inductances cancel each other, which means they are \[180^\circ \] out of phase with each other, the circuit is said to be purely resistive. And no current can flow through it.