Question

For a R-L-C series circuit under resonance :
A. Current in the circuit is in phase with the applied voltage
B. Voltage drop across capacitor \(C\) and inductor \(L\) are equal in magnitude
C. Voltage across the capacitor is equal in magnitude to the applied voltage
D. Current in the circuit is maximum Choose the correct answer from the options given below :

• A. A only
• B. B, D and E only
• C. A, B and D only
• D. C, D and E only

Hint:

At resonance in a series RLC circuit: \[ X_L=X_C \] Hence:
• impedance is minimum,
• current is maximum,
• power factor becomes unity,
• current and voltage are in phase.

Answer 1

The Correct Option is C

Concept: In a series RLC circuit, resonance occurs when: \[ X_L=X_C \] where:
• \(X_L=\omega L\) is inductive reactance,
• \(X_C=\dfrac{1}{\omega C}\) is capacitive reactance. At resonance:
• net reactance becomes zero,
• impedance becomes minimum,
• current becomes maximum,
• current and voltage become in phase. The resonance frequency is: \[ f_r=\frac{1}{2\pi\sqrt{LC}} \]

Step 1: Checking statement \(A\). Statement \(A\): \[ \text{Current in the circuit is in phase with the applied voltage} \] At resonance: \[ X_L=X_C \] Hence net reactance: \[ X=X_L-X_C=0 \] Thus impedance becomes purely resistive: \[ Z=R \] In a purely resistive circuit: \[ \text{Current and voltage are in phase} \] Hence statement \(A\) is correct.

Step 2: Checking statement \(B\). Statement \(B\): \[ \text{Voltage drop across capacitor and inductor are equal in magnitude} \] At resonance: \[ X_L=X_C \] Therefore: \[ V_L=IX_L \] and \[ V_C=IX_C \] Since: \[ X_L=X_C \] we get: \[ V_L=V_C \] Hence statement \(B\) is correct.

Step 3: Checking statement \(C\). Statement \(C\): \[ \text{Voltage across capacitor is equal to applied voltage} \] This is not necessarily true. In resonance condition:
• \(V_L\) and \(V_C\) may individually become much larger than source voltage,
• they cancel each other because they are \(180^\circ\) out of phase. Thus capacitor voltage is not necessarily equal to supply voltage. Hence statement \(C\) is incorrect.

Step 4: Checking statement \(D\). Statement \(D\): \[ \text{Current in the circuit is maximum} \] At resonance: \[ Z=R \] Since impedance becomes minimum: \[ I=\frac{V}{R} \] Thus current reaches maximum value. Hence statement \(D\) is correct.

Step 5: Selecting the correct option. Correct statements are: \[ A,\;B,\;D \] Hence the correct option is: \[ \boxed{(3)\; A,B,D\text{ only}} \]