Assertion (A): Peak voltage across the resistance can be greater than the peak voltage of the source in a series \(LCR\) circuit.
Reason (R): Peak voltage across the inductor can be greater than the peak voltage of the source in an series \(LCR\) circuit.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Peak voltage across resistor is \(V_R = I_0 R\). Peak source voltage is \(V_0 = I_0 Z\). Since \(Z \ge R\), \(V_R \le V_0\). So A is false. Peak voltage across inductor is \(V_L = I_0 X_L\). At resonance, if \(X_L > R\), then \(V_L\) can be greater than \(V_0\) (voltage magnification). So R is true. Thus, A is false and R is true.
Assertion (A): The power rating of an element in \(AC\) circuit refers to average power rating.
Reason (R): A given value for \(AC\) voltage or current is usually its average value.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Power rating of \(AC\) devices always refers to average power. However, \(AC\) voltage or current values (e.g., 220V) are typically Root Mean Square (RMS) values, not average values. For a full cycle, the average value of sinusoidal \(AC\) is zero.
Assertion (A): Average power consumed in a circuit is never negative.
Reason (R): Instantaneous power is always positive.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Average power consumed by passive circuits is non-negative. Instantaneous power \(p = vi\) can be negative during parts of an \(AC\) cycle, especially in reactive circuits, when energy is temporarily returned to the source.
Assertion (A): At an airport, a person is made to walk through the doorway of a metal detector.
Reason (R): Metal detector works on the principle of resonance in \(AC\) circuits.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Metal detectors use \(LC\) resonant circuits. When a metal object enters the coil's magnetic field, it changes the coil's inductance, altering the circuit's resonant frequency and triggering detection.
Assertion (A): Smaller the band width, sharper the resonance and easier it is to tune an \(LCR\) circuit.
Reason (R): Resonant frequency is arithmetic mean of half power frequencies.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Smaller bandwidth implies a higher quality factor \(Q\) and sharper resonance, leading to better frequency selectivity (easier tuning). Resonant frequency \(\omega_0\) is the *geometric mean* \(sqrt{\omega_1 \omega_2}\,\) not the arithmetic mean of half-power frequencies.
Assertion (A): At resonance in AC circuits current and emf are in phase.
Reason (R): At resonance in AC circuits, current is maximum.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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At resonance, the inductive reactance \(X_L\) equals the capacitive reactance \(X_C\), making the total impedance purely resistive \(Z=R\). This results in the current and emf being in phase. Since impedance is minimal \(Z=R\), the current is maximal. Therefore, R is the correct explanation of A.
Assertion (A): At frequency greater than resonant frequency, circuit is inductive in nature.
Reason (R): Reciprocal of reactance is called susceptance.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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For frequencies greater than resonant frequency, inductive reactance \(X_L = \omega L\) becomes greater than capacitive reactance \(X_C = 1/(\omega C)\), making the circuit inductive. The reciprocal of reactance is defined as susceptance. Both statements are true, but R does not explain A.
Assertion (A): If the resistance of a series resonant LCR circuit is decreased, then the peak current versus frequency graph will be taller and narrower.
Reason (R): If the resistance of a series resonant LCR circuit decreased, then its resonance will be unaffected.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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When resistance \(R\) in a series \(LCR\) circuit decreases, the peak current \(I_{max} = V/R\) at resonance increases, making the peak taller. The quality factor \(Q = (\omega_0 L)/R\) increases, leading to a narrower resonance curve. The resonant frequency \(omega_0 = 1/\sqrt{LC}\) remains unchanged, but the overall resonance behavior (sharpness, peak current) is affected. Therefore, (A) is true and (R) is false.
Assertion (A): The impedance of series L-C-R circuit can be greater, equal or less than the resistance.
Reason (R): The minimum impedance of series LCR circuit depends over angular frequency of applied emf.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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The impedance of a series \(LCR\) circuit is given by \(Z = \sqrt{R^2 + (X_L - X_C)^2}\). Since \((X_L - X_C)^2\) is always non-negative, \(Z\) is always greater than or equal to \(R\). Thus, (A) is false. The minimum impedance occurs at resonance, where \(Z_{min} = R\). This minimum value depends only on \(R\) and not on the angular frequency \(omega\). Thus, (R) is also false.
Assertion (A): A capacitor of suitable capacitance can be used in an A.C. circuit in place of the choke coil.
Reason (R): A capacitor blocks D.C. and allows A.C. only.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
A choke coil is an inductor with high inductance and low resistance, used to limit AC current without much power loss. A capacitor also provides reactance \(X_C = 1/(\omega C)\) in an AC circuit, limiting AC current without dissipating significant power. Thus, a capacitor of suitable capacitance can indeed replace a choke coil for AC current limiting applications, so (A) is true. Reason (R) states that a capacitor blocks DC and allows AC, which is a fundamental property of a capacitor. This property (allowing AC) is why it can function as a reactive element in AC circuits, including current limiting, similar to a choke coil. Therefore, (R) is the correct explanation for (A).