Assertion (A): If an iron rod is inserted into a steady current carrying solenoid, the current in solenoid decreases.
Reason (R): Magnetic flux linked with solenoid increases.
Inserting an iron rod increases magnetic flux \( \Phi \). By Lenz's law, this induces an opposing \( \text{EMF} \), causing current \( \text{I} \) to decrease during insertion.
Assertion (A): ac current flows through a bulb and a solenoid connected in series. If a soft iron core is inserted in the solenoid, the bulb glows much brighter.
Reason (R): The inductance of solenoid decreases on inserting soft iron core in it.
Inserting a soft iron core increases solenoid inductance \(L\), thus increasing inductive reactance \(X_L = omega L\). This increases circuit impedance \(Z\), reducing current \(I = V/Z\) and making the bulb dimmer. Both Assertion (A) and Reason (R) are false.
Assertion (A): In a series \(LCR\) circuit at resonance, the voltage across the capacitor or inductor may be more than the applied voltage.
Reason (R): At resonance in a series \(LCR\) circuit, the voltages across inductor and capacitor are out of phase.
At resonance, \(V_L = V_C\) but they are \(180^\circ\) out of phase. The applied voltage is \(V = IR\). Due to voltage magnification (high \(Q\) factor), \(V_L\) or \(V_C\) can be much greater than \(V\). Reason is true, but it doesn't explain *why* they can be larger than applied voltage, it explains why they cancel out to make \(V=IR\).
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.
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): 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.
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.
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.
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.
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.
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.
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.