Problem 57 - Master Resource Book in JEE Main Physics

Question

Assertion In series \(L-C-R\) circuit resonance can take place. Reason Resonance takes place if inductive and capacitive reactances are equal and opposite.

Step-by-Step Solution

Answer

The assertion is true because resonance occurs when inductive and capacitive reactances are equal and opposite, as stated in the reason.

Step 1: Understanding the Assertion

An L-C-R circuit (inductor, capacitor, resistor) can resonate when certain conditions are met. Resonance in the circuit is achieved when the overall impedance is minimized, usually implying that the circuit is able to pass the maximum current for a given voltage.

Step 2: Understanding the Reason

Resonance occurs in an L-C-R circuit when the inductive reactance ( X_L = ωL ) equals the capacitive reactance ( X_C = 1/( ωC) ). At this point, these reactances cancel each other out because they are equal in magnitude and opposite in phase.

Step 3: Formulating Resonance Condition

For resonance to occur, set the inductive reactance equal to the capacitive reactance: X_L = X_C. This gives the equation ωL = 1/( ωC). Solving for ω (angular frequency) gives the resonant frequency: ω = 1/ √( LC).

Step 4: Conclusion

The assertion that resonance can happen in an L-C-R circuit holds true because when the inductive and capacitive reactances are equal and opposite, resonance occurs. Therefore, the reason is also correct as it provides the condition under which resonance takes place.

Key Concepts

Inductive Reactance

Inductive reactance is a key property of inductors within an LCR circuit, directly affecting the circuit's resonance behavior. It is represented by the symbol \(X_L\). This property measures how much an inductor opposes the change in current through it. The formula for inductive reactance is \(X_L = \omega L\), where \(\omega\) is the angular frequency and \(L\) is the inductance of the coil. Inductive reactance increases linearly with frequency:

• The higher the frequency of the alternating current, the larger the inductive reactance.
• Conversely, at lower frequencies, the reactance is smaller.

This opposition changes phase with frequency, influencing how the circuit responds to different signals.

Capacitive Reactance

Capacitive reactance, designated by \(X_C\), is the property of capacitors that affects how they impede alternating current in an LCR circuit. It defines how much a capacitor resists the flow of alternating current at a certain frequency. Mathematically, it is given by the formula \(X_C = \frac{1}{\omega C}\), where \(C\) is the capacitance and \(\omega\) is the angular frequency.

• As the frequency increases, the capacitive reactance decreases.
• Similarly, if the frequency decreases, the reactance increases.

This inverse relationship with frequency means that capacitors let more current pass with increasing frequency, thereby interacting with the circuit dynamics in complex ways.

Resonant Frequency

Resonant frequency is the specific frequency at which an LCR circuit naturally oscillates with maximum amplitude. It is a critical condition that ensures the circuit's efficient operation by minimizing impedance. At this frequency, the inductive reactance \((X_L)\) and the capacitive reactance \((X_C)\) are equal and opposite, effectively canceling each other out. The formula for determining this frequency is \[\omega = \frac{1}{\sqrt{LC}}\].

• This is where the impedance of the entire circuit is at its lowest, allowing maximum current to flow.
• At resonant frequency, the circuit behaves like a pure resistive circuit without reactive components.

Resonant frequency is a foundational concept in tuning circuits for specific frequencies in applications like radio transmission and reception.

Impedance Minimization

Impedance minimization in an LCR circuit plays a vital role when achieving resonance. Impedance is the total opposition a circuit presents to the flow of alternating current, represented by \(Z\). For an LCR circuit, the expression for impedance combines resistive (\(R\)) and reactive components (inductive and capacitive reactance) as \(Z = \sqrt{R^2 + (X_L - X_C)^2}\).

• Resonance is achieved when \(X_L = X_C\), thus \(Z\) is minimized to simply \(R\).
• This results in the largest possible current for a given voltage.

Impedance minimization ensures efficient power transfer and is crucial in designing devices like tuners and filters to optimize performance at specific frequencies. Understanding this concept is crucial for engineering circuits that need to handle high frequencies effectively.