Problem 64 - Master Resource Book in JEE Main Physics

Question

The reduce the resonant frequency in an \(L-C-R\) series circuit with a generator \(\quad\) [NCERT Exemplar] (a) the generator frequency should be reduced (b) another capacitor should be added in parallel to the first [c) the iron core of the inductor should be removed (d) dielectric in the capacitor should be removed

Step-by-Step Solution

Answer

Option (b) is correct: Adding a capacitor in parallel reduces the resonant frequency.

Step 1: Understanding the Resonant Frequency Formula

In an L-C-R series circuit, the resonant frequency \( f_0 \) is given by the formula \( f_0 = \frac{1}{2\pi\sqrt{LC}} \), where \( L \) is the inductance and \( C \) is the capacitance. The goal is to reduce this frequency.

Step 2: Evaluating Generator Frequency Adjustment

Option (a) suggests reducing the generator frequency. However, the generator frequency does not affect the resonant frequency of the circuit as the resonant frequency is a property of the components \( L \) and \( C \) in the circuit.

Step 3: Investigating the Effect of Adding a Capacitor

Option (b): Adding another capacitor in parallel to the first increases the total capacitance \( C \). The formula for total capacitance in parallel is \( C_{total} = C_1 + C_2 \). Increasing \( C \) results in lower \( f_0 \) as seen from \( f_0 = \frac{1}{2\pi\sqrt{LC}} \).

Step 4: Examining the Impact of Removing the Iron Core

Option (c): Removing the iron core from the inductor decreases its inductance \( L \). A lower \( L \) results in a higher resonant frequency. Therefore, removing the iron core does not help in reducing \( f_0 \).

Step 5: Assessing the Effect of Removing Dielectric

Option (d): Removing the dielectric from a capacitor decreases its capacitance \( C \), which causes an increase in the resonant frequency. Therefore, option (d) does not help to reduce \( f_0 \).

Step 6: Conclusion on Best Option

After evaluating all the options, only option (b), adding another capacitor in parallel, effectively reduces the resonant frequency by increasing the total capacitance \( C \).

Key Concepts

LCR Series Circuit

The LCR series circuit is a type of electrical circuit that consists of three key components: an inductor (L), a capacitor (C), and a resistor (R). These components are connected in a series, meaning the same current passes sequentially through each one. This configuration makes it particularly important in many electrical applications, where it can filter signals or determine the behavior of alternating current (AC) in circuits.

In an LCR circuit, the resonant frequency is a pivotal concept because it helps to understand how the circuit responds to different frequencies. At resonance, the reactive effects of the inductor and capacitor cancel each other out, leading to minimal impedance and thus a stronger circuit response at this particular frequency.

The formula to calculate the resonant frequency is derived from the properties of both the inductance and capacitance:

• Resonant frequency is given by: \( f_0 = \frac{1}{2\pi\sqrt{LC}} \)
• This formula shows that the resonant frequency depends inversely on the square root of the product of L and C.

Understanding the interplay between L, C, and resonance can help optimize circuit performance for specific electronic applications.

Inductance and Capacitance

Inductance and capacitance are fundamental electrical properties that characterize inductors and capacitors respectively. These components are crucial for the operation of LCR circuits and impact the circuit's frequency response, especially at resonance.

**Inductance (L):**

• Inductance is a property of coils that opposes changes in current. It is typically measured in henrys (H).
• The presence of a magnetic core, often iron, in an inductor can greatly enhance its inductance.
• In an LCR circuit, a higher inductance generally means a lower resonant frequency, according to the formula \( f_0 = \frac{1}{2\pi\sqrt{LC}} \).

**Capacitance (C):**

• Capacitance measures a capacitor's ability to store an electric charge and is measured in farads (F).
• The size of the plates and the material between them (the dielectric) directly impacts the capacitance.
• Larger capacitance results in a lower resonant frequency, reinforcing the relationship observed in the resonance formula.

Both inductance and capacitance are manipulated in circuits to achieve the desired performance characteristics in various electronic and signal processing scenarios.

Parallel Capacitance

Parallel capacitance involves connecting capacitors side by side so that they share the same voltage but not necessarily the same charge. Understanding how parallel capacitance works is vital because it offers a way to alter the total capacitance value in a circuit without changing the physical configuration drastically.

When capacitors are connected in parallel, the total capacitance is simply the sum of the individual capacitances. This is mathematically expressed as:

• \( C_{total} = C_1 + C_2 + C_3 + \ldots \)
• This formula highlights how adding capacitors can significantly increase the capacitance in the circuit.

The advantage of using parallel capacitors in circuits, such as the LCR series circuit, is that it allows for the resonant frequency to be reduced, as seen from the resonant frequency formula. Increasing the overall capacitance through parallel configuration effectively decreases the resonant frequency, making parallel capacitance adjustments a practical solution for achieving specific circuit performance.

Furthermore, parallel capacitors help in providing a robust & flexible method for tuning circuits without extensive reconfiguration or redesign.