Problem 27 - Physics

Question

The resonant frequency of a series \(\mathrm{RCL}\) circuit is \(9.3 \mathrm{kHz}\). The inductance and capacitance of the circuit are each tripled. What is the new resonant frequency?

Step-by-Step Solution

Answer

The new resonant frequency is 3.1 kHz.

Step 1: Understanding the Formula

The resonant frequency of a series RCL circuit is given by the formula \( f_0 = \frac{1}{2\pi\sqrt{LC}} \), where \( L \) is the inductance and \( C \) is the capacitance.

Step 2: Applying Changes to L and C

Since both the inductance \( L \) and the capacitance \( C \) are tripled, the new values become \( 3L \) and \( 3C \) respectively.

Step 3: Calculating the New Resonant Frequency

Substitute \( 3L \) and \( 3C \) into the formula for resonant frequency: \[ f_0' = \frac{1}{2\pi\sqrt{3L \cdot 3C}} = \frac{1}{2\pi\sqrt{9LC}} = \frac{1}{3} \cdot \frac{1}{2\pi\sqrt{LC}} = \frac{1}{3}f_0 \]Therefore, the new resonant frequency is one-third of the original resonant frequency.

Step 4: Final Calculation

Since the original resonant frequency \( f_0 \) is \( 9.3 \text{ kHz} \), the new resonant frequency \( f_0' \) is \[ f_0' = \frac{1}{3} \times 9.3 \text{ kHz} = 3.1 \text{ kHz} \].

Key Concepts

RCL Circuit

An RCL circuit is a fundamental component of electrical engineering, consisting of three basic elements: a resistor (R), a capacitor (C), and an inductor (L). These elements are connected in series, forming a single loop through which the electric current flows. The circuit's behavior changes depending on the frequency of the supplied voltage, and it's commonly used in tuning applications like radio receivers.

In an RCL circuit, each component plays a critical role:

• The resistor dampens the current flow, influencing the overall resistance.
• The capacitor stores energy in the electric field, introducing reactance.
• The inductor stores energy in the magnetic field, also contributing reactance.

These elements interact with each other to produce a resonant frequency, which is the frequency at which the circuit naturally oscillates with the minimum energy loss.

Inductance

Inductance is a fundamental property of an inductor in an electric circuit. It is the measure of an inductor's ability to store energy in its magnetic field. The unit of inductance is the henry (H), named after Joseph Henry. The inductor resists changes in current, causing a phase shift in the current when alternating current is used.

Key facts about inductance include:

• It is proportional to the turns of the coil and the permeability of the core.
• Inductance is directly related to the strength of the magnetic field created.
• An inductor's opposition to current change is called inductive reactance.

When the inductance in a circuit is altered, it affects the resonant frequency, as seen when we triple the inductance in an RCL circuit.

Capacitance

Capacitance is the ability of a capacitor to store electrical energy in an electric field. It is measured in farads (F), and it quantifies the amount of charge stored per unit voltage. Capacitors are essential in circuits for functions like filtering, oscillation, and coupling.

Important aspects of capacitance are:

• Capacitance depends on the surface area of the plates, the distance between them, and the dielectric material.
• When voltage is applied, capacitors get charged, storing energy.
• The voltage stored across a capacitor influences the current flow in a circuit.

Changes in capacitance, such as tripling it, can significantly alter the resonant frequency in an RCL circuit, influencing its operation and efficiency.

Formula for Resonant Frequency

The resonant frequency is the frequency at which an RCL circuit oscillates with minimum energy loss. It is a crucial concept for applications like radio tuning, setting the circuit to a preferred frequency. The formula to determine the resonant frequency is:\[ f_0 = \frac{1}{2\pi\sqrt{LC}} \]Here, \( L \) is the inductance and \( C \) is the capacitance. This formula tells us how these components interact to determine the natural frequency of the circuit.

Key points to note with this formula include:

• If either inductance or capacitance increases, the resonance frequency decreases.
• The formula involves the square root of the product of \( L \) and \( C \), contributing to its sensitivity to changes.
• Altering \( L \) and \( C \) by a factor affects the frequency accordingly, as demonstrated by tripling both to achieve a new frequency.

This understanding allows engineers to design circuits that function optimally at desired frequencies.