Problem 30 - Physics

Question

A series RCL circuit has a resonant frequency of \(1500 \mathrm{~Hz}\). When operating at a frequency other than \(1500 \mathrm{~Hz}\), the circuit has a capacitive reactance of \(5.0 \Omega\) and an inductive reactance of \(30.0 \Omega .\) What are the values of (a) \(L\) and (b) \(C ?\)

Step-by-Step Solution

Answer

L is approximately 3.18 mH and C is approximately 21.22 μF.

Step 1: Understand the Relationship of Frequency and Reactance

The resonant frequency \( f_0 \) of an RCL circuit is defined as \( f_0 = \frac{1}{2\pi\sqrt{LC}} \). At the resonant frequency, capacitive (\( X_C \)) and inductive reactance (\( X_L \)) are equal. However, the problem provides reactances at a non-resonant frequency, hence \( X_C = \frac{1}{2\pi f C} \) and \( X_L = 2\pi f L \) can be used when not at resonance.

Step 2: Calculate the Difference in Reactance

The difference in reactance \( X_L - X_C = 30.0 \Omega - 5.0 \Omega = 25.0 \Omega \). This difference helps us find the resonant frequency relationship \( f_0 \) by setting equations for non-resonant frequencies.

Step 3: Use the Resonant Frequency Formula

With the resonant frequency provided as \( f_0 = 1500 \text{ Hz} \), we have the equation \( f_0 = \frac{1}{2\pi\sqrt{LC}} \). We need to now solve for either \( L \) or \( C \) using the provided reactance data.

Step 4: Determine Inductance \( L \)

Using the inductive reactance formula \( X_L = 2 \pi f L \) at the resonant frequency can give \( L = \frac{X_L}{2 \pi f} = \frac{30.0 \Omega}{2 \pi \cdot 1500 \text{ Hz}} \approx 0.00318 \text{ H or } 3.18 \text{ mH} \).

Step 5: Determine Capacitance \( C \)

Using the capacitive reactance formula \( X_C = \frac{1}{2 \pi f C} \), we rearrange for \( C = \frac{1}{2 \pi f X_C} = \frac{1}{2 \pi \cdot 1500 \text{ Hz} \cdot 5.0 \Omega} \approx 21.22 \mu\text{F} \).

Key Concepts

Resonant Frequency

In an RCL circuit, the resonant frequency is the frequency at which the inductive reactance equals the capacitive reactance. This results in the circuit exhibiting purely resistive behavior because the reactive components cancel each other out.
At resonance, the circuit tends to oscillate at its natural frequency without any external frequency input. This can be determined using the formula:

• \( f_0 = \frac{1}{2\pi\sqrt{LC}} \)

Here, \( L \) is the inductance and \( C \) is the capacitance. The resonant frequency is crucial in applications like radio receivers and filters, as it determines how the circuit responds to different frequencies.

Capacitive Reactance

Capacitive reactance represents the opposition a capacitor offers to alternating current (AC). It is given by the formula:

• \( X_C = \frac{1}{2\pi f C} \)

where \( f \) is the frequency of the applied AC, and \( C \) is the capacitance. Capacitive reactance, measured in ohms, decreases with an increase in frequency and vice versa.
This inversely proportional relationship means that at higher frequencies, capacitors allow more current to pass through, acting very much like short circuits. On the contrary, at lower frequencies, they behave more like open circuits, blocking the current.

Inductive Reactance

Inductive reactance quantitatively states how much an inductor opposes the flow of alternating current. The formula used to calculate it is:

• \( X_L = 2\pi f L \)

where \( f \) is the frequency and \( L \) is the inductance. Inductive reactance is directly proportional to frequency; thus, it increases with an increase in frequency.
At high frequencies, inductors oppose AC strongly, acting almost like open circuits. However, at low frequencies, they offer much less resistance, allowing current to flow more easily, similar to short circuits. This property is exploited in devices like transformers and chokes.

Inductance

Inductance is the property of an electrical conductor by which a change in current through it induces an electromotive force (EMF). It primarily depends on factors like coil structure and core material.
The symbol for inductance is \( L \) and the unit of measurement is Henry (H). High inductance means a large EMF is induced for a change in current, making it crucial for functions like filtering signals and energy storage.

• Inductors store energy in a magnetic field.
• They resist changes in current.

The ability to oppose changes in current makes them vital in tuning and oscillation circuits.

Capacitance

Capacitance is the ability of a component or circuit to store an electrical charge. Capacitors store energy electrostatically in an electric field when they are charged.
The capacitance is represented by the symbol \( C \) and is measured in farads (F). The capacitance value indicates how much charge a capacitor can hold at a given voltage. Key aspects of capacitance include:

• The larger the capacitance, the more charge it can store.
• Capacitors block direct current while allowing alternating current to pass.

This characteristic is useful in applications involving filtering, buffering, and timing circuits.