Problem 10 - Fundamentals of Physics

Question

\(L C\) oscillators have been used in circuits connected to loudspeakers to create some of the sounds of electronic music. What inductance must be used with a \(6.7 \mu \mathrm{F}\) capacitor to produce a frequency of \(10 \mathrm{kHz}\), which is near the middle of the audible range of frequencies?

Step-by-Step Solution

Answer

The inductance must be approximately 2.37 mH.

Step 1: Identify the Formula

To find the inductance, we will use the formula for the resonant frequency of an LC circuit: \( f = \frac{1}{2 \pi \sqrt{LC}} \), where \( f \) is the frequency, \( L \) is the inductance, and \( C \) is the capacitance.

Step 2: Rearrange the Formula for Inductance

Rearrange the formula to solve for the inductance \( L \): \( L = \frac{1}{(2 \pi f)^2 C} \).

Step 3: Substitute the Known Values

Substitute the given values into the formula. The frequency \( f = 10 \text{kHz} = 10,000 \text{Hz} \) and capacitance \( C = 6.7 \mu \text{F} = 6.7 \times 10^{-6} \text{F} \).

Step 4: Perform the Calculation

Calculate using the substituted values: \[ L = \frac{1}{(2 \pi \times 10000)^2 \times 6.7 \times 10^{-6}} \] Calculate \( 2 \pi \times 10000 \) which equals approximately \( 62831.853 \).Square the result to get approximately \( 3.948 \times 10^{9} \).Finally, divide 1 by \( 3.948 \times 10^{9} \times 6.7 \times 10^{-6} \).

Step 5: Calculate Final Result

Complete the calculation to find the inductance:\[ L = \frac{1}{3.948 \times 10^{9} \times 6.7 \times 10^{-6}} \approx 2.37 \times 10^{-3} \text{Henries} \]Thus, the required inductance \( L \) is approximately \( 2.37 \text{mH} \).

Key Concepts

Resonant Frequency

In an LC circuit, the resonant frequency is the frequency at which the system naturally oscillates. This concept is central to understanding how LC circuits function within electronic devices such as loudspeakers and radios. Resonant frequency is a critical point where the inductive and capacitive reactances in the circuit are equal, leading to maximum voltage across the components and minimal energy loss.
The resonant frequency \( f \) of an LC circuit can be determined from the equation:\[f = \frac{1}{2 \pi \sqrt{LC}}\]Here:

• \( f \) is the resonant frequency in Hertz (Hz)
• \( L \) is the inductance in Henrys (H)
• \( C \) is the capacitance in Farads (F)

At resonance, the circuit can oscillate with the highest amplitude possible, which is useful for tuning circuits to specific frequencies. Resonant frequency allows LC circuits to be applied in filters and oscillators, playing an integral role in communication and audio equipment.

Inductance Calculation

Inductance is a measure of an inductor's ability to store energy in a magnetic field. Calculating it is crucial when designing circuits to ensure that they function properly at the intended frequency. In the given exercise, we need to find the inductance needed for a specific resonant frequency, which is important in audio applications.
The inductance \( L \) can be calculated using the rearranged formula:\[L = \frac{1}{(2 \pi f)^2 C}\]To determine \( L \), follow these steps:

• Convert all values to the correct units. For frequency, 10 kHz = 10,000 Hz and for capacitance, 6.7 μF = 6.7 × 10^{-6} F.
• Substitute the values into the formula.
• Calculate \( 2 \pi f \), square the result, and plug into the equation.
• Perform the final calculation to find the inductance.

By understanding how to manipulate the formula, you can solve for \( L \) and ensure that the circuit will perform optimally at the intended frequency. This calculation ensures that our electronic circuits function smoothly and efficiently.

Capacitance

Capacitance is the ability of a component to store and release electrical energy in the form of an electric field. In the context of LC circuits, capacitance is a crucial factor that affects the resonant frequency and the overall behavior of the circuit.
In the LC circuit resonant frequency formula, capacitance \( C \) plays a central role. It determines how much energy the capacitor can store and also influences how the circuit responds to changes in frequency. Capacitors are measured in Farads (F), though practical capacitors often use microfarads (μF), as seen in the exercise.
Understanding capacitance:

• Capacitance increases the amount of charge a capacitor can hold for a given voltage.
• The unit is typically too large for everyday circuits, so μF or pF (picofarads) are commonly used.
• Adjusting the capacitance in a circuit will shift the resonant frequency, affecting how the circuit responds to AC signals.

Controlling the capacitance in LC circuits allows engineers to design filters and tuners, ensuring electronic devices operate precisely and efficiently within desired frequency bands.