Problem 39 - College Physics

Question

A tuning circuit in a radio receiver has a fixed inductance of \(0.50 \mathrm{mH}\) and a variable capacitor. (a) If the circuit is tuned to a radio station broadcasting at \(980 \mathrm{kHz}\) on the AM dial, what is the capacitance of the capacitor? (b) What value of capacitance is required to tune into a station broadcasting at \(1280 \mathrm{kHz}\) ?

Step-by-Step Solution

Answer

(a) 5.22 pF for 980 kHz, (b) 2.44 pF for 1280 kHz.

Step 1: Understand the Resonance Frequency Formula

A L-C circuit resonates at a frequency given by the formula \( f = \frac{1}{2\pi\sqrt{LC}} \), where \( L \) is the inductance and \( C \) is the capacitance. This formula will be used to solve the problem.

Step 2: Convert Given Frequencies to Hertz

For part (a), the frequency \( f_1 \) is 980 kHz, which needs to be converted to hertz: \( f_1 = 980,000 \) Hz. For part (b), the frequency \( f_2 \) is 1280 kHz: \( f_2 = 1,280,000 \) Hz.

Step 3: Calculate Capacitance for 980 kHz

Rearrange the resonance frequency formula to solve for capacitance: \( C = \frac{1}{(2\pi f)^2 L} \). Substitute \( f_1 = 980,000 \) Hz and \( L = 0.0005 \) H (since 0.50 mH = 0.0005 H) into the formula:\[ C_1 = \frac{1}{(2\pi \times 980,000)^2 \times 0.0005} \]This calculates to approximately \( C_1 \approx 5.22 \times 10^{-12} \) F or 5.22 pF.

Step 4: Calculate Capacitance for 1280 kHz

Using the same rearranged formula, substitute \( f_2 = 1,280,000 \) Hz into the formula:\[ C_2 = \frac{1}{(2\pi \times 1,280,000)^2 \times 0.0005} \]This calculates to approximately \( C_2 \approx 2.44 \times 10^{-12} \) F or 2.44 pF.

Step 5: Interpret Results from Calculations

The capacitance needed to tune to 980 kHz is approximately 5.22 pF. For 1280 kHz, the required capacitance is approximately 2.44 pF. With increasing frequency, the required capacitance decreases, which aligns with the behavior of an LC circuit.

Key Concepts

Inductance

Inductance is a fundamental concept in the study of electromagnetism, particularly in LC circuits. It is the property of a conductor by which a change in current flowing through it induces an electromotive force. The standard unit of inductance is the henry (H).

• In an LC circuit, inductance works alongside capacitance to allow the circuit to resonate at a particular frequency.
• The fixed inductance effectively sets one of the two parameters needed to tune the circuit to a specific frequency.

Understanding the role of inductance in an LC circuit helps to grasp how radios can tune into specific frequencies, such as AM radio stations.

Capacitance

Capacitance is the ability of a system to store an electric charge. Like inductance, it is a key feature in LC circuits, influencing the circuit's ability to resonate at specific frequencies. Capacitance is measured in farads (F).

• A variable capacitor in a radio's tuning circuit allows you to change the circuit's resonant frequency to match the frequency of the desired radio station.
• By adjusting the capacitance, you are effectively shifting the range of frequencies the circuit can tune into.

The relationship between inductance, capacitance, and resonant frequency is given by the formula: \[ f = \frac{1}{2\pi\sqrt{LC}} \]This formula shows that as capacitance changes, the resonant frequency of the circuit also changes, making capacitance a critical player in the tuning process.

AM Radio Frequencies

AM radio frequencies range from about 530 kHz to 1700 kHz and are commonly used for broadcasting music, news, and sports. These frequencies are allocated to AM (Amplitude Modulation) radio stations, each occupying a specific spot on the radio dial.

• The exercise involves calculating the capacitance needed to tune into stations at 980 kHz and 1280 kHz within this range.
• Different stations are broadcast on different frequencies, requiring tuning circuits in radios to adjust their settings to receive the correct signals.

Understanding AM radio frequencies is crucial because it highlights the practice of frequency modulation and how we can isolate specific channels for clear listening experiences.

Radio Tuning

Radio tuning involves adjusting the resonant frequency of a radio circuit to receive signals at the desired frequency, typically by changing the capacitance in the tuning circuit. This process allows listeners to select different radio stations.

• The variable capacitor in the tuning circuit plays a significant role in changing the frequency the circuit is tuned to, allowing for precise adjustments.
• As reflected in the exercise, different frequencies require different capacitances, as shown by the resonance equation.

By mastering radio tuning, one steps closer to understanding how electronic devices like radios, transistors, and other communication gadgets function to provide desired signals efficiently.