Problem 5 - College Physics

Question

\(\bullet\) A radio inductor. You want the current amplitude through a \(0.450-\mathrm{mH}\) inductor (part of the circuitry for a radio receiver) to be 2.60 \(\mathrm{m}\) A when a sinusoidal voltage with amplitude 12.0 \(\mathrm{V}\) is applied across the inductor. What frequency is required?

Step-by-Step Solution

Answer

The required frequency is approximately 17.0 kHz.

Step 1: Understand the Problem

We are given an inductor with inductance \( L = 0.450 \text{ mH} \) and a voltage amplitude \( V = 12.0 \text{ V} \). The current amplitude \( I = 2.60 \text{ mA} \). We need to find the frequency of the sinusoidal voltage.

Step 2: Use the Inductive Reactance Formula

The inductive reactance \( X_L \) is given by the formula \( X_L = 2 \pi f L \), where \( f \) is the frequency. Rewrite the formula for frequency: \( f = \frac{X_L}{2 \pi L} \).

Step 3: Apply Ohm's Law for AC Circuits

For an inductor, Ohm's Law in AC circuits is given by \( V = I X_L \). Therefore, \( X_L = \frac{V}{I} \). Substitute \( V = 12.0 \text{ V} \) and \( I = 2.60 \text{ mA} \) into the equation to find \( X_L \).

Step 4: Calculate the Inductive Reactance

Convert current to amperes: \( I = 2.60 \text{ mA} = 2.60 \times 10^{-3} \text{ A} \). Calculate \( X_L \) using \( X_L = \frac{12.0 \text{ V}}{2.60 \times 10^{-3} \text{ A}} \).

Step 5: Calculate the Frequency

Use the inductive reactance value from Step 4 and the formula from Step 2. Inductance \( L \) must be converted to henrys: \( L = 0.450 \text{ mH} = 0.450 \times 10^{-3} \text{ H} \). Substitute into \( f = \frac{X_L}{2 \pi L} \) to find \( f \).

Step 6: Simplify and Solve the Equation

Calculate the expression \( f = \frac{\frac{12.0}{2.60 \times 10^{-3}}}{2 \pi \times 0.450 \times 10^{-3}} \). Simplify and solve for \( f \).

Key Concepts

Inductance

Inductance is a property of an electrical component, typically a coil or inductor, which helps it store energy in a magnetic field when electric current flows through it. The concept was first identified by Michael Faraday and is measured in units called henrys (H). Inductance arises in circuits mainly because of the magnetic coupling between the turns of a coil or because of an inductor's shape and material composition.
When a current passes through an inductor, a magnetic field is created around it. As the current changes, so does the strength of the magnetic field. An essential feature of inductors is their ability to resist changes in current due to the changing magnetic field they produce. This makes them incredibly useful in AC circuits where currents are constantly shifting.

• Inductors can be used to filter signals, store energy temporarily, and divide frequencies in radio receivers, like the one in our exercise.
• The ability of the inductor to oppose changes in current is what makes it valuable for various applications including tuning circuits, transformers, and power supplies.

Understanding inductance is crucial for grasping how complex AC circuits function because it provides the foundational behavior of how inductors interact with alternating currents.

Ohm's Law

Ohm's Law is a fundamental principle that relates the current flowing through a conductor with the voltage and resistance. For AC circuits, Ohm's Law is adapted to consider alternating currents and reactance, which affects the impedance of a circuit.
For inductors, which are prevalent in AC circuits, Ohm's Law is expressed as:\[ V = IX_L \]where:

• \( V \) is the voltage across the inductor,
• \( I \) is the current through the inductor,
• \( X_L \) is the inductive reactance.

In the context of our radio circuitry problem, the understanding of Ohm's Law is pivotal. The given voltage and desired current through the inductor are used to determine the inductive reactance (\( X_L \)), which further helps in finding what frequency the inductor needs to work at. The law explains the linear relationship between these electrical quantities in a straightforward manner, making it an indispensable tool for solving AC circuit problems.

Inductive Reactance

Inductive reactance is a measure of the opposition that an inductor presents to an alternating current's change. Unlike resistance, which affects both AC and DC, reactance is dependent on the frequency of the applied voltage. It is quantified in ohms, like resistance, and varies directly with the frequency of the alternating current and the inductor's inductance.
Mathematically, inductive reactance \( X_L \) is calculated using the formula:\[ X_L = 2 \pi f L \]where:

• \( f \) is the frequency of the AC signal,
• \( L \) is the inductance.

In our given exercise, this formula helps identify the correct frequency needed for the inductor to achieve a particular current with a specific voltage. As frequency increases, so does the inductive reactance, meaning greater frequency results in more opposition to current changes.
Inductive reactance is crucial because it demonstrates how inductors behave in AC circuits. It ensures the proper functioning of radios, as precise frequency tuning is necessary to select the desired signal while filtering out others. Understanding \( X_L \) allows engineers to design circuits that precisely control how signals pass through or are blocked, which is essential for effective communication systems.