Problem 8 - Physics

Question

A 0.047 -H inductor is wired across the terminals of a generator that has a voltage of 2.1 V and supplies a current of 0.023 A. Find the frequency of the generator.

Step-by-Step Solution

Answer

The frequency of the generator is approximately 308.24 Hz.

Step 1: Understand the Given Values

Identify the given values: the inductance \( L = 0.047 \text{ H} \), the voltage \( V = 2.1 \text{ V} \), and the current \( I = 0.023 \text{ A} \). We are tasked to find the frequency \( f \) of the generator.

Step 2: Use the Impedance Formula for an Inductor

The impedance \( Z \) of an inductor can be expressed as \( Z = 2\pi f L \). Ohm's Law tells us \( V = IZ \), therefore \( Z = \frac{V}{I} \).

Step 3: Solve for Impedance

Calculate the impedance using the voltage and current: \( Z = \frac{2.1}{0.023} \approx 91.30 \text{ Ohms} \).

Step 4: Set Up the Frequency Equation

The impedance formula \( Z = 2\pi f L \) can be rearranged to solve for the frequency: \( f = \frac{Z}{2\pi L} \).

Step 5: Calculate the Frequency

Substitute the known values into the frequency equation: \[ f = \frac{91.30}{2\pi \times 0.047} \approx 308.24 \text{ Hz} \].

Key Concepts

Inductance Calculation

Inductance is a fundamental property of an inductor that reflects its ability to resist changes in electric current. It is measured in henries (H) and is denoted as \(L\). The calculation of inductance relates to the coil's design, the number of turns in the coil, and the core material. However, in many applications, such as the given exercise, we are concerned with using the inductance value directly to determine other electrical properties. In our scenario, the inductance of the coil is given as 0.047 H.

• Inductance (\(L\)): 0.047 H

Understanding inductance helps in calculating the impedance and analyzing the behavior of circuits with alternating current (AC). It plays a critical role in making sure that energy is effectively stored and released in the circuit.

Ohm's Law

Ohm's Law is a foundational concept in electrical engineering and electronics, which relates the voltage (\(V\)), current (\(I\)), and resistance (\(R\)) of a circuit. The basic formula is \( V = IR \). However, when dealing with the impedance of inductors (or capacitors), we extend this concept to include impedance (\(Z\)) for AC circuits, where \( V = IZ \).

• Voltage (\(V\)): 2.1 V
• Current (\(I\)): 0.023 A

In the context of our exercise, Ohm's Law connects the known voltage and current to allow us to solve for the impedance of the inductor. By using \( Z = \frac{V}{I} \), we calculate the impedance necessary for determining the frequency of the generator.

Impedance of an Inductor

Impedance is the measure of opposition that a circuit presents to the passage of a current when a voltage is applied. For inductors, impedance is not merely resistance; it also includes a component that depends on frequency. The impedance (\(Z\)) of an inductor is calculated using the formula \( Z = 2\pi f L \), where \( f \) is frequency and \( L \) is inductance. From the exercise, we know:

• Calculated Impedance (\(Z\)): 91.30 Ω

This formula highlights how the impedance of an inductor increases with frequency and inductance. The calculated impedance is crucial for rearranging the formula to solve for frequency in systems involving AC signals.

Frequency Calculation

The frequency of a generator tells us how many cycles per second the AC waveform completes. It is measured in hertz (Hz) and is a critical parameter in AC circuits. To find the frequency (\(f\)) of the generator from our exercise, we rearrange the impedance formula to \( f = \frac{Z}{2\pi L} \). Substituting the known values:

• Impedance (\(Z\)): 91.30 Ω
• Inductance (\(L\)): 0.047 H

The calculation becomes:\[f = \frac{91.30}{2\pi \times 0.047} \approx 308.24 \text{ Hz}\]This final step shows us how to determine the frequency based on impedance and inductance values, thereby completing the analysis with the accurate frequency of the generator.