Problem 5 - Applied Physics

Question

What is the inductive reactance (in ohms) of a 425- \(\mu\) H inductance at a frequency of \(15.0 \mathrm{MHz}\) ?

Step-by-Step Solution

Answer

The inductive reactance is 40.13 ohms.

Step 1: Identify Relevant Formula

The inductive reactance (in ohms) of an inductor can be calculated using the formula \( X_L = 2\pi f L \), where \( X_L \) is the inductive reactance, \( f \) is the frequency in Hertz, and \( L \) is the inductance in Henrys.

Step 2: Convert Units

Given that the inductance \( L \) is 425 \( \mu \)H, convert it to Henrys by multiplying by \( 10^{-6} \). Thus, \( L = 425 \times 10^{-6} \) H.

Step 3: Plug Values into Formula

Substitute \( f = 15.0 \times 10^6 \) Hz and \( L = 425 \times 10^{-6} \) H into the formula \( X_L = 2\pi f L \).

Step 4: Calculate Inductive Reactance

Perform the calculation: \[ X_L = 2 \times \pi \times 15.0 \times 10^6 \times 425 \times 10^{-6} \]Simplify the expression to find \( X_L = 40.13 \) ohms.

Key Concepts

Inductor

An inductor is a fundamental component of electrical circuits, primarily used to store energy in a magnetic field. It is typically made by winding a wire into a coil. The key characteristic of an inductor is its inductance, which is measured in units called Henrys (H). This property governs how an inductor opposes changes in the current flowing through it, creating a potential that we call inductive reactance.

Inductors are widely used in alternating current (AC) circuits to control the current flow and to filter signals. The ability of an inductor to resist changes in current can be harnessed for various applications, including tuning circuits, power supply filters, and transformers. Understanding the inductor's role is crucial as it influences both the frequency response and behavior of the circuit in which it is used.

Frequency

Frequency refers to the number of cycles a periodic wave completes in one second and is usually expressed in Hertz (Hz). When it comes to inductor reactance, frequency plays a pivotal role. The higher the frequency of the electrical signal passing through an inductor, the greater the inductive reactance it exhibits.

In essence, inductive reactance increases linearly with frequency. This behavior makes inductors useful for filtering applications. At higher frequencies, the inductor offers more resistance to the flow of AC, while at lower frequencies, it permits easier current flow. This unique characteristic is used in radio receivers and other electronic devices to isolate or pass specific frequency bands.

Formula for Reactance

The formula for inductive reactance is essential for understanding how inductors behave in circuits. It is given by the equation: \( X_L = 2\pi f L \).

• **\( X_L \)** is the inductive reactance (measured in ohms), representing the opposition to the AC current flow.
• **\( f \)** is the frequency of the AC signal (in Hertz), dictating the number of cycles occurring every second.
• **\( L \)** is the inductance (in Henrys), a measure of the inductor's capacity to store energy.

This formula shows that the reactance depends on both the frequency of the applied signal and the inductor's inherent properties. Understanding this relationship is key to designing circuits that rely on the dynamic behavior of inductors.

Unit Conversion

Unit conversion is crucial in electrical calculations, especially when dealing with components like inductors, which often use smaller units like microhenrys (\(\mu H\)). In engineering tasks, converting these smaller units into more standard units like Henrys (H) is necessary to work effectively with formulas such as the inductive reactance equation.

The conversion from microhenrys to Henrys involves multiplying the value by \(10^{-6}\), as 1 microhenry equals \(1\times10^{-6}\) Henrys. This step is important because it ensures dimensional consistency in your calculations, allowing accurate results. Without proper unit conversion, the application of mathematical formulas could lead to errors and inaccurate outcomes. Learning to convert units confidently is a basic yet vital skill in physics and engineering.