Problem 10 - Applied Physics

Question

Solve each formula for the quantity given. $$ X_{L}=2 \pi f L \text { for } f $$

Step-by-Step Solution

Answer

The frequency \( f \) is given by \( f = \frac{X_{L}}{2\pi L} \).

Step 1: Understand the Formula

The given formula is the inductive reactance formula, expressed as \( X_{L} = 2\pi f L \). Here, \( X_{L} \) is the inductive reactance, \( f \) is the frequency, and \( L \) is the inductance.

Step 2: Isolate the Target Variable

Since we need to solve for \( f \), we start by isolating \( f \) in the equation. The given equation is:\[ X_{L} = 2\pi f L \] Divide both sides by \( 2\pi L \) to isolate \( f \):\[ f = \frac{X_{L}}{2\pi L} \]

Step 3: Simplified Expression

The equation is now solved for \( f \). We have the frequency expressed as:\[ f = \frac{X_{L}}{2\pi L} \].

Key Concepts

Frequency Calculation

In the world of electrical engineering, frequency plays a crucial role, particularly when dealing with alternating current (AC) circuits. The frequency denotes how often the current changes direction per second. It is typically measured in hertz (Hz). When calculating frequency in the context of inductive reactance, we lean on the formula for inductive reactance:

• Inductive reactance
• Frequency
• Inductance

In this specific exercise, we are given the formula \[ X_{L} = 2\pi f L \] where \( X_{L} \) represents inductive reactance, \( f \) stands for frequency, and \( L \) indicates inductance in henrys. Our goal is to calculate \( f \), the frequency.
The relationship between these variables illustrates how changes in inductive reactance or inductance affect frequency. By manipulating the formula and isolating \( f \), we realize the frequency's dependency on both the inductance and the inductive reactance.

Isolation of Variables

Isolation of variables is a fundamental mathematical technique used to solve for unknown quantities in equations. In this exercise, we're tasked with isolating the frequency \( f \) from the equation \[ X_{L} = 2\pi f L \].
To isolate \( f \), we need to rearrange the equation to express \( f \) as the subject. A step-by-step method allows us to focus on each component of the equation, bringing clarity to the problem-solving process.

• Start by identifying the target variable, \( f \), in the equation.
• Consider the operations applied to \( f \) - it's currently being multiplied by \( 2\pi L \).
• To isolate \( f \), perform the inverse operation, which in this case is dividing both sides by \( 2\pi L \).

This results in the equation: \[ f = \frac{X_{L}}{2\pi L} \]. This rearrangement allows us to directly calculate the frequency as a function of inductive reactance and inductance.

Mathematical Formulas

Mathematical formulas serve as the backbone for calculations in physics and engineering. These formulas are like recipes, guiding us through complex concepts with precision and simplicity. The formula we faced, \[ X_{L} = 2\pi f L \], is a representation of the relationship between inductive reactance, frequency, and inductance.
It embodies the principle that as frequency increases, so does the inductive reactance, if inductance stays constant. Mathematical formulas are powerful tools because they:

• Summarize complex relationships concisely.
• Allow predictions and insights into physical behavior.
• Enable computations across different parameters.

In this exercise, transforming the formula to solve for \( f \) emphasized the role of mathematical formulas in isolation and rearrangement of variables. Remember, working with formulas requires attention to detail, careful manipulation of symbols, and an understanding of the underlying concepts.