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Chapter 18: Problem 29

Find the required quantities from the given proportions. In an electric instrument called a "Wheatstone bridge," electric resistances are related by $$\begin{aligned} &\frac{R_{1}}{R_{2}}=\frac{R_{3}}{R_{4}}\\\ &\text { Find } R_{2} \text { if } R_{1}=6.00 \Omega\\\ &R_{3}=62.5 \Omega, \text { and }\\\ &R_{4}=15.0 \Omega . \text { See Fig. } 18.4 \end{aligned}$$

Short Answer

Expert verified
The resistance \( R_{2} \) is approximately \( 1.44 \Omega \).

Step by step solution

01

Understand the Proportion Formula

The given problem involves a proportion:\[ \frac{R_{1}}{R_{2}} = \frac{R_{3}}{R_{4}} \]Our goal is to find the unknown resistance \( R_{2} \). We have the values of \( R_{1}, R_{3}, \) and \( R_{4} \).
02

Rearrange the Formula for R2

To solve for \( R_{2} \), rearrange the proportion formula:\[ R_{2} = \frac{R_{1} \times R_{4}}{R_{3}} \]This isolates \( R_{2} \) on one side of the equation.
03

Substitute Known Values

Now, plug in the values you have:- \( R_{1} = 6.00 \Omega \)- \( R_{3} = 62.5 \Omega \)- \( R_{4} = 15.0 \Omega \)The equation becomes:\[ R_{2} = \frac{6.00 \times 15.0}{62.5} \]
04

Calculate R2

Perform the multiplication in the numerator:\[ 6.00 \times 15.0 = 90.0 \]Now divide by \( R_{3} = 62.5 \):\[ R_{2} = \frac{90.0}{62.5} \approx 1.44 \Omega \]
05

Verify the Solution

Double-check the calculations to ensure they are correct. Substitute back into the original proportion to confirm consistency:\[ \frac{6.00}{1.44} \approx 4.17 \quad \text{and} \quad \frac{62.5}{15.0} = 4.17 \]Both sides of the original equation are equivalent, confirming that \( R_{2} = 1.44 \Omega \) is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wheatstone Bridge
A Wheatstone Bridge is an essential tool in electrical engineering, used to measure unknown electrical resistance by balancing two legs of a bridge circuit. The circuit typically consists of four resistors, denoted as \( R_1, R_2, R_3, \) and \( R_4 \). These resistors are arranged in a diamond shape, with a galvanometer used to detect the balance point of the bridge. When the bridge is balanced, the ratio of one pair of resistors equals the ratio of the other pair. This relationship is expressed in the equation \( \frac{R_1}{R_2} = \frac{R_3}{R_4} \).

This balance condition makes the Wheatstone Bridge an accurate and reliable tool for measuring resistance. Since the bridge can detect very small changes in resistance, it's highly sensitive and very useful for laboratory experiments. When working with a Wheatstone Bridge, you can achieve high precision by carefully setting the resistors and adjusting the bridge until it reaches a state of balance. At this point, unknown resistors can easily be calculated using the given ratio.
Proportion Formula
The proportion formula is crucial when dealing with equations in a Wheatstone Bridge setup because it establishes the relationship between different resistors within the bridge. This formula is expressed as \( \frac{R_1}{R_2} = \frac{R_3}{R_4} \). It indicates that the ratio of resistances \( R_1 \) to \( R_2 \) is equal to the ratio of resistances \( R_3 \) to \( R_4 \).

The main advantage of using a proportion formula is its ability to simplify the process of finding an unknown resistance. To solve for an unknown, such as \( R_2 \) in the given formula, you can rearrange the equation to solve for the unknown value. For example, to find \( R_2 \), rearrange the formula to \( R_2 = \frac{R_1 \times R_4}{R_3} \).
  • This rearrangement is possible because the properties of proportions allow for multiplying across the equal sign.
  • After substitution of known values, calculate the product of the parameters on the right-hand side and divide by the parameter on the left to isolate for \( R_2 \).
This straightforward approach ensures a clear pathway to finding unknown resistors, aiding in efficient problem-solving within electrical networks.
Resistor Calculations
Resistor calculations are an integral part of working with electrical circuits, especially in applications like the Wheatstone Bridge. In resistor calculations, you aim to determine unknown resistances using known values and given relationships. When solving for a specific resistor, like \( R_2 \), the method typically follows these steps:

  • Identify the known values: Start by noting the given resistor values, such as \( R_1=6.00 \Omega \), \( R_3=62.5 \Omega \), and \( R_4=15.0 \Omega \).
  • Apply the proportion formula: Use the equation \( \frac{R_1}{R_2} = \frac{R_3}{R_4} \) to set up the calculation.
  • Rearrange to solve for the unknown: Modify the equation to solve for the unknown resistor, resulting in \( R_2 = \frac{R_1 \times R_4}{R_3} \).
  • Substitute and compute: Plug in the known values and perform the multiplication and division to find \( R_2 \).

These calculations not only help in identifying the required resistance but also in maintaining the balance within electrical circuits. A correct calculation will confirm consistency when checked against initial conditions, ensuring all components work harmoniously within the electric network.

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Most popular questions from this chapter

Answer the given questions by setting up and solving the appropriate proportions. In testing for quality control, it was found that 17 of every 500 computer chips produced by a company in a day were defective. If a total of 595 defective parts were found, what was the total number of chips produced during that day?

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Solve the given applied problems involving variation. The time \(t\) required to test a computer memory unit varies directly as the square of the number \(n\) of memory cells in the unit. If a unit with 4800 memory cells can be tested in 15.0 s, how long does it take to test a unit with 8400 memory cells?
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