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Chapter 1: Problem 77

The formula \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\) gives the total resistance \(R\) in an electric circuit due to three resistances, \(R_{1}, R_{2}\), and \(R_{3}\), connected in parallel. If \(10 \leq R_{1} \leq 20,20 \leq R_{2} \leq 30\), and \(30 \leq R_{3} \leq 40\), find the range of values for \(R\).

Short Answer

Expert verified
The range of values for \(R\) is \([5.45, 9.23]\) ohms.

Step by step solution

01

Understand the Formula

The formula \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\) is used for resistors in parallel. It states that the reciprocal of total resistance \(R\) is equal to the sum of the reciprocals of individual resistances \(R_1, R_2,\) and \(R_3\).
02

Determine Boundary Values of R1, R2, R3

Identify the minimum and maximum values for each resistor given the constraints. For \(R_1: 10 \leq R_1 \leq 20\), \(R_2: 20 \leq R_2 \leq 30\), and \(R_3: 30 \leq R_3 \leq 40\).
03

Calculate Maximum Total Resistance R (Minimum of Reciprocals)

To achieve the maximum value of \(R\), minimize the sum of the reciprocals by choosing the maximum values of \(R_1, R_2, R_3\). Calculate \(\frac{1}{20} + \frac{1}{30} + \frac{1}{40}\), then take the reciprocal to get \(R_{\text{max}}\).
04

Calculate Minimum Total Resistance R (Maximum of Reciprocals)

To achieve the minimum value of \(R\), maximize the sum of the reciprocals by choosing the minimum values of \(R_1, R_2, R_3\). Calculate \(\frac{1}{10} + \frac{1}{20} + \frac{1}{30}\), then take the reciprocal to get \(R_{\text{min}}\).
05

Compute the Results

Calculate the actual values from the expressions in Steps 3 and 4: For maximum \(R\), calculate \(\frac{1}{20} + \frac{1}{30} + \frac{1}{40} \approx 0.10833\), giving \(R_{\text{max}} \approx 9.23\). For minimum \(R\), calculate \(\frac{1}{10} + \frac{1}{20} + \frac{1}{30} = 0.1833\), giving \(R_{\text{min}} = 5.45\).
06

State the Range of R

Combine the results to state the range: The total resistance \(R\) ranges from \(5.45\) ohms to \(9.23\) ohms.

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Key Concepts

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

Electrical Resistance
Electrical resistance is a property of a material that opposes the flow of electric current. The resistance is measured in ohms (\(\Omega\)). When multiple resistors are present in an electrical circuit, their resistance affects the overall current flow. Parallel circuits, in particular, have specific rules for calculating their total resistance. Unlike series circuits where resistance adds up directly, parallel circuits combine resistances using a reciprocal formula. Understanding how each type of circuit affects total resistance is crucial when analyzing or designing electronic circuits.
Reciprocal Calculations
Reciprocal calculations play a significant role when dealing with resistors in parallel. In this context, the reciprocal function is used to make the computation of total resistance easier and more straightforward.Given the formula \(\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\), the task is to find the reciprocal of each resistor's value and then sum these results. This provides the reciprocal of the total resistance. To find \(R\), you simply take the reciprocal of this sum. Using reciprocals is essential when handling parallel vs. series circuits, as it reflects how current can flow through different paths simultaneously.
Range of Values
When discussing the range of values for the total resistance in a parallel circuit, it becomes necessary to identify the maximum and minimum possible resistances. This is done by employing the extremes for each resistor's value within the given range.To find the maximum total resistance, each resistor should be at its highest value, thereby minimizing \(\frac{1}{R}\). Conversely, for finding the minimum total resistance, each resistor should be at its lowest value, maximizing \(\frac{1}{R}\). After calculating these values, the range provides an understanding of how the circuit's resistance might vary. In this case, the resistance ranges from approximately 5.45 ohms to 9.23 ohms.
Physics Concepts
Resistors in circuits are a practical application of key physics concepts. Understanding how they work helps us grasp broader ideas in physics, such as energy conservation and electronic flow. Parallel circuits allow multiple pathways for current to travel. This means that even if one pathway encounters high resistance, the current can continue to flow through another path. This principle supports the concept of network reliability in electronics, explaining why parallel configurations are commonly used in building stable and efficient circuits. Knowing the role and impact of resistance in these settings is essential for anyone studying physics or electrical engineering.

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