Problem 203 - Calculus Volume 3

Question

The electrical resistance \(R\) produced by wiring resistors \(R_{1}\) and \(R_{2}\) in parallel can be calculated from the formula \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). If \(R_{1}\) and \(R_{2}\) are measured to be \(7 \Omega\) and \(6 \Omega,\) respectively, and if these measurements are accurate to within \(0.05 \Omega\), estimate the maximum possible error in computing \(R\). (The symbol \(\Omega\) represents an ohm, the unit of electrical resistance.)

Step-by-Step Solution

Answer

The maximum possible error in computing \( R \) is approximately \( 0.0248 \, \Omega \).

Step 1: Understand the Problem

We need to estimate the maximum possible error in computing the total resistance \( R \) for resistors \( R_1 = 7 \ \Omega \) and \( R_2 = 6 \ \Omega \) connected in parallel. Given their measurement inaccuracies of \( 0.05 \ \Omega \), we'll use differential error analysis to find this error in resistance \( R \).

Step 2: Write Down the Formula

The formula for the equivalent resistance \( R \) in parallel is given by:\[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \]We will use this formula to calculate \( R \) and differentiate it for error analysis.

Step 3: Differentiate the Formula

Differentiate both sides of the formula with respect to their variables:\[ -\frac{1}{R^2} dR = -\frac{1}{R_1^2} dR_1 - \frac{1}{R_2^2} dR_2 \]Our goal is to isolate \( dR \), which represents the change or error in \( R \).

Step 4: Substitute Known Values

First, solve the original equation for \( R \):\[ \frac{1}{R} = \frac{1}{7} + \frac{1}{6} = \frac{6 + 7}{42} = \frac{13}{42} \]Thus, \( R = \frac{42}{13} \approx 3.23 \ \Omega \).

Step 5: Compute Derivatives and Substitute Errors

Substitute \( R_1 = 7 \), \( R_2 = 6 \), and their respective differentials \( dR_1 = 0.05 \) and \( dR_2 = 0.05 \) into the differentiated equation:\[ dR = R^2 \left( \frac{1}{R_1^2} dR_1 + \frac{1}{R_2^2} dR_2 \right) \]\[ dR = (3.23)^2 \left( \frac{1}{49} \times 0.05 + \frac{1}{36} \times 0.05 \right) \]

Step 6: Calculate the Maximum Error

Now compute the maximum error:\[ dR = 10.4329 \times \left( \frac{0.05}{49} + \frac{0.05}{36} \right) \approx 10.4329 \times (0.0010204 + 0.0013889) \approx 0.0248 \ \Omega \]The maximum possible error in \( R \) is approximately \( 0.0248 \ \Omega \).

Key Concepts

Understanding Electrical Resistance

Electrical resistance is a fundamental concept in the field of electronics. It is a measure of the difficulty electrons face when trying to pass through a conductor. Resistance is analogous to friction for electrons in a wire. The unit of electrical resistance is ohms, symbolized by \(\Omega\).

Factors that impact electrical resistance include:

• The material of the conductor (e.g., copper has low resistance, while rubber has high resistance).
• The cross-sectional area of the conductor (larger areas allow more ease of electron flow).
• The length of the conductor (longer wires mean longer paths for electrons to travel, thus more resistance).
• The temperature of the conductor (higher temperatures increase resistance).

Understanding how these factors work together helps in designing circuits & controlling the flow of electrons effectively.

Parallel Resistors in a Circuit

When resistors are connected in parallel, the overall effect in the circuit is different than when they are connected in series. In a parallel configuration, the total resistance diminishes. This happens because each resistor provides an additional path for current to travel, thereby making the entire current flow more accessible.The formula to calculate total resistance \( R \) in parallel is:\[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n} \]This formula suggests that the total resistance is less than any individual resistance in the circuit.

For example, if two resistors, \( R_1\) and \( R_2 \), have values of \( 7 \, \Omega \) and \( 6 \, \Omega \) respectively, the total resistance \( R \) would be less than \( 6 \, \Omega \) in this configuration.

Benefits of parallel resistors include more efficient current distribution in circuits and protection against circuit failures. If one path fails, current can still travel through the other paths, maintaining function.

Estimating Maximum Error in Calculations

In measurements and calculations, there is always the possibility of errors. Estimating the maximum error helps predict the accuracy and reliability of calculated results.

Maximum error estimation is crucial in engineering fields when precision is paramount. Using differential error analysis, we can estimate the maximum possible error by looking at the partial derivatives of a function. This method helps determine how small errors in measurement translate to errors in computed results.In our problem, resistors \( R_1 \) and \( R_2 \) were both measured with a possible error of \( 0.05 \, \Omega \). By applying differential calculus to the parallel resistor equation, the total possible error in the combined resistance calculation was estimated to be approximately \( 0.0248 \, \Omega \).

This provides insights into the precision of our equipment and measurements. Understanding and minimizing maximum errors is key to achieving better and more dependable results in electronic circuit designs.