Problem 63 - Calculus Early Transcendentals

Question

Three electrical resistors Extending Exercise \(62,\) when three electrical resistors with resistance \(R_{1}>0, R_{2}>0\) and \(R_{3}>0\) are wired in parallel in a circuit (see figure), the combined resistance \(R\), measured in ohms \((\Omega),\) is given by \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}} .\) Estimate the change in \(R\) if \(R_{1}\) increases from \(2 \Omega\) to \(2.05 \Omega, R_{2}\) decreases from \(3 \Omega\) to \(2.95 \Omega,\) and \(R_{3}\) increases from \(1.5 \Omega\) to \(1.55 \Omega\).

Step-by-Step Solution

Answer

Question: Estimate the change in combined resistance when the individual resistances change as follows: \(R_1\) increases from \(2\Omega\) to \(2.05\Omega\), \(R_2\) decreases from \(3\Omega\) to \(2.95\Omega\), and \(R_3\) increases from \(1.5\Omega\) to \(1.55\Omega\). Answer: The combined resistance has decreased by approximately \(0.8205\Omega\).

Step 1: Calculate the initial combined resistance

Using the given formula and the initial values of \(R_1 = 2\Omega\), \(R_2 = 3\Omega\), and \(R_3 = 1.5\Omega\), we calculate the initial combined resistance: \(\frac{1}{R} = \frac{1}{2}+\frac{1}{3}+\frac{1}{1.5} = \frac{1}{2}+\frac{1}{3}+\frac{2}{3}\) \(\frac{1}{R} = \frac{1+1+2}{6}=\frac{4}{6}=\frac{2}{3}\) Now we can find \(R\) by taking the reciprocal: \(R = \frac{1}{\frac{2}{3}} = \frac{3}{2}=1.5\Omega\) The initial combined resistance is \(1.5\Omega\).

Step 2: Calculate the new combined resistance

Now we apply the changes to the resistances: - \(R_1\) increases to \(2.05\Omega\) - \(R_2\) decreases to \(2.95\Omega\) - \(R_3\) increases to \(1.55\Omega\) We now use the formula with these new resistances to find the new combined resistance: \(\frac{1}{R'} = \frac{1}{2.05}+\frac{1}{2.95}+\frac{1}{1.55}\) Use a calculator to find the decimal approximations for the new combined resistance: \(\frac{1}{R'} \approx 0.4878 + 0.3389+0.6451 = 1.4718\) Now we can find \(R'\) by taking the reciprocal: \(R' = \frac{1}{1.4718} \approx 0.6795\Omega\) The new combined resistance is approximately \(0.6795\Omega\).

Step 3: Estimate the change in combined resistance

Now we can find the change in the combined resistance by subtracting the initial combined resistance from the new combined resistance: \(\Delta R = R' - R \approx 0.6795\Omega - 1.5\Omega \approx -0.8205\Omega\) So the combined resistance has decreased by approximately \(0.8205\Omega\).

Key Concepts

Resistance Calculation

To calculate the total resistance in certain circuits, we must understand the rules for combining resistances. In this case, we are dealing with resistors in parallel. Calculating resistance for parallel circuits utilizes a unique formula that involves reciprocals. This formula is crucial for accurate calculations.The combined resistance \( R \) of resistors in parallel is calculated with the following equation:\[\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\]To find the combined or total resistance from this expression, follow these steps:

• First, compute the reciprocal of each resistor's resistance.
• Sum the reciprocals to get the reciprocal of the total resistance.
• Take the reciprocal of this sum to find the total resistance \( R \).

This approach can seem counterintuitive because, unlike series circuits where resistances add up, in parallel circuits, more paths lower the overall resistance.

Parallel Circuits

Parallel circuits have a distinct characteristic: the voltage is the same across all branches. This setup allows multiple paths for the current to flow, which reduces the overall resistance compared to configurations where components are in series. In a parallel circuit:

• The total resistance always diminishes with additional pathways.
• Each resistor contributes a part of the total current, based on its value.
• The sum of the currents through each path equals the total current flowing from the source.

When you encounter problems requiring resistance calculations in such setups, remember that parallel circuits yield lower resistance. This means that the resistance you calculate will be less than the smallest resistor in the group. This quality of parallel circuits is especially useful for designing circuits where lower total resistance and, thus, higher currents are desirable.

Change in Resistance

Changes in the resistance of individual components impact the overall resistance in intriguing ways—particularly in parallel circuits. As seen in the problem, when we altered the resistances of three parallel resistors, it substantially changed the total resistance. Here's how to think about it:

• Increasing a resistance in the parallel arrangement causes a minor increase in total resistance, as the change only affects one term of several in the reciprocal sum.
• Conversely, decreasing a resistance lowers the overall resistance more noticeably because the reciprocal of a smaller resistance is larger.
• The net effect on total resistance depends on balance among all resistor changes—sometimes it even reduces the total resistance, as demonstrated in the exercise.

By mastering resistance calculations and understanding their behavior in parallel circuits, you can predict how variations in individual resistors affect the entire circuit, which is crucial for managing and optimizing electrical designs.