Problem 74 - Physics for Scientists and Engineers with Modern Physics

Question

How many \(\frac{1}{2}\) -W resistors, each of the same resistance, must be used to produce an equivalent \(3.2-\mathrm{k} \Omega, 3.5-\mathrm{W}\) resistor? What is the resistance of each, and how must they be connected? Do not exceed \(P=\frac{1}{2} \mathrm{~W}\) in each resistor.

Step-by-Step Solution

Answer

Use 7 resistors of 457.14 \(\Omega\) each, connected in series.

Step 1: Understanding the Problem

We need to determine how many resistors of a given power rating (\( \frac{1}{2} \) W) and the same resistance value can be combined to result in an equivalent resistor of \(3.2\,\mathrm{k} \Omega\) and 3.5 W power. Each resistor must not exceed \(\frac{1}{2}\,\mathrm{W}\).

Step 2: Calculating Number of Resistors for Power

To meet the power requirement of 3.5 W with each resistor capable of handling only \(\frac{1}{2}\) W, we need: \[ n_{\text{resistors}} = \frac{3.5\, \mathrm{W}}{\frac{1}{2}\,\mathrm{W}} = 7 \] So, 7 resistors are required to achieve the necessary power rating.

Step 3: Calculating Resistor Value for Series Connection

If the resistors are to be used together to produce a total resistance of \(3.2\,\mathrm{k}\Omega\), we can use them in series. The resistance of each resistor will therefore be: \[ R = \frac{3.2\,000\,\Omega}{7} = 457.14\,\Omega \]

Step 4: Verifying Connection Type Consistency

Given that each resistor handles only \(\frac{1}{2}\) W, and we've calculated 457.14 \(\Omega\) for each in a series configuration, this checks out with the initial resistance and power constraints. No parallel connection is needed for resistance calculation.

Key Concepts

Understanding Equivalent Resistance

Equivalent resistance is a fundamental concept in electronics that describes a single resistance value replacing a combination of resistors, offering the same electrical effect in a circuit. When resistors are combined, whether in series or parallel, they create a new resistance measurement.

• In series, the resistances simply add up: \(R_{eq} = R_1 + R_2 + R_3 + ... + R_n\).
• In parallel, the reciprocal formula applies:\(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}\).

For our exercise, an equivalent resistance of \(3.2\,\text{k}\Omega\) is achieved by adding the resistors in series. This means the total resistance is spread equally across all components in the series.

Series Connection of Resistors

A series connection links components end-to-end, ensuring the same current flows through each one. This makes it the simplest method to achieve a higher equivalent resistance.

• The total resistance is equal to the sum of individual resistances.
• Adding resistors in series increases overall resistance without affecting the power each can dissipate.

For the task at hand, each of the 7 resistors needed to contribute equally to reach a total of \(3.2\,\text{k}\Omega\). Thus, each resistor in the series carried a resistance of \(457.14\,\Omega\).
This method not only meets the required resistance but also adheres to the constraint that no individual resistor exceeds its power rating.

Understanding Power Rating

A resistor's power rating tells us how much power it can safely dissipate. It is critical to ensure resistors within a circuit do not exceed this limit, as it prevents overheating and potential damage.

• The power rating is derived from: \(P = \frac{V^2}{R}\) or \(P = I^2R\), where \(P\) is power, \(V\) is voltage, and \(I\) is current.
• In our scenario, each resistor must not exceed \(\frac{1}{2} \text{W}\).
• By using 7 resistors, we distribute the total power requirement of \(3.5 \text{W}\) equally: \(n_{resistors} = \frac{3.5\,\text{W}}{\frac{1}{2}\,\text{W}} = 7\).

This strategy ensures each resistor remains within its safe operating limits, even while collectively achieving the total desired power and resistance.