Problem 10 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Three \(1.70-\mathrm{k} \Omega\) resistors can be connected together in four different ways, making combinations of series and/or parallel circuits. What are these four ways, and what is the net resistance in each case?

Step-by-Step Solution

Answer

The four net resistances are 5.10 kΩ, 0.567 kΩ, 1.133 kΩ, and 2.55 kΩ.

Step 1: Understanding the Problem

We have three resistors of 1.70 kΩ each, and we need to find four different ways to connect them in series and/or parallel combinations and then calculate the net resistance for each configuration.

Step 1: All in Series

When resistors are connected in series, the total resistance is the sum of the individual resistances. For three resistors, each with a resistance of 1.70 kΩ, the total resistance is: \[ R = 1.70 + 1.70 + 1.70 = 5.10 ext{ k} \Omega \]

Step 2: All in Parallel

For resistors in parallel, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances. For three resistors, each with a resistance of 1.70 kΩ, the calculation is: \[ \frac{1}{R} = \frac{1}{1.70} + \frac{1}{1.70} + \frac{1}{1.70} \] \[ \frac{1}{R} = \frac{3}{1.70} \] \[ R = \frac{1.70}{3} = 0.567 ext{ k} \Omega \]

Step 3: Two in Series, One in Parallel

Two resistors in series have their resistances summed. A third resistor in parallel with this series combination has a total resistance calculated as: \[ R_{series} = 1.70 + 1.70 = 3.40 ext{ k} \Omega \] \[ \frac{1}{R_{total}} = \frac{1}{3.40} + \frac{1}{1.70} \] \[ \frac{1}{R_{total}} = \frac{1}{3.40} + \frac{2}{3.40} \] \[ R_{total} = \frac{3.40}{3} = 1.133 ext{ k} \Omega \]

Step 4: Two in Parallel, One in Series

Two resistors in parallel have a combined resistance of: \[ \frac{1}{R_{parallel}} = \frac{1}{1.70} + \frac{1}{1.70} \] \[ \frac{1}{R_{parallel}} = \frac{2}{1.70} \] \[ R_{parallel} = \frac{1.70}{2} = 0.85 ext{ k} \Omega \] Adding this parallel combination in series with the third resistor: \[ R_{total} = 0.85 + 1.70 = 2.55 ext{ k} \Omega \]

Key Concepts

Series Circuits

In electronics, a series circuit is a type of electrical circuit where components are connected end-to-end, forming a single, continuous path for the flow of current throughout the circuit.
It is important to understand this setup because the characteristics of series circuits directly affect how resistors behave when combined.
Here are the key aspects of series circuits:

• Single Path: There is only one path in a series circuit. This means that the same current flows through each component in the circuit.
• Summation of Resistance: The total or net resistance of resistors in series is simply the sum of all the individual resistances. This means that the overall resistance increases when more resistors are added in series.
• Voltage Across Each Resistor: The voltage drop across each resistor varies depending on the resistor's value, but the sum of all voltage drops equals the total source voltage.

For example, if you connect three resistors, each of 1.70 kΩ, in series, the total resistance is calculated as follows: \[ R = 1.70 + 1.70 + 1.70 = 5.10 \text{ k} \Omega \].
This total reflects how resistors add up in a series to increase the net resistance.

Parallel Circuits

In parallel circuits, the configuration allows components to be connected across common points or junctions, creating multiple paths for current. Understanding how resistors work in this setup is crucial for calculating the overall resistance effectively.
Here are the pivotal features of parallel circuits:

• Multiple Paths: Since there are various branches, current can travel through more than one path. This results in a division of current across the different paths.
• Decreasing Resistance: Contrary to series circuits, the total resistance decreases in a parallel arrangement because the reciprocal of the total resistance is the sum of the reciprocals of individual resistances.
• Equal Voltage: The voltage across each resistor in parallel remains the same as the source voltage.

When you put three resistors of 1.70 kΩ each in parallel, the total resistance is calculated as follows: \[ \frac{1}{R} = \frac{1}{1.70} + \frac{1}{1.70} + \frac{1}{1.70} \].
Thus, solving for \( R \) gives us:\[ R = \frac{1.70}{3} = 0.567 \text{ k} \Omega \].
This shows how the net resistance reduces in a parallel setup.

Net Resistance Calculation

Calculating net resistance involves finding the total resistance for combinations of series and parallel circuits, which are common in more complex circuits.
To grasp this concept, consider two configurations of mixed series and parallel connections, highlighting how resistors can be arranged differently to alter the net resistance.

In the first configuration, two resistors are in series, while a third resistor is in parallel with them. Firstly, you sum the two resistors:

• \[ R_{series} = 1.70 + 1.70 = 3.40 \text{ k} \Omega \].
• Then, the total resistance with a parallel resistor is calculated as:
• \[ \frac{1}{R_{total}} = \frac{1}{3.40} + \frac{1}{1.70} \] giving \[ R_{total} = 1.133 \text{ k} \Omega \].

In an alternative arrangement, two resistors are placed in parallel first, and this combination is connected in series with the third resistor:

• \[ \frac{1}{R_{parallel}} = \frac{1}{1.70} + \frac{1}{1.70} \], resulting in \( R_{parallel} = 0.85 \text{ k} \Omega \).
• Finally, adding the third resistor gives:
• \[ R_{total} = 0.85 + 1.70 = 2.55 \text{ k} \Omega \].

By mastering these calculations, you gain deeper insights into how to control and predict circuit behaviors, crucial for both theoretical understanding and practical applications.