Problem 109 - Physics

Question

Eight different values of resistance can be obtained by connecting together three resistors \((1.00,2.00\), and \(3.00 \Omega\) ) in all possible ways. What are they?

Step-by-Step Solution

Answer

Possible resistances: 0.545, 0.83, 1.33, 1.50, 2, 3, 4, 5, 6 ohms.

Step 1: Understand Parallel and Series Combinations

In electrical circuits, resistors can be connected in series or parallel. The total resistance of resistors in series is the sum of their resistances: \(R_{total} = R_1 + R_2 + R_3\). For parallel resistors, the reciprocal of total resistance is the sum of the reciprocals of each resistor: \(\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\).

Step 2: Calculate Series Combinations

When all three resistors are in series: \(R_{total} = 1.00 + 2.00 + 3.00 = 6.00 \Omega\). Two resistors in series with the third one open: \(1+2=3\), \(1+3=4\), \(2+3=5\). Hence, the series combination resistances are 3, 4, 5, and 6 ohms.

Step 3: Calculate Pairwise Series and Parallel Combinations

Calculate the resistance for all pairs in series and their parallel combination with the third:1. Series for \(1.00\) and \(2.00\): \(R_{series} = 3.00\), parallel with \(3.00\) is \(\frac{1}{\frac{1}{3} + \frac{1}{3}} = 1.50\) ohms.2. Series for \(1.00\) and \(3.00\): \(R_{series} = 4.00\), parallel with \(2.00\) is \(\frac{1}{\frac{1}{4} + \frac{1}{2}} = 1.33\) ohms.3. Series for \(2.00\) and \(3.00\): \(R_{series} = 5.00\), parallel with \(1.00\) is \(\frac{1}{\frac{1}{5} + 1} = 0.83\) ohms.

Step 4: Calculate All Parallel Combination of Three Resistors

When all resistors are in parallel: \(\frac{1}{R_{total}} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} = \frac{11}{6}\). So, \(R_{total} = \frac{6}{11} \approx 0.545\) ohms.

Step 5: Identify Unique Resistance Values

Combine series and parallel outcomes: 0.545, 0.83, 1.33, 1.50, 2, 3, 4, 5, 6 ohms. Remove repeated values, resulting in the unique set: 0.545, 0.83, 1.33, 1.50, 2, 3, 4, 5, 6 ohms. This set contains all possible unique resistances.

Key Concepts

Resistors in Series

When you connect resistors in series, the total or equivalent resistance is simply the sum of all individual resistances. Picture resistors as little hurdles blocking the path of current. When you line up hurdles, the total "height" or resistance increases. If you place a 1 ohm, a 2 ohm, and a 3 ohm resistor in series, the total resistance will be 6 ohms:

• Formula: \( R_{total} = R_1 + R_2 + R_3 \)
• Example: For 1, 2, and 3 ohms in series: \( R_{total} = 1 + 2 + 3 = 6 \) ohms

Series resistors offer a straightforward way to increase resistance within a circuit, making it an elementary concept crucial for understanding electrical circuits.

Resistors in Parallel

Resistors in parallel function quite differently from series connections. Imagine cars driving on a multi-lane road. When you add more lanes (or parallel resistors), you reduce the overall traffic congestion (or resistance). The formula to calculate total resistance in parallel is:

• Formula: \( \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \)
• Example: Three resistors of 1 ohm, 2 ohms, and 3 ohms parallel: \( \frac{1}{R_{total}} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} \)
• Solving gives \( R_{total} \approx 0.545 \) ohms

Parallel resistance lowers the total resistance, often resulting in a resistance value lesser than the smallest resistor in the group.

Total Resistance Calculation

Total resistance in a circuit depends on the combination of series and parallel configurations. Sometimes, to find the total resistance in a mixed circuit, you must perform calculations in steps:

• Calculate series combinations and note their resistance values.
• Determine parallel combinations and use their reciprocal formula.
• Evaluate mixed configurations by breaking them into simpler series or parallel groups and calculating them step by step.

For example, a series connection of 2 ohms and 3 ohms gives 5 ohms. This 5 ohms in parallel with 1 ohm leads to a total resistance:

• Step 1: \( R_{series} = 2 + 3 = 5 \) ohms
• Step 2: \( \frac{1}{R_{total}} = \frac{1}{5} + \frac{1}{1} = 1 + 0.2 = 1.2 \)
  Convert: \( R_{total} = \frac{1}{1.2} \approx 0.83 \) ohms

This method helps in finding precise resistance measurements in complex circuits.

Unique Resistance Values

When combining multiple resistors in various series and parallel configurations, it's possible to achieve different resistance values. From our three resistors (1 ohm, 2 ohms, and 3 ohms), multiple unique total resistances emerged: 0.545, 0.83, 1.33, 1.50, 2, 3, 4, 5, and 6 ohms.
By arranging these resistors:

• Identify and calculate every possible configuration.
• Verify each configuration to ensure no duplicate values are listed.
• Note that each unique resistance provides different circuit functionalities.

Through experimentation and calculation, this exercise helps in grasping the significance of distinct resistance values and their applications in electrical design.