Problem 55 - College Physics

Question

You absentmindedly solder a 69.8 \(\mathrm{k} \Omega\) resistor into a circuit where a 36.5 \(\mathrm{k} \Omega\) should be. How can you get the proper resistance without replacing the bigger resistor or removing anything from the circuit?

Step-by-Step Solution

Answer

Add a 76.52 kΩ resistor in parallel with the 69.8 kΩ resistor.

Step 1: Understanding the Problem

You have mistakenly used a 69.8 kΩ resistor instead of a 36.5 kΩ resistor. You need to adjust the resistance without removing the 69.8 kΩ resistor or adding extra components to the setup.

Step 2: Consider Parallel Resistance

To reduce the effective resistance to 36.5 kΩ, consider adding another resistor in parallel with the 69.8 kΩ. This approach allows the total resistance to decrease.

Step 3: Use Parallel Resistance Formula

For resistors in parallel, the formula is \( R_{ ext{total}} = \frac{R_1 \times R_2}{R_1 + R_2} \). Here, \( R_1 = 69.8 \) kΩ and you want \( R_{ ext{total}} = 36.5 \) kΩ. You need to solve for \( R_2 \).

Step 4: Set Up the Equation

Set the equation using the values: \[ 36.5 = \frac{69.8 \times R_2}{69.8 + R_2} \]. Rearrange this to solve for \( R_2 \).

Step 5: Solve for R2

Rearrange the equation: \( 36.5 (69.8 + R_2) = 69.8 R_2 \), which simplifies to \( 36.5 \times 69.8 + 36.5 R_2 = 69.8 R_2 \). Simplify further to \( 36.5 \times 69.8 = (69.8 - 36.5) R_2 \).

Step 6: Calculate the Result

Calculate \( R_2 \): \( R_2 = \frac{36.5 \times 69.8}{69.8 - 36.5} \). Doing the math: \( 36.5 \times 69.8 = 2548.7 \) and \( 69.8 - 36.5 = 33.3 \). Then, \( R_2 = \frac{2548.7}{33.3} \approx 76.52 \) kΩ.

Key Concepts

Parallel Resistance

Understanding parallel resistance is crucial when working with circuits. When resistors are connected in parallel, it means that they are connected across the same two points and share the same voltage. The effective resistance of resistors in parallel is different than that of resistors placed in series.
In a parallel configuration, the total resistance is lower than the smallest resistor in the group. This happens because the current has multiple paths to take, effectively decreasing the overall opposition, or resistance, in the circuit.
Key Points about Parallel Resistance:

• Parallel resistors offer alternate paths for current flow.
• The total equivalent resistance is less than any individual resistor in the setup.
• The formula used is: \( R_{total} = \frac{R_1 \times R_2}{R_1 + R_2}\)

Using the formula, you can always adjust a circuit's resistance by adding new parallel resistors, thus fine-tuning your circuit's performance without removing existing components.

Circuit Troubleshooting

Circuit troubleshooting involves applying logical and systematic methods to identify and solve issues within an electrical circuit.
Whether due to human error, like in the given problem where the wrong resistor was installed, or other issues such as a damaged component.
Steps for Effective Circuit Troubleshooting:

• Understand the circuit layout and the intended function of each component.
• Identify and isolate the problem area within the circuit.
• Consider potential solutions such as adjusting resistor values or adding components like parallel resistors.
• Implement the simplest solution first and test the circuit.
• Refine and test further if the initial solution does not fully resolve the issue.

In this exercise, adjusting the resistance with a parallel resistor offers a solution. It's important to account for how changes can affect the entire circuit; sometimes minor tweaks can significantly enhance or disrupt functionality.

Resistance Calculations

Resistance calculations are foundational to circuit design and troubleshooting. Calculating the right value ensures the circuit operates as intended.
In our specific problem, one must calculate net resistance using parallel resistance concepts to achieve a specific goal.
Step-by-Step Resistance Calculations:

• Identify the resistors involved and their configurations within the circuit.
• Apply the appropriate formulas, such as:\[R_{total} = \frac{R_1 \times R_2}{R_1 + R_2}\]for parallel resistors. This formula helps resolve the effective resistance issue caused by the misplaced resistor.
• Rearrange and solve equations carefully to find unknown resistor values.
• Check calculations several times to ensure accuracy. Even small mistakes can lead to different results, impacting the circuit's function.
• Verify by physically testing in a non-critical circuit environment before finalizing.

Accurate resistance calculations assure that circuits deliver the desired performance, preserving both efficiency and safety.