Problem 47 - College Physics

Question

Calculate the (a) maximum and (b) minimum values of resistance that can be obtained by combining resistors of \(36 \Omega,\) \(47 \Omega,\) and 51\(\Omega .\)

Step-by-Step Solution

Answer

Max resistance: 134 Ω in series; Min resistance: ~14.56 Ω in parallel.

Step 1: Identify Possible Combinations

To find the maximum and minimum resistance values, we need to consider the possible combinations of arranging the resistors: all in series, all in parallel, or combinations of both.

Step 2: Calculate Maximum Resistance (Series Combination)

In a series combination, the total resistance is simply the sum of all resistor values. So, calculate:\[R_{max} = R_1 + R_2 + R_3 = 36 \, \Omega + 47 \, \Omega + 51 \, \Omega\]Calculate the sum to find the maximum resistance.

Step 3: Solution to Series Calculation

Sum the resistors:\[ R_{max} = 36 + 47 + 51 = 134 \, \Omega \]Therefore, the maximum resistance obtainable by connecting the resistors in series is 134 Ω.

Step 4: Calculate Minimum Resistance (Parallel Combination)

For resistors in parallel, the reciprocal of the total resistance is the sum of the reciprocals of each resistance:\[\frac{1}{R_{min}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\]Substitute the known values:\[\frac{1}{R_{min}} = \frac{1}{36} + \frac{1}{47} + \frac{1}{51}\]Calculate the values to find the reciprocal of the minimum resistance.

Step 5: Solution to Parallel Calculation

Perform the calculations:\[\frac{1}{R_{min}} \approx \frac{1}{36} + \frac{1}{47} + \frac{1}{51} \approx 0.0278 + 0.0213 + 0.0196 \approx 0.0687\]Therefore, the minimum resistance is:\[ R_{min} = \frac{1}{0.0687} \approx 14.56 \, \Omega \]

Step 6: Conclusion

By arranging the resistors in series, the maximum resistance obtained is 134 Ω. By arranging them in parallel, the minimum resistance obtained is approximately 14.56 Ω.

Key Concepts

Understanding Series Resistance

When resistors are connected in a series, they are aligned end-to-end so that the current flows through each resistor one after the other. The key principle here is that the total resistance of the series is simply the sum of each individual resistor's resistance. This is because each additional resistor adds more opposition to the flow of electric current.

Series resistance can be calculated using the following formula:

• \[ R_{total} = R_1 + R_2 + R_3 + \ldots \]

In our original exercise, the resistors with values 36 Ω, 47 Ω, and 51 Ω are combined in series to achieve the maximum possible resistance. By adding these values together, we find that the total resistance is \( 134 \, \Omega \), meaning the circuit resists the current flow the most when arranged in this way.
Understanding this concept makes it easier to design circuits where high resistance is required, such as in limiting current flow to protect sensitive components.

Exploring Parallel Resistance

In a parallel circuit, resistors are connected across the same two points, providing multiple paths for the electrical charge to flow. Here, the total resistance decreases as more resistors are added. This is because a parallel arrangement allows for more pathways for the current, reducing the load on each resistor.

The formula used to calculate total resistance in parallel is the reciprocal of the sum of the reciprocals of each resistor:

• \[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots \]

In our exercise, by applying this formula to resistors 36 Ω, 47 Ω, and 51 Ω, we calculated the total resistance to be approximately \( 14.56 \, \Omega \). This shows the minimum resistance because multiple paths allow for more current to pass through the circuit.
Parallel resistances are useful in applications where low resistance is desired, such as in ensuring power is supplied efficiently over long distances.

Performing Resistance Calculations

Resistance calculations are fundamental in electrical engineering to determine how resistors affect a circuit. Calculating resistance accurately is crucial for designing circuits that function correctly and safely.

When calculating resistance:

• For series resistors, simply sum them up using: \( R_{total} = R_1 + R_2 + \ldots \)
• For parallel resistors, use the reciprocal formula: \( \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots \)

These calculations allow you to foresee how changes in resistor configurations will influence the overall resistance of the circuit. Accurate calculations are vital for ensuring each component receives the intended voltage and current, protecting the devices from potential damage.
Whether designing a simple circuit or a robust electrical system, understanding how to perform these basic resistance calculations will help improve your designs and enhance reliability.