Problem 8 - Physics for Scientists and Engineers with Modern Physics

Question

(I) How many \(10-\Omega\) resistors must be connected in series to give an equivalent resistance to five \(100-\Omega\) resistors connected in parallel?

Step-by-Step Solution

Answer

2 resistors must be connected in series.

Step 1: Understand the Problem

We need to find the number of 10-ohm resistors connected in series that will provide the same equivalent resistance as five 100-ohm resistors connected in parallel.

Step 2: Calculate Equivalent Resistance of Parallel Resistors

The formula for calculating the equivalent resistance (R_{eq}) of resistors connected in parallel is given by:\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} \]Since each of the five resistors has a resistance of 100 ohms, we substitute:\[ \frac{1}{R_{eq}} = \frac{1}{100} + \frac{1}{100} + \frac{1}{100} + \frac{1}{100} + \frac{1}{100} \]\[ \frac{1}{R_{eq}} = \frac{5}{100} \]\[ R_{eq} = \frac{100}{5} = 20 \, \Omega \]

Step 3: Determine the Series Resistance Required

In a series circuit, the total resistance (R_{total}) is simply the sum of the individual resistances. Therefore, to achieve an equivalent resistance of 20 ohms using 10-ohm resistors connected in series, we set up the equation:\[ R_{total} = 10n = 20 \]where R_{total} = 20 \, \Omega and n is the number of 10-ohm resistors.

Step 4: Solve for the Number of Resistors

Solving for n:\[ 10n = 20 \]\[ n = \frac{20}{10} \]\[ n = 2 \]

Step 5: Conclusion

Therefore, 2 resistors of 10 ohms each connected in series will have an equivalent resistance of 20 ohms, which matches the resistance of five 100-ohm resistors in parallel.

Key Concepts

Equivalent Resistance

The concept of equivalent resistance is crucial when dealing with circuits that involve multiple resistors either in series or parallel. Equivalent resistance essentially means the total resistance that a single resistor would have to have in order to replace a combination of resistors without altering the current in the circuit.

When resistors are connected in parallel, the overall equivalent resistance is calculated using the formula:

• \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} \)

This equation shows that the reciprocal of the total resistance is equal to the sum of the reciprocals of the individual resistances.

For series connections, it's more straightforward as you simply add up all the resistances:

• \( R_{total} = R_1 + R_2 + \cdots + R_n \)

In the original exercise, calculating the equivalent resistance of the parallel resistors helped determine how many 10-ohm resistors were necessary in series to match this resistance.

Ohm's Law

Ohm's Law is a fundamental principle in the study of electricity and circuits, stating that the current through a conductor between two points is directly proportional to the voltage across the two points. It is given by:

• \( V = IR \)

Where:

• \( V \) is the voltage (in Volts)
• \( I \) is the current (in Amperes)
• \( R \) is the resistance (in Ohms)

Ohm's Law is used throughout electronics to relate the voltage, current, and resistance in a circuit. Understanding this relationship is essential for determining one of these values when the other two are known.

It’s important to grasp that when using Ohm's Law in circuits with combinations of resistors, the total or equivalent resistance affects the total current flow. Knowing how to calculate equivalent resistance accurately allows for correct applications of Ohm’s Law in complex circuits.

Resistance Calculation

Resistance calculation is a key skill in ensuring circuits operate correctly and safely. When calculating resistance:

• Identify if you are working with a series or parallel circuit.
• Use the appropriate formula to find the equivalent resistance (as previously covered in the equivalent resistance concept).

For example, in a series circuit, the total resistance \( R_{total} \) is simply the sum of each individual resistance. In a parallel circuit, you make use of the reciprocal formula to find \( R_{eq} \).

In practical terms, knowing how to calculate total resistance informs on how much of a load a circuit can handle and helps prevent overloading that can lead to overheating or failure. It also assists in determining the number of components, as in the original exercise where calculating the resistance helped find how many resistors were needed in series to achieve a particular resistance.