Problem 27 - University Physics with Modern Physics

Question

A strand of wire has resistance 5.60\(\mu \Omega .\) Find the net resistance of 120 such strands if they are (a) placed side by side to form a cable of the same length as a single strand, and (b) connected end to end to form a wire 120 times as long as a single strand.

Step-by-Step Solution

Answer

(a) 0.0467\(\mu\Omega\); (b) 672\(\mu\Omega\).

Step 1: Understanding Parallel Resistance

When resistances are placed side by side, they are in parallel. The formula to calculate the equivalent resistance \( R_{ ext{parallel}} \) for parallel connections of identical resistors is: \[ R_{ ext{parallel}} = \frac{R}{n} \] where \( R \) is the resistance of one strand and \( n \) is the number of strands. Here, \( R = 5.60 \mu \Omega \) and \( n = 120 \).

Step 2: Calculating Parallel Resistance

Using the formula for parallel resistance: \[ R_{ ext{parallel}} = \frac{5.60 \mu \Omega}{120} \]Calculating, \( R_{ ext{parallel}} = 0.0467 \mu \Omega \).

Step 3: Understanding Series Resistance

For resistances connected end to end, they are in series. The formula to calculate the equivalent resistance \( R_{ ext{series}} \) for series connections is: \[ R_{ ext{series}} = R \times n \] where \( R \) is the resistance of one strand and \( n \) is the number of strands.

Step 4: Calculating Series Resistance

Using the formula for series resistance: \[ R_{ ext{series}} = 5.60 \mu \Omega \times 120 \] Calculating, \( R_{ ext{series}} = 672 \mu \Omega \).

Key Concepts

Parallel Resistance

When we connect resistors side by side, it is known as a parallel connection. This concept is pivotal in electrical circuits because it affects how we calculate the total resistance.

• Each strand shares the total current flowing through the circuit.
• The voltage across each resistor remains the same.

To find the equivalent resistance in parallel (\( R_{\text{parallel}} \)), use the formula:\[R_{\text{parallel}} = \frac{R}{n}\]Here the resistance of a single strand is \( 5.60 \mu \Omega \), and the number of strands \( n \) is 120. Substituting in the formula, you get: \( R_{\text{parallel}} = \frac{5.60 \mu \Omega}{120} = 0.0467 \mu \Omega \). The result shows that the more resistors you add in parallel, the lower the overall resistance becomes. This is because there's an increased number of paths for the current to flow through.

Series Resistance

Resistances that are connected end to end have a series configuration. This configuration impacts how resistance accumulates in a circuit.

• All current flows through each strand sequentially.
• The total voltage is the sum of the voltages across each resistor.

For series resistance (\( R_{\text{series}} \)), the formula is simpler:\[R_{\text{series}} = R \times n\]Using a strand resistance of \( 5.60 \mu \Omega \) and \( n = 120 \) strands, we calculate:\( R_{\text{series}} = 5.60 \mu \Omega \times 120 = 672 \mu \Omega \).As you can see, the total resistance increases with each additional resistor when they're in series, as each provides more resistance to the flow of electrical current.

Resistance Calculation

Calculating resistance is essential to understanding how electrical circuits function. Let's review the steps to ensure clarity in both parallel and series setups.

• For parallel circuits, you divide the resistance of a single resistor by the number of resistors: \( R_{\text{parallel}} = \frac{R}{n} \).
• In series circuits, simply multiply the resistance by the number of resistors: \( R_{\text{series}} = R \times n \).

These calculations offer insight into how individual components affect overall circuit behavior. They help predict how a circuit will perform once merged with other components or loads. Understanding parallel and series resistance calculations can therefore guide in crafting circuits with desired electrical characteristics.