Calculating New Resistance After Twisting Wire Halves

Introduction

Resistors are fundamental components in electrical circuits, and understanding how their resistances behave under different conditions is crucial for various engineering applications. This article explores a specific scenario involving a piece of wire and the changes in resistance when the wire is bent and twisted. We will delve into the concepts of resistance calculation and the behavior of resistors in parallel, providing a clear, step-by-step explanation.

Understanding the Problem

Consider a 4 ohm resistive piece of wire that is bent at its midpoint such that the two halves are twisted together. The question is: what is the new resistance of the wire?

Concepts Involved

This problem revolves around the principles of resistor behavior when connected in parallel and the relationship between resistivity, length, and cross-sectional area.

Method A: Using the Resistor Formula

The Original Calculation

The initial resistivity #961;original of the wire is constant. The formula for resistance is given by:

{displaystyle R rho frac{L}{A}}

where:

R is the resistance in ohms (Ω) #961; is the resistivity of the material in ohm-meters (Ω?m) L is the length of the wire in meters (m) A is the cross-sectional area of the wire in square meters (m2)

In our case, the wire is bent at its midpoint, effectively halving the length L while doubling the cross-sectional area A. Therefore, the new resistance can be calculated as:

{displaystyle R_{new} rho frac{L/2}{2A} frac{1}{4} rho frac{L}{A} frac{1}{4} R}

Since the original resistance R is 4 ohms, the new resistance is:

{displaystyle R_{new} frac{1}{4} times 4 Omega 1 Omega}

Method B: Using Parallel Resistance Calculation

Halves in Parallel Calculation

Alternatively, we can treat each half of the wire as a resistor. Each half has a resistance of:

{R_1 R_2 frac{R}{2} 2 Omega}

The formula for the total resistance R_{text{total}} of resistors in parallel is:

{frac{1}{R_{text{total}}} frac{1}{R_1} frac{1}{R_2}}

Substituting the values:

{frac{1}{R_{text{total}}} frac{1}{2 Omega} frac{1}{2 Omega} frac{1}{2} frac{1}{2} 1}

Hence, the total resistance is:

{R_{text{total}} 1 Omega}

Conclusion

The new resistance of the twisted wire is 1 ohm. This calculation demonstrates the principle of resistors in parallel and reinforces the importance of practice and understanding in problem-solving. It is essential to approach homework not just for completing tasks but for mastering the underlying concepts.

Additional Insights

This problem highlights the behavior of resistors in different configurations and the practical applications of electrical principles. Understanding such basics is crucial for more advanced electrical engineering and physics studies.

References and Further Reading

For readers interested in further exploration, the following resources may be helpful:

General Science: Britannica - Physics Electrical Engineering: All About Circuits - Resistors in Parallel Resistor Concepts: Electronics Products - Resistivity and Conductivity