91Ó°ÊÓ

Resistivity is a fundamental property of materials that affects how easily electric current can pass through them. It is denoted by the Greek letter \( \rho \) and is measured in ohm-meters (\( \Omega \cdot m \)). The concept of resistivity is crucial for understanding electrical resistance, as it indicates how much a material opposes the flow of electric current. High resistivity means that the material does not conduct electricity well, while low resistivity indicates that it allows electric current to flow more freely.
In practical terms, resistivity helps us determine the resistance of a wire using the formula:

• \( R = \frac{\rho L}{A} \)

where \( R \) is the resistance, \( L \) is the length of the wire, and \( A \) is the cross-sectional area. Each material has a different resistivity, which affects how we choose materials for wiring based on their ability to conduct electricity efficiently and their ability to withstand various electrical and environmental conditions.

The cross-sectional area of a wire plays a significant role in determining its resistance. In simple terms, the cross-sectional area is the surface area of the wire's end, which can be thought of as a slice of the wire. For a circular wire, the cross-sectional area \( A \) is calculated using the formula:

• \( A = \frac{\pi d^2}{4} \)

where \( d \) is the diameter of the wire. A larger cross-sectional area allows more electric current to pass through, resulting in lower resistance. Conversely, a smaller cross-sectional area restricts the flow of current, increasing resistance.
Understanding this concept is important when designing circuits, as the wire’s cross-section must be appropriate for the intended current flow to ensure efficient operation without overheating. Thus, the choice of wire diameter is directly influenced by the desired electrical characteristics.

Understanding the diameter ratio is essential in exercises where wires of varying lengths have the same resistance, like the one given. In this specific problem, both wires have equal resistance, but their lengths differ, with one being twice as long as the other. Therefore, we need to calculate the ratio of their diameters.
We start by using the resistance formula \( R = \frac{\rho L}{A} \). Given that the resistance \( R \) is the same for both wires, we set up the equation:

• \( \frac{L_1}{d_1^2} = \frac{2L_1}{d_2^2} \)

Cross-multiplying yields: \( d_2^2 = 2d_1^2 \). By taking the square root of both sides, we find that:

• \( \frac{d_2}{d_1} = \sqrt{2} \)

This calculation shows that the diameter of the longer wire is \( \sqrt{2} \) times that of the shorter wire. This ratio ensures that despite differing lengths, both wires possess the same resistance. This principle is crucial in applications where maintaining constant electrical characteristics is important, even when physical dimensions change.