91Ó°ÊÓ

Temperature coefficient of resistivity, denoted by \( \alpha \), is a crucial concept in understanding how materials react to temperature changes. As temperature varies, the resistivity of a material typically changes too. This is particularly important for materials like tungsten wire, known for its stability and high melting points.
This coefficient \( \alpha \) is defined as the change in resistivity per degree of temperature change relative to the resistivity at a reference temperature, often 20°C for calculations. In the exercise, \( \alpha \) for tungsten is given as \( 4.5 \times 10^{-3} \left( \mathrm{C}^{\circ} \right)^{-1} \).
When heating a wire from 20°C to 1320°C, the resistivity significantly increases due to this temperature dependence. The formula for resistance at a different temperature is given by:
\[ R = R_0 (1 + \alpha \Delta T) \]
where \( R_0 \) is the original resistance and \( \Delta T \) is the temperature change. Understanding \( \alpha \) helps predict how electrical properties will change with temperature, which is vital for designing circuits and materials for varying thermal environments.

Ohm's Law is a fundamental principle in electronics and electrical engineering. It establishes a simple relationship between voltage (V), current (I), and resistance (R) as follows:
\[ V = I \cdot R \]
This law states that the voltage across a conductor is proportional to the current flowing through it, and the constant of proportionality is the resistance. In the given exercise, with an applied voltage of 120 V and a current of 1.5 A, the resistance can be calculated directly:
\[ R = \frac{V}{I} = \frac{120\,V}{1.5\,A} = 80\,\Omega \]
Understanding Ohm's Law aids in solving the exercise by enabling us to compute the resistance at higher temperatures, a step crucial in determining the wire's length. This principle is foundational for analyzing how designs will perform when operating under different conditions.

Thermal expansion describes how the size of an object changes in response to temperature fluctuations. As temperature increases, most materials expand, affecting their dimensions.
In the context of this exercise, thermal expansion could influence the length of the tungsten wire. However, the exercise directed us to neglect these effects—a simplifying assumption that allows us to focus on electrical properties like resistivity and resistance.
Despite this simplification, understanding thermal expansion is essential because it impacts real-world applications. Different materials expand at different rates, characterized by their coefficient of linear expansion. For practical purposes such as measuring precision or constructing infrastructure, accounting for thermal expansion ensures accuracy and safety.

Calculating the cross-sectional area of a wire is necessary for understanding its electrical properties, as resistance is directly related to both length and this cross-sectional area. For a circular wire, the cross-sectional area \( A \) is calculated using the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius.
For the tungsten wire in the exercise, with a radius of \( 0.075 \text{ mm} \), converting to meters results in \( r = 0.075 \times 10^{-3} \text{ m} \). Plugging into the formula gives:
\[ A = \pi (0.075 \times 10^{-3})^2 \approx 1.767 \times 10^{-8} \text{ m}^2 \]
This computed cross-sectional area allows us to further calculate the wire's length using the resistance formula \( R_0 = \frac{\rho_0 L}{A} \). By understanding these calculations, you can unravel how physical dimensions influence electrical resistance, aiding in the design and analysis of electronic components.