Problem 14 - College Physics

Question

What is the resistance of a NichromeTM wire at \(0.0^{\circ} \mathrm{C}\) if its resistance is 100.00\(\Omega\) at \(11.5^{\circ} \mathrm{C}\) ? The temperature coefficient of resistivity for Nichrome is 0.00040\(\left(\mathrm{C}^{\circ}\right)^{-1}\)

Step-by-Step Solution

Answer

The resistance of the Nichrome wire at 0.0°C is approximately 99.54 Ω.

Step 1: Identify Variables

First, we identify the given variables in the problem. We have:- The resistance at 11.5°C, \( R = 100.00 \, \Omega \).- The temperature \( T = 11.5^{\circ}C \).- The temperature coefficient of resistivity \( \alpha = 0.00040 \, (\mathrm{C}^{\circ})^{-1} \). We need to find the resistance at 0.0°C.

Step 2: Use the Resistance Temperature Formula

The resistance of a wire at a certain reference temperature can be calculated using the formula:\[ R_T = R_0 (1 + \alpha \cdot \Delta T) \]where:- \( R_T \) is the resistance at temperature \( T \), which is \(100.00 \, \Omega \).- \( R_0 \) is the resistance at the reference temperature (0°C), which we need to find.- \( \Delta T \) is the change in temperature, \( T - 0^{\circ}C = 11.5^{\circ}C \).

Step 3: Calculate Resistance at Reference Temperature

Now, we rearrange the formula to find \( R_0 \):\[ R_0 = \frac{R_T}{1 + \alpha \cdot \Delta T} \]Substitute the known values into the equation:\[ R_0 = \frac{100.00}{1 + 0.00040 \cdot 11.5} \]

Step 4: Perform the Calculations

Calculate the value of \( 1 + \alpha \cdot \Delta T \):\[ 1 + 0.00040 \cdot 11.5 = 1 + 0.0046 = 1.0046 \]Then calculate \( R_0 \):\[ R_0 = \frac{100.00}{1.0046} \approx 99.54 \, \Omega \]

Step 5: Summarize the Result

Therefore, the resistance of the Nichrome wire at \(0.0^{\circ}C\) is approximately \(99.54 \, \Omega \).

Key Concepts

Temperature Coefficient of Resistivity

The temperature coefficient of resistivity is an important concept when studying how a material's resistance changes with temperature. This coefficient (denoted as \( \alpha \)) tells us how much the resistivity of a material changes per degree Celsius.
- For example, in the problem above, Nichrome has a temperature coefficient of 0.00040 (\( \text{C}^{\circ} \text{)}^{-1} \)).
- This means for each degree change in temperature, the resistance will change by 0.00040 times its initial value at the reference temperature.

Understanding the temperature coefficient helps predict how the resistance of an object will vary. This can be crucial in designing circuits, ensuring that components will function properly under different thermal conditions. The formula used is:
\[ R = R_0 (1 + \alpha \Delta T) \]
where:

• \( R \) is the resistance at temperature \( T \).
• \( R_0 \) is the resistance at the reference temperature.
• \( \Delta T \) is the temperature change.
• \( \alpha \) is the temperature coefficient of resistivity.

Nichrome Wire

Nichrome is an alloy primarily composed of nickel and chromium. It is popular in applications that require good resistance to heat and electricity. One of its common uses is in heating elements such as toasters and hair dryers. Here are some characteristics of Nichrome:

• Heat Resistance: Nichrome can withstand high temperatures without oxidizing rapidly, making it ideal for hot environments.
• Stability: Its resistance remains relatively constant across different temperature ranges, apart from predictable changes dictated by its temperature coefficient.
• Durability: Being an alloy, it resists wear and corrosion over time.

In the exercise, the properties of Nichrome allow us to apply the temperature coefficient to calculate its resistance change with temperature, showcasing its reliability and adaptability in various applications.

Ohm's Law

Ohm's Law is a fundamental principle used extensively in electrical engineering and physics. It describes the relationship between voltage, current, and resistance in an electrical circuit:
\[ V = I \cdot R \]
where:

• \( V \) is the voltage across the conductor.
• \( I \) is the current through the conductor.
• \( R \) is the resistance of the conductor.

Although Ohm's Law is not directly used in the calculation of temperature effects on resistance in this exercise, understanding this principle is crucial. It allows you to understand how changes in resistance (like those caused by temperature variations) affect the overall behavior of electrical circuits. For instance, a decrease in resistance, due to a decrease in temperature, would typically result in an increase in current, provided the voltage remains constant.

Thermal Effects on Resistance

The thermal effects on resistance in materials, such as Nichrome, play a critical role in various applications. When the temperature of a conductor changes, its atoms vibrate more or less, affecting how easily electrons can flow. Here's how:

• When temperature increases, typically, resistance increases because atoms vibrate more, causing more collisions with moving electrons.
• Conversely, when the temperature decreases, resistance usually decreases as there are fewer atomic vibrations and consequently, fewer collisions.

This relationship is particularly important in the exercise to find the resistance at 0°C from the given resistance at 11.5°C. By understanding the thermal effects on resistance and using the temperature coefficient of resistivity, one can accurately predict and adjust for changes, ensuring better performance and reliability of electrical systems.