Problem 17 - Physics

Question

A toaster uses a Nichrome heating wire. When the toaster is turned on at \(20^{\circ} \mathrm{C}\), the initial current is \(1.50 \mathrm{~A}\). A few seconds later, the toaster warms up and the current now has a value of \(1.30 \mathrm{~A}\). The average temperature coefficient of resistivity for Nichrome wire is \(4.5 \times 10^{-4}\left(\mathrm{C}^{0}\right)^{-1} .\) What is the temperature of the heating wire?

Step-by-Step Solution

Answer

The temperature of the heating wire is approximately \(248^{\\circ}\mathrm{C}\).

Step 1: Understand the Problem

We need to determine the temperature of the heating wire after it heats up, given the initial and final current and the temperature coefficient of resistivity for Nichrome.

Step 2: Recall Ohm's Law

Ohm's Law states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor: \( I = \frac{V}{R} \).

Step 3: Relate Resistance and Temperature

The resistance \( R \) at a temperature \( T \) can be related to the initial resistance \( R_0 \) at a reference temperature \( T_0 \) by the formula: \( R = R_0 (1 + \alpha \Delta T) \), where \( \alpha \) is the temperature coefficient of resistivity and \( \Delta T \) is the change in temperature.

Step 4: Calculate Initial Resistance

From Ohm's Law, the initial resistance \( R_0 \) can be found using the initial current: \( R_0 = \frac{V}{I_0} \). Since the problem doesn't provide voltage, we'll use relative changes: \( R_0 = \frac{1}{1.50} \).

Step 5: Calculate Final Resistance

Similarly, the resistance after heating \( R \) can be calculated with the final current: \( R = \frac{V}{I} = \frac{1}{1.30} \).

Step 6: Use Temperature Relation Formula

Since \( R = R_0 (1 + \alpha \Delta T) \), substitute \( R \) and \( R_0 \) into the equation: \( \frac{1}{1.30} = \frac{1}{1.50}(1 + 4.5 \times 10^{-4} \Delta T) \).

Step 7: Solve for \( \Delta T \)

Rearrange the equation to solve for \( \Delta T \): \( \Delta T = \frac{\left(\frac{1}{1.30} - \frac{1}{1.50}\right)}{4.5 \times 10^{-4}} \). Calculate \( \Delta T \).

Step 8: Calculate Final Temperature

Now add \( \Delta T \) to the initial temperature to find the final temperature: \( T = 20 + \Delta T \).

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering, describing how voltage, current, and resistance are related. The law can be summarized with the formula:

• \( I = \frac{V}{R} \)

This equation tells us that the current (\( I \)) through a conductor is directly proportional to the voltage (\( V \)) across it and inversely proportional to the resistance (\( R \)) of the conductor.
In simpler terms, if the voltage is increased, the current increases, given that resistance remains constant. Conversely, if the resistance increases while the voltage stays constant, the current decreases.
Ohm's Law is essential because it helps us understand how changing one of these factors affects the others. In this context, when a Nichrome wire heats up, its resistance changes, affecting the current flowing through it.

Resistivity

Resistivity is a material property that measures how much a material opposes the flow of electric current. It is denoted by the Greek letter \( \rho \) and is influenced by factors such as material composition and temperature.
The resistance \( R \) of an object is directly proportional to its resistivity \( \rho \). This is expressed in the equation:

• \( R = \rho \frac{L}{A} \)

where \( L \) is the length and \( A \) is the cross-sectional area of the material.
When discussing resistivity in terms of temperature, we often use the temperature coefficient of resistivity, \( \alpha \), which quantifies how much resistivity changes with temperature. For most conductors, resistivity increases with a rise in temperature.
In our example, the Nichrome wire has a temperature coefficient of resistivity that helps us calculate how its resistance changes as it heats up.

Nichrome Wire

Nichrome wire is an alloy made primarily of nickel and chromium, known for its high resistance and ability to withstand high temperatures. These properties make Nichrome wire an excellent choice for applications like toasters and other heating elements.
One of the distinguishing characteristics of Nichrome is its relatively high temperature coefficient of resistivity, defined as \( 4.5 \times 10^{-4} \left( \text{C}^{0} \right)^{-1} \). This means that for every degree Celsius increase in temperature, its resistance changes by this factor.
In practical electrical applications, knowing the temperature coefficient allows engineers and designers to predict how much resistance will change with fluctuations in temperature; thus, maintaining efficiency and safety in electrical circuits.

• Durable and reliable in high-temperature environments
• Commonly used in heating elements
• Predictable resistance changes with temperature

Current and Resistance Relationship

The relationship between current and resistance is key to understanding how electrical circuits operate, especially when materials within the circuit change temperature. Ohm's Law gives us a direct way to observe this:

• \( I = \frac{V}{R} \)

In this relationship, as resistance (\( R \)) increases, current (\( I \)) decreases if the voltage (\( V \)) remains constant. This inverse relationship is crucial when materials, like Nichrome wire, undergo temperature changes.
In our scenario with a toaster, the initial higher current at a lower temperature means lower resistance. As the temperature rises due to the wire heating, resistance increases, thus reducing the current.
Understanding this relationship allows us to anticipate how devices will behave under various conditions. Predicting these changes ensures circuits can operate safely without exceeding design limits or causing malfunctions. This knowledge is especially useful in designing appliances that regulate heat effectively.