Problem 55 - Fundamentals of Physics

Question

A Nichrome heater dissipates \(500 \mathrm{~W}\) when the applied potential difference is \(110 \mathrm{~V}\) and the wire temperature is \(800^{\circ} \mathrm{C}\). What would be the dissipation rate if the wire temperature were held at \(200^{\circ} \mathrm{C}\) by immersing the wire in a bath of cooling oil? The applied potential difference remains the same, and \(\alpha\) for Nichrome at \(800^{\circ} \mathrm{C}\) is \(4.0 \times 10^{-4} \mathrm{~K}^{-1}\)

Step-by-Step Solution

Answer

The dissipation rate at 200°C is approximately 620 W.

Step 1: Understand the formula for resistance with temperature

The resistance of a wire at any temperature can be calculated using the formula \( R = R_0 (1 + \alpha (T - T_0)) \), where \( R_0 \) is the resistance at a reference temperature \( T_0 \), \( \alpha \) is the temperature coefficient of resistance, and \( T \) is the new temperature.

Step 2: Calculate initial resistance at 800°C

Find the resistance using the given power and voltage at 800°C. Using the formula for power \( P = \frac{V^2}{R} \), rearrange to find \( R = \frac{V^2}{P} = \frac{110^2}{500} \approx 24.2 \, \Omega \). This is the resistance at 800°C.

Step 3: Calculate resistance at new temperature 200°C

Now find what the resistance would be at 200°C. Using the formula from Step 1, substitute \( R_0 = 24.2 \, \Omega \), \( \alpha = 4.0 \times 10^{-4} \), \( T_0 = 800 \, ^\circ\mathrm{C} \), and \( T = 200 \, ^\circ\mathrm{C} \). Then, \[ R_{200} = 24.2 \times (1 + 4.0 \times 10^{-4} \times (200 - 800)) \approx 19.5 \, \Omega \].

Step 4: Calculate new power dissipation at 200°C

Using the resistance found for 200°C, calculate the new power dissipation. Use the power formula again: \[ P_{200} = \frac{110^2}{R_{200}} = \frac{110^2}{19.5} \approx 620 \, \mathrm{W} \].

Key Concepts

Nichrome Wire

Nichrome is a popular material for making heating elements due to its high electrical resistance and durability. It is an alloy made primarily of nickel and chromium, which provides excellent resistance to oxidation and corrosion at high temperatures. Here are some attributes of Nichrome wires that make them suitable for applications like heating circuits:

• High melting point: Nichrome can withstand extreme temperatures without melting, making it ideal for use in heating appliances.
• Consistent resistance: Its resistance remains relatively constant over a wide temperature range, which translates to efficient and predictable heating performance.
• Durability and flexibility: These wires can be shaped and coiled, allowing easy integration into complex heating setups.

Understanding these properties can help comprehend why Nichrome is an effective choice for resistive heating, such as in the exercise given.

Temperature Coefficient of Resistance

The temperature coefficient of resistance (\(\alpha\)) is a factor that indicates how much the resistance of a material changes with temperature. Metals, including Nichrome, typically have positive temperature coefficients, meaning their resistance increases with temperature. To calculate how resistance changes with temperature, we use the formula:

\[ R = R_0 (1 + \alpha (T - T_0)) \]

• \(R\): Resistance at temperature \(T\).
• \(R_0\): Resistance at a reference temperature \(T_0\).
• \(\alpha\): Temperature coefficient of resistance.

This relationship is crucial for determining the performance of devices operating at different temperatures. For instance, in our exercise, the resistance of the Nichrome wire changes when the temperature drops from 800°C to 200°C. By applying the formula, it’s possible to predict how much resistance changes and how this affects the power dissipation of the wire.

Power Dissipation in Resistors

Power dissipation in resistors refers to the process of converting electrical energy into heat energy within the resistor. This occurs due to the resistance to the flow of electric current, and it's calculated using the formula:

\[ P = \frac{V^2}{R} \]
where

• \(P\) is the power dissipated,
• \(V\) is the voltage across the resistor,
• and \(R\) is the resistance.

In resistive materials like Nichrome, power dissipation results in the generation of heat, which is exploited in applications like electric heaters. A key aspect of power dissipation to remember is that it's directly related to both the voltage applied and the resistance of the material. Hence, if the resistance of the wire changes due to temperature variations, the power dissipation will also change. Using the same voltage, we applied the formula in the exercise to calculate the change in power dissipation at two different temperatures, showing how altering resistance affects heat output.