Problem 26 - Physics

Question

A piece of Nichrome wire has a radius of \(6.5 \times 10^{-4} \mathrm{~m}\). It is used in a laboratory to make a heater that uses \(4.00 \times 10^{2} \mathrm{~W}\) of power when connected to a voltage source of \(120 \mathrm{~V}\). Ignoring the effect of temperature on resistance, estimate the necessary length of wire.

Step-by-Step Solution

Answer

The necessary length of Nichrome wire is approximately 43.47 meters.

Step 1: Identify Given Values

We are given the radius of the Nichrome wire, \( r = 6.5 \times 10^{-4} \text{ m} \), the power used by the heater, \( P = 400 \text{ W} \), and the voltage applied, \( V = 120 \text{ V} \).

Step 2: Recall the Formula for Power

The power consumed by the heater can be expressed using the formula \( P = \frac{V^2}{R} \), where \( V \) is the voltage and \( R \) is the resistance of the wire. Rearrange the formula to solve for resistance: \( R = \frac{V^2}{P} \).

Step 3: Calculate the Resistance

Substitute the given values into the formula \( R = \frac{120^2}{400} \) to find the resistance of the wire. Simplifying gives \( R = \frac{14400}{400} = 36 \, \Omega \).

Step 4: Recall the Formula for Resistance of a Wire

The resistance \( R \) of a wire is given by \( R = \rho \frac{L}{A} \), where \( \rho \) is the resistivity of Nichrome, \( L \) is the length of the wire, and \( A \) is the cross-sectional area. The resistivity of Nichrome is approximately \( \rho = 1.10 \times 10^{-6} \Omega \cdot \text{m} \).

Step 5: Calculate the Cross-Sectional Area

The cross-sectional area \( A \) of the wire, assuming it is a cylinder, is calculated as \( A = \pi r^2 \). Substitute the radius: \( A = \pi (6.5 \times 10^{-4})^2 = \pi \times 4.225 \times 10^{-7} \text{ m}^2 \).

Step 6: Rearrange Resistance Formula for Length

Solve for the length \( L \) using the resistance formula: \( L = \frac{R \cdot A}{\rho} \). Substitute \( R = 36 \Omega \), \( A = \pi \times 4.225 \times 10^{-7} \text{ m}^2 \), and \( \rho = 1.10 \times 10^{-6} \Omega \cdot \text{m} \).

Step 7: Solve for Length of Wire

Calculate \( L = \frac{36 \times \pi \times 4.225 \times 10^{-7}}{1.10 \times 10^{-6}} \). First calculate \( \pi \times 4.225 \times 10^{-7} = 1.3283 \times 10^{-6} \), then \( L = \frac{36 \times 1.3283 \times 10^{-6}}{1.10 \times 10^{-6}} \approx 43.47 \text{ m}\).

Key Concepts

Nichrome Wire

Nichrome wire is a popular material for making heating elements. It's an alloy composed mainly of nickel and chromium, which gives it unique properties that make it useful in high-temperature applications.
One of its key characteristics is high resistivity, meaning it doesn't conduct electricity as well as other metals, like copper. This property makes Nichrome effective at converting electrical energy into heat, which is why it's often used in electric heaters and toasters.
Besides, Nichrome has a relatively stable resistance across a wide range of temperatures, which is beneficial for maintaining consistent heating. This stability makes calculations easier when designing systems like laboratory heaters. In calculations involving Nichrome, we typically assume its resistivity does not change with temperature.

Power and Voltage Relation

Understanding the relationship between power and voltage is critical in electrical engineering. Power, denoted as \(P\), is the rate at which energy is used or generated. It's expressed in watts (W). Voltage, \(V\), is the potential difference that drives the current through a conductor, expressed in volts (V).
There's a formula connecting these two:

• \(P = \frac{V^2}{R}\)

where \(R\) is resistance in ohms (Ω). This shows that the power consumed in a circuit is directly proportional to the square of the voltage, when resistance is constant.
In practical terms, for a heater consuming 400 watts when connected to a 120-volt source, you can determine the resistance of the heater element using the formula: \(R = \frac{V^2}{P}\). This understanding is important for designing circuits and ensuring that all components operate safely and efficiently.

Resistance Formula

The resistance of a wire is determined by several factors and can be calculated using the formula:

• \( R = \rho \frac{L}{A} \)

In this formula, \(R\) represents resistance, \(\rho\) is the resistivity of the material, \(L\) is the length of the wire, and \(A\) is the cross-sectional area.
Resistivity \(\rho\) is a material-specific property, indicating how strongly a material opposes the flow of electric current. For Nichrome, this is approximately \(1.10 \times 10^{-6}\, \Omega \cdot \text{m}\).
Length \(L\) and area \(A\) are geometric factors. Longer wires have greater resistance, while thicker wires (larger area) have less resistance. This formula allows you to design wires for specific applications by choosing the right material, length, and thickness to achieve the desired resistance.

Cross-Sectional Area Calculation

Calculating the cross-sectional area of a wire is an essential step in determining resistance, as shown in the formula \( R = \rho \frac{L}{A} \). For a cylindrical wire, such as the Nichrome wire in this example, the cross-sectional area \(A\) is calculated using

• \( A = \pi r^2 \)

where \(r\) is the radius of the wire.
For a wire with a radius of \(6.5 \times 10^{-4}\, \text{m}\), the area calculation becomes \( A = \pi (6.5 \times 10^{-4})^2 = \pi \times 4.225 \times 10^{-7}\, \text{m}^2\).
This calculated area is crucial for further calculations, such as determining resistance and the necessary length of wire needed in a heater design. By understanding how to find the cross-sectional area, engineers can better predict how a wire will perform in an electrical circuit.