Problem 17 - Fundamentals of Physics

Question

A wire of Nichrome (a nickel-chromium-iron alloy commonly used in heating elements) is \(1.0 \mathrm{~m}\) long and \(1.0 \mathrm{~mm}^{2}\) in cross- sectional area. It carries a current of 4.0 A when a \(2.0 \mathrm{~V}\) potential difference is applied between its ends. Calculate the conductivity \(\sigma\) of Nichrome.

Step-by-Step Solution

Answer

The conductivity \( \sigma \) of the Nichrome wire is \( 2 \times 10^6 \text{ S/m} \).

Step 1: Understand the Problem

We are given a Nichrome wire of specific dimensions and electrical characteristics, and we need to find its conductivity using the current, potential difference, and dimensions provided.

Step 2: Write Down the Known Values

Length of the wire, \( L = 1.0 \text{ m} \); cross-sectional area, \( A = 1.0 \text{ mm}^2 = 1.0 \times 10^{-6} \text{ m}^2 \); current, \( I = 4.0 \text{ A} \); potential difference, \( V = 2.0 \text{ V} \).

Step 3: Use Ohm's Law to Find Resistance

Ohm's Law states \( V = IR \), where \( R \) is the resistance. Rearrange to find \( R \):\[ R = \frac{V}{I} = \frac{2.0}{4.0} = 0.5 \text{ ohms} \].

Step 4: Apply the Formula for Resistivity

We use the formula for resistivity, \( \rho \):\[ \rho = \frac{RA}{L} = \frac{0.5 \times 1.0 \times 10^{-6}}{1.0} = 0.5 \times 10^{-6} \text{ ohm} \cdot \text{m} \].

Step 5: Calculate Conductivity

Conductivity \( \sigma \) is the reciprocal of resistivity:\[ \sigma = \frac{1}{\rho} = \frac{1}{0.5 \times 10^{-6}} = 2 \times 10^{6} \text{ S/m} \].

Step 6: Complete the Calculation

The conductivity of the Nichrome wire is calculated as \( 2 \times 10^6 \text{ S/m} \).

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in the field of electronics and physics. It helps to describe the relationship between voltage, current, and resistance in an electrical circuit. The law is expressed with the formula \( V = IR \), where \( V \) is the voltage across the circuit, \( I \) is the current flowing through the circuit, and \( R \) is the resistance offered by the circuit.

When you know any two of these three quantities, you can easily calculate the third. For example, in our exercise, we were given the current (4.0 A) and the voltage (2.0 V) across a Nichrome wire. By rearranging Ohm's Law, we calculated the resistance \( R \) as \( \frac{V}{I} \).

Ohm's Law is widely applicable in numerous electrical and electronic devices used in our daily lives, making it an essential tool for understanding and designing electrical circuits.

Resistivity

Resistivity is a property of materials that indicates how strongly a material opposes the flow of electric current. It's a specific property inherent to the material's structure, independent of its shape or size.

The formula to calculate resistivity \( \rho \) is \( \rho = \frac{RA}{L} \), where \( R \) is the resistance, \( A \) is the cross-sectional area, and \( L \) is the length of the material.

A high resistivity means the material does not allow electric current to pass easily, while a low resistivity means it conducts electricity well. In our Nichrome wire, we calculated resistivity using the calculated resistance and known dimensions, which helped in understanding its conductive properties further.

Cross-sectional Area

The cross-sectional area of a wire is crucial when determining its ability to conduct electricity. It refers to the area of the cut surface if the wire were sliced perpendicular to its length.

In the exercise, the given cross-sectional area is \( 1.0 \text{ mm}^2 \), which we converted to \( 1.0 \times 10^{-6} \text{ m}^2 \) for consistency in calculations with other metric units.

The area impacts the resistance of the wire: a larger cross-sectional area allows more electrons to flow, reducing resistance; a smaller area restricts flow, increasing resistance. Hence, the cross-sectional area is a crucial factor in designing and evaluating electric circuits.

Nichrome

Nichrome is an acronym for a nickel-chromium blend, often including iron, used for its remarkable resistive properties. It is widely used in applications like heating elements and toasters due to its ability to withstand high temperatures while steadily maintaining resistance.

Its resistivity is a key factor in these applications, as it determines how much heat the Nichrome wire can generate when electric current courses through it. In our exercise, you explored Nichrome's resistive quality, which plays an integral role in its functionality in products that depend on heat production from electrical resistance.

Electrical Resistance

Electrical resistance quantifies how strongly a material opposes the flow of electric current. This concept plays a vital role in controlling and managing an electrical circuit's current flow.

Resistance is measured in ohms (\( \Omega \)) and is calculated using Ohm's Law: \( R = \frac{V}{I} \). In the problem, we found resistance to be 0.5 ohms in the Nichrome wire. Understanding resistance helps determine how much energy is lost as heat in a circuit (converted from electrical energy).

Factors affecting resistance include material type, temperature, length, and cross-sectional area. This makes it a critical concept for designing electronic components and systems.