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 \(4.0 \mathrm{~A}\) 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 of Nichrome is \(2 \times 10^{6} \, \text{S/m}\).

Step 1: Understand the Formula for Conductivity

Conductivity (\(\sigma\)) is given by the formula \(\sigma = \frac{1}{\rho}\), where \(\rho\) is the resistivity. Resistivity can be calculated using Ohm's Law and the relation between resistivity, resistance, length, and cross-sectional area.

Step 2: Use Ohm's Law to Find Resistance

Ohm's Law states that\(V = I \times R\), where \(V\) is the potential difference, \(I\) is the current, and \(R\) is the resistance. Here, \(V = 2.0\) V and \(I = 4.0\) A, so we can rearrange the formula to find \(R\): \[ R = \frac{V}{I} = \frac{2.0}{4.0} = 0.5 \, \Omega \].

Step 3: Relate Resistivity to Resistance, Length, and Area

Resistivity (\(\rho\)) can be calculated using the formula\(\rho = R \times \frac{A}{L}\), where \(A\) is the cross-sectional area and \(L\) is the length of the wire. Substituting \(R = 0.5 \, \Omega\), \(A = 1.0 \times 10^{-6} \, \text{m}^2\), and \(L = 1.0 \, \text{m}\), we get: \[ \rho = 0.5 \times \frac{1.0 \times 10^{-6}}{1.0} = 0.5 \times 10^{-6} \, \Omega \, \text{m} \].

Step 4: Calculate the Conductivity

Finally, calculate the conductivity \(\sigma\) using the relation \(\sigma = \frac{1}{\rho}\). With \(\rho = 0.5 \times 10^{-6} \,\Omega \,\text{m}\), the conductivity is: \[ \sigma = \frac{1}{0.5 \times 10^{-6}} = 2 \times 10^{6} \, \text{S/m} \].

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in the study of electricity, defining the relationship between voltage, current, and resistance. The law is expressed by the equation \( V = I \times R \), where \( V \) stands for voltage (or potential difference), \( I \) represents current, and \( R \) denotes electrical resistance. Understanding this law is crucial because it sets the groundwork for analyzing electrical circuits.

Think of voltage as the force pushing the electric current through a conductor, while resistance is like a barrier that slows the flow of current. Current is the flow of electric charge, facilitated by conducting materials like wires. In practical terms:

• If the resistance in a circuit increases while the voltage remains constant, the current will decrease.
• Conversely, if resistance decreases, the current will rise.

This relationship helps engineers and electricians design safe and efficient electronic devices and systems, making Ohm's Law a pillar of electrical engineering.

Resistivity

Resistivity is a material property that measures how much a material opposes the flow of electric current. Unlike resistance, which depends on the shape and size of the material, resistivity is an intrinsic property, meaning it remains constant for a pure substance under specific conditions.

Resistivity is denoted by the symbol \( \rho \) (rho) and is expressed in ohm-meters (\( \Omega \cdot \text{m} \)). The relationship between resistance \( R \), resistivity \( \rho \), the length of the conductor \( L \), and cross-sectional area \( A \) is given by the formula:

• \( \rho = R \times \frac{A}{L} \)

Materials with low resistivity, such as copper and silver, are excellent conductors of electricity, while materials with high resistivity, like rubber and glass, are good insulators. Engineers need to consider resistivity when choosing materials for electrical cables and circuits to ensure optimal performance.

Electrical Resistance

Electrical resistance is a measure of how much a material resists the flow of electric current. It's represented by the symbol \( R \) and measured in ohms (\( \Omega \)). Resistance arises mainly from collisions between the electrons moving through the conductor and the atoms of the material, which impede their flow.

Several factors influence electrical resistance, including:

• Material type: Conductors have low resistance, while insulators have high resistance.
• Length: Longer wires have higher resistance.
• Cross-sectional area: Thicker wires have lower resistance.
• Temperature: In many materials, resistance increases with temperature.

By understanding these factors, one can tailor the resistance of electrical components to suit specific applications, such as creating precise resistors used in electronic circuits.

Current

Current, symbolized by \( I \), is the rate of flow of electric charge through a conductor. It is measured in amperes (A), which signifies the amount of charge passing through the point in a circuit each second. The flow of current results from potential differences and is analogous to the flow of water being driven by a pump through a hose.

There are two types of current: direct current (DC) and alternating current (AC).

• **Direct Current (DC):** Flows in a single direction, common in batteries.
• **Alternating Current (AC):** Changes direction periodically, used in household power supplies.

Understanding current is crucial for designing circuits and systems, ensuring they function correctly and safely. Proper management of current underpins all modern electronics, from small devices to large power systems.

Potential Difference

Potential difference, or voltage, is the work needed to move a unit charge from one point to another, measured in volts (V). It represents the electric potential energy per charge between two points in a circuit. Without a potential difference, there will be no current flow in the circuit.

Here's a simple way to think about it:

• Voltage is like the pressure of water in a hose, which pushes water flow.
• Current is the actual flow of water.

The source of voltage could be a battery or power supply. Its role is fundamental, as it drives the current through the circuit, powering components such as lamps, motors, and electronic devices. In household circuits, the typical potential difference is 120 or 240 volts, allowing various appliances to function efficiently.