Problem 5 - College Physics

Question

Copper has \(8.5 \times 10^{28}\) electrons per cubic meter. (a) How many electrons are there in a 25.0 \(\mathrm{cm}\) length of 12 -gauge copper wire (diameter 2.05 \(\mathrm{mm} ) ?\) (b) If a current of 1.55 \(\mathrm{A}\) is flowing in the wire, what is the average drift speed of the electrons along the wire? (There are \(6.24 \times 10^{18}\) electrons in 1 coulomb of charge.)

Step-by-Step Solution

Answer

(a) Approximately \(7.032 \times 10^{22}\) electrons. (b) Average drift speed is \(3.416 \times 10^{-4} \text{ m/s}\).

Step 1: Calculate the Volume of the Copper Wire

First, we need to convert the diameter from millimeters to meters. The diameter is given as 2.05 mm, which equals 0.00205 meters. Next, the radius \( r \) is half of the diameter: \( r = \frac{0.00205}{2} \) = 0.001025 meters.The length \( L \) of the wire is 25.0 cm, which is equivalent to 0.25 meters. The volume \( V \) of the copper wire, which is cylindrical, is calculated using the formula for the volume of a cylinder: \[ V = \pi r^2 L \]Plugging in the values, we have:\[ V = \pi (0.001025)^2 \times 0.25 \approx 8.272 \times 10^{-7} \, \text{m}^3 \]

Step 2: Calculate the Number of Electrons

We know the density of electrons in copper is \( 8.5 \times 10^{28} \) electrons per cubic meter. Given the volume of the wire from Step 1, the total number of electrons \( N \) is:\[ N = (8.5 \times 10^{28}) \times (8.272 \times 10^{-7}) \]Calculating this gives:\[ N \approx 7.032 \times 10^{22} \] Thus, there are approximately \( 7.032 \times 10^{22} \) electrons in the copper wire.

Step 3: Determine the Cross-sectional Area of the Wire

Before calculating the drift speed, we need to find the cross-sectional area \( A \) of the wire, which is the area of the circular end of the cylinder:\[ A = \pi r^2 \]Using the radius in meters:\[ A = \pi (0.001025)^2 \approx 3.300 \times 10^{-6} \, \text{m}^2 \]

Step 4: Calculate the Drift Speed of Electrons

The drift speed \( v_d \) can be calculated using the formula for current \( I \): \[ I = n e A v_d \]where \( n \) is the number of electrons per cubic meter, \( e \) is the charge of an electron (\( 1.6 \times 10^{-19} \) C), and \( A \) is the cross-sectional area of the wire.Rearranging the formula to find \( v_d \):\[ v_d = \frac{I}{n e A} \]Substituting the known values:\[ v_d = \frac{1.55}{(8.5 \times 10^{28})(1.6 \times 10^{-19})(3.300 \times 10^{-6})} \]Calculating this gives:\[ v_d \approx 3.416 \times 10^{-4} \, \text{m/s} \]This is the average drift speed of the electrons.

Key Concepts

Electron Drift Speed

The electron drift speed refers to the average velocity that electrons attain as they move through a conductor due to an electric field. It's an essential concept in understanding how current flows in a wire.
When a current is applied, electrons, which normally move randomly, gain a net flow in one direction, albeit slowly. This movement is characterized by the drift speed, which contrasts with the speed of the electric field itself, which propagates close to the speed of light. Drift speed is much slower, typically in the order of millimeters per second.
The drift speed can be calculated using the formula:

• \( I = n e A v_d \)
• where \( I \) is the current, \( n \) is the electron density, \( e \) is the charge of an electron, \( A \) is the cross-sectional area, and \( v_d \) is the drift speed.

Understanding drift speed helps us relate how electrical energy is transmitted efficiently and is foundational for designing electrical circuits.

Cylindrical Volume

Volume measurement of wires is often done using the formula for cylindrical volumes because most wires have a cylindrical shape. Calculating this volume accurately is critical to assessing the number of atoms, electrons, or even the thermal conductivity of the wire.
For a cylinder, the volume \( V \) can be calculated using:

• \[ V = \pi r^2 L \]
• where \( r \) is the radius, and \( L \) is the length of the cylinder.

In practical applications, ensuring the correct unit conversions is essential. For example, converting the wire diameter from millimeters to meters aids in calculating the radius needed for the volume formula. Knowing the volume of the wire is the first step in determining how densely packed it is with electrons.

Cross-sectional Area

The cross-sectional area of a wire is important because it affects resistance and current flow. Like the volume, this area is based on the circular cross-section typical of wires.
To find the cross-sectional area \( A \), use the formula:

• \[ A = \pi r^2 \]
• where \( r \) is the radius of the wire.

This area is crucial because it influences how easily current can pass through the wire. A larger cross-sectional area means lower resistance and more spot for electrons to move through. It's analogous to the width of a river, where a wider river can carry more water with less obstruction.

Copper Wire Electrons

Copper is commonly used in electrical wiring due to its high electrical conductivity. The number of free electrons contributes significantly to its conductivity, enabling it to transfer electrical charge effectively.
In a given volume of copper wire, the electron density, measured as electrons per cubic meter, helps calculate how many electrons are available to carry charge. For example, copper has an electron density of about \(8.5 \times 10^{28}\) electrons per cubic meter, as mentioned in the exercise.
This immense number of electrons facilitates efficient current flow, making copper an ideal choice for electrical circuits. By multiplying electron density by the wire's volume, we can find the total number of electrons in a segment of wire.

Charge of Electron

Electrons carry a fundamental electric charge denoted by \( e \). Each electron has a charge of \( 1.6 \times 10^{-19} \) coulombs. This small unit is fundamental in understanding how much charge is transferred per second during electrical conduction.
In the context of current electricity, the electron's charge is crucial for calculations such as current \( I \), given by:

• \( I = n e A v_d \)

where \( n \) is electron density, \( A \) is the cross-sectional area, and \( v_d \) is the drift velocity.
The charge of an electron is a constant that links to various realms of physics, playing a crucial role in how electricity is measured and applied, from microchips to power lines.