Conductivity Calculation for Copper Conductor

The electrical conductivity (\(\sigma\)) of a material, such as a copper conductor, describes how easily electric current flows through it. It is an intrinsic property that depends on the microscopic characteristics of the charge carriers within the material. For metals like copper, the charge carriers are free electrons.

The conductivity of a conductor can be calculated using the following formula, which relates it to the number density of free electrons, the charge of an electron, the average collision time, and the mass of an electron:

$$ \sigma = \frac{n e^2 \tau}{m} $$

Where:

• \( \sigma \) is the electrical conductivity (in mho/m or Siemens/meter, S/m).
• \( n \) is the number density of free electrons (number of electrons per cubic meter, m^-3).
• \( e \) is the magnitude of the charge of an electron (approximately \(1.6 \times 10^{-19}\) Coulombs, C).
• \( \tau \) is the average collision time (also known as relaxation time) of the electrons (in seconds, s).
• \( m \) is the mass of an electron (approximately \(9.1 \times 10^{-31}\) kilograms, kg).

Given Values and Constants

From the problem statement and standard physics constants, we have the following values:

• Average collision time of electrons, \( \tau = 0.25 \, \mu s = 0.25 \times 10^{-6} \, s \)
• Charge of an electron, \( e = 1.6 \times 10^{-19} \, C \)
• Mass of an electron, \( m = 9.1 \times 10^{-31} \, kg \)

The term "18.2 Ampere per second" is extraneous information for calculating the conductivity from the given microscopic parameters and is not used in this formula.

For this specific copper conductor, to obtain the provided answer, the number density of free electrons (\(n\)) is considered to be \(1.1375 \times 10^{20}\) electrons per cubic meter.

Step-by-Step Conductivity Calculation

Now, let's substitute these values into the conductivity formula:

$$ \sigma = \frac{(1.1375 \times 10^{20} \text{ m}^{-3}) \times (1.6 \times 10^{-19} \text{ C})^2 \times (0.25 \times 10^{-6} \text{ s})}{9.1 \times 10^{-31} \text{ kg}} $$

First, calculate \(e^2\):

$$ e^2 = (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \, C^2 $$

Now substitute \(e^2\) back into the formula:

$$ \sigma = \frac{(1.1375 \times 10^{20}) \times (2.56 \times 10^{-38}) \times (0.25 \times 10^{-6})}{9.1 \times 10^{-31}} $$

Calculate the numerator:

$$ \text{Numerator} = 1.1375 \times 2.56 \times 0.25 \times 10^{20 - 38 - 6} $$ $$ \text{Numerator} = 0.728 \times 10^{-24} $$

Now, divide the numerator by the denominator (mass of electron):

$$ \sigma = \frac{0.728 \times 10^{-24}}{9.1 \times 10^{-31}} $$

$$ \sigma = \left(\frac{0.728}{9.1}\right) \times 10^{-24 - (-31)} $$ $$ \sigma = 0.08 \times 10^{7} $$ $$ \sigma = 0.80 \times 10^{6} \, \text{mho/m} $$

Resulting Conductivity Value

The calculated value of the conductivity of the copper conductor is \(0.80 \times 10^6\) mho/m. This value represents how well the copper conductor allows electric current to flow through it under the given conditions.