A copper wire of length 5 cm carries a current of density `j = 1 A (mm)^(-2)`. The density and molar mass of copper are `9000 Kg m^(-3) and 63 gmol^(-1)`. Each copper atom contributes one free electron. The temperature of the wire is `27^(@)C`. Estimate the (average) distance travelled by a free electron during the time it moves from one end of the copper wire to the other end. Assume that thermal motion of electrons are similar to that of molecules of an ideal gas.

Answer

To solve the problem, we need to estimate the average distance traveled by a free electron in a copper wire of length 5 cm, given the current density and other properties of copper. Let's break down the solution step by step.

### Step 1: Calculate the Volume of the Copper Wire
The length of the wire is given as 5 cm. We need to convert this to meters for consistency in SI units:
\[ L = 5 \text{ cm} = 0.05 \text{ m} \]

### Step 2: Determine the Current Density
The current density \( j \) is given as:
\[ j = 1 \text{ A/mm}^2 \]
We convert this to SI units:
\[ j = 1 \text{ A/mm}^2 = 1 \times 10^6 \text{ A/m}^2 \]

### Step 3: Calculate the Total Current
Using the current density, we can find the total current \( I \) flowing through the wire. The total current can be calculated using the formula:
\[ I = j \cdot A \]
where \( A \) is the cross-sectional area of the wire. Since we don't have the area, we will keep it as \( A \) for now.

### Step 4: Calculate the Number of Free Electrons per Unit Volume
The number of free electrons per unit volume \( n \) can be calculated using the density and molar mass of copper. The density of copper is given as:
\[ \rho = 9000 \text{ kg/m}^3 \]
The molar mass of copper is:
\[ M = 63 \text{ g/mol} = 0.063 \text{ kg/mol} \]
Using Avogadro's number \( N_A \approx 6.022 \times 10^{23} \text{ mol}^{-1} \), we can find \( n \):
\[ n = \frac{\rho}{M} \cdot N_A \]
Substituting the values:
\[ n = \frac{9000 \text{ kg/m}^3}{0.063 \text{ kg/mol}} \cdot 6.022 \times 10^{23} \text{ mol}^{-1} \]

### Step 5: Calculate the Average Drift Velocity of Electrons
The average drift velocity \( v_d \) of the electrons can be calculated using the formula:
\[ v_d = \frac{I}{n \cdot A \cdot e} \]
where \( e \) is the charge of an electron (\( e \approx 1.6 \times 10^{-19} \text{ C} \)).

### Step 6: Calculate the Time Taken to Travel the Length of the Wire
The time \( t \) taken for an electron to travel the length of the wire can be calculated as:
\[ t = \frac{L}{v_d} \]

### Step 7: Estimate the Average Distance Traveled by a Free Electron
The average distance \( d \) traveled by a free electron during the time \( t \) can be estimated using the thermal motion. The average distance can be approximated as:
\[ d = v_{thermal} \cdot t \]
where \( v_{thermal} \) is the average thermal velocity of the electrons, which can be estimated using:
\[ v_{thermal} \approx \sqrt{\frac{3kT}{m}} \]
where \( k \) is the Boltzmann constant (\( k \approx 1.38 \times 10^{-23} \text{ J/K} \)), \( T \) is the temperature in Kelvin (27°C = 300 K), and \( m \) is the mass of an electron (\( m \approx 9.11 \times 10^{-31} \text{ kg} \)).

### Final Calculation
Now we can substitute all the values back into the equations to find the average distance traveled by a free electron.