A copper wire of diameter 1 mm carries a current of 1.1A. The drift speed of electrons in (given, density of `Cu=9g//cm^(3)`, atomic weight of Cu = 63 g and one electron is contributed by each Cu atom)

Options

A. 0.1 mm/s
B. 0.2 mm/s
C. 0.3 mm/s
D. 0.2 cm/s

Answer

To solve the problem of finding the drift speed of electrons in a copper wire, we can follow these steps:

### Step 1: Calculate the number density of conduction electrons (n)

The number density of conduction electrons \( n \) can be calculated using the formula:

\[ n = \frac{N_A \cdot \rho}{M} \]

Where:
- \( N_A = 6.023 \times 10^{23} \, \text{atoms/mol} \) (Avogadro's number)
- \( \rho = 9 \, \text{g/cm}^3 = 9 \times 10^3 \, \text{kg/m}^3 \) (density of copper)
- \( M = 63 \, \text{g/mol} = 63 \times 10^{-3} \, \text{kg/mol} \) (molar mass of copper)

Substituting the values:

\[ n = \frac{6.023 \times 10^{23} \times 9 \times 10^3}{63 \times 10^{-3}} \]

Calculating this gives:

\[ n \approx 0.86 \times 10^{29} \, \text{m}^{-3} \]

### Step 2: Calculate the cross-sectional area (A) of the wire

The cross-sectional area \( A \) of the wire can be calculated using the formula for the area of a circle:

\[ A = \pi r^2 \]

Where the radius \( r \) is half of the diameter. Given the diameter is 1 mm:

\[ r = \frac{1 \, \text{mm}}{2} = 0.5 \, \text{mm} = 0.5 \times 10^{-3} \, \text{m} \]

Now substituting for \( A \):

\[ A = \pi (0.5 \times 10^{-3})^2 \]

Calculating this gives:

\[ A \approx \pi \times (0.25 \times 10^{-6}) \approx 0.785 \times 10^{-6} \, \text{m}^2 \]

### Step 3: Use the formula for current (I) to find drift speed (v_d)

The formula for current is given by:

\[ I = n \cdot A \cdot e \cdot v_d \]

Where:
- \( I = 1.1 \, \text{A} \) (current)
- \( e = 1.6 \times 10^{-19} \, \text{C} \) (charge of an electron)

Rearranging the formula to solve for drift speed \( v_d \):

\[ v_d = \frac{I}{n \cdot A \cdot e} \]

Substituting the values we have:

\[ v_d = \frac{1.1}{(0.86 \times 10^{29}) \cdot (0.785 \times 10^{-6}) \cdot (1.6 \times 10^{-19})} \]

### Step 4: Calculate drift speed

Calculating the denominator:

\[ n \cdot A \cdot e \approx (0.86 \times 10^{29}) \cdot (0.785 \times 10^{-6}) \cdot (1.6 \times 10^{-19}) \approx 1.08 \times 10^{5} \]

Now substituting back into the drift speed formula:

\[ v_d \approx \frac{1.1}{1.08 \times 10^{5}} \approx 0.00001019 \, \text{m/s} = 0.1 \, \text{mm/s} \]

### Final Answer

The drift speed of electrons in the copper wire is approximately:

\[ v_d \approx 0.1 \, \text{mm/s} \]