A wire of diameter 0.02 metre contains 1028 free electrons per cubic metre. For an electrical current of 100 A , the drift velocity of the free electrons in the wire is nearly

Options

A. `1 xx 10^(-19) m//s`
B. `5 xx 10^(-10) m//s`
C. `2 xx 10^(-4) m//s`
D. `8 xx10^(3) m//s`

Answer

To find the drift velocity of free electrons in a wire, we can use the formula that relates current (I), number of free electrons per unit volume (n), charge of an electron (e), cross-sectional area of the wire (A), and drift velocity (v_d):

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

From this, we can rearrange the formula to solve for drift velocity (v_d):

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

### Step 1: Given Values
- Diameter of the wire, \(d = 0.02 \, \text{m}\)
- Current, \(I = 100 \, \text{A}\)
- Number of free electrons per cubic meter, \(n = 10^{28} \, \text{m}^{-3}\)
- Charge of an electron, \(e = 1.6 \times 10^{-19} \, \text{C}\)

### Step 2: Calculate the Radius of the Wire
The radius \(r\) is half of the diameter:

\[ r = \frac{d}{2} = \frac{0.02}{2} = 0.01 \, \text{m} \]

### Step 3: Calculate the Cross-Sectional Area (A)
The cross-sectional area \(A\) of the wire can be calculated using the formula for the area of a circle:

\[ A = \pi r^2 = \pi (0.01)^2 = \pi \times 0.0001 \approx 3.14 \times 0.0001 = 3.14 \times 10^{-4} \, \text{m}^2 \]

### Step 4: Substitute Values into the Drift Velocity Formula
Now we can substitute the values into the drift velocity formula:

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

Substituting the known values:

\[ v_d = \frac{100}{(10^{28}) \cdot (1.6 \times 10^{-19}) \cdot (3.14 \times 10^{-4})} \]

### Step 5: Calculate the Denominator
Calculating the denominator:

\[ n \cdot e \cdot A = (10^{28}) \cdot (1.6 \times 10^{-19}) \cdot (3.14 \times 10^{-4}) \]

Calculating step by step:

1. \(10^{28} \cdot 1.6 \times 10^{-19} = 1.6 \times 10^{9}\)
2. \(1.6 \times 10^{9} \cdot 3.14 \times 10^{-4} = 5.024 \times 10^{5}\)

### Step 6: Calculate Drift Velocity
Now substituting back into the drift velocity formula:

\[ v_d = \frac{100}{5.024 \times 10^{5}} \approx 1.99 \times 10^{-4} \, \text{m/s} \]

### Final Answer
Thus, the drift velocity of the free electrons in the wire is approximately:

\[ v_d \approx 2 \times 10^{-4} \, \text{m/s} \]