A current of 10 A is maintained in a conductor of cross-section `1 cm^(2) ` . If the free electron density in the conductor is `9 xx 10^(28) m^(-3) ` , then drift velocity of free electrons is

Options

A. `6.94 xx 10^(-6) m //s `
B. ` 5.94 xx 10^(-2) m //s `
C. ` 1.94 xx 10^(-3) m//s `
D. ` 2.94 xx 10^(-4) m//s `

Answer

To find the drift velocity of free electrons in a conductor, we can use the formula for current:

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

Where:
- \( I \) = current (in Amperes)
- \( n \) = free electron density (in electrons per cubic meter)
- \( A \) = cross-sectional area of the conductor (in square meters)
- \( e \) = charge of an electron (approximately \( 1.6 \times 10^{-19} \) Coulombs)
- \( V_d \) = drift velocity of the electrons (in meters per second)

### Step 1: Convert the cross-sectional area from cm² to m²
Given that the cross-sectional area \( A = 1 \, \text{cm}^2 \):

\[ A = 1 \, \text{cm}^2 = 1 \times 10^{-4} \, \text{m}^2 \]

### Step 2: Identify the values
From the question, we have:
- Current \( I = 10 \, \text{A} \)
- Free electron density \( n = 9 \times 10^{28} \, \text{m}^{-3} \)
- Charge of an electron \( e = 1.6 \times 10^{-19} \, \text{C} \)

### Step 3: Rearrange the formula to solve for drift velocity \( V_d \)
We can rearrange the formula to find \( V_d \):

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

### Step 4: Substitute the known values into the formula
Now we can substitute the values into the equation:

\[ V_d = \frac{10}{(9 \times 10^{28}) \cdot (1 \times 10^{-4}) \cdot (1.6 \times 10^{-19})} \]

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

\[ n \cdot A \cdot e = (9 \times 10^{28}) \cdot (1 \times 10^{-4}) \cdot (1.6 \times 10^{-19}) \]

Calculating step-by-step:

1. \( 9 \times 1.6 = 14.4 \)
2. \( 14.4 \times 10^{28} \times 10^{-4} = 14.4 \times 10^{24} \)

So,

\[ n \cdot A \cdot e = 14.4 \times 10^{24} \]

### Step 6: Substitute back into the drift velocity formula
Now substituting back into the drift velocity formula:

\[ V_d = \frac{10}{14.4 \times 10^{24}} \approx 6.94 \times 10^{-6} \, \text{m/s} \]

### Final Result
The drift velocity of free electrons is approximately:

\[ V_d \approx 6.94 \times 10^{-6} \, \text{m/s} \]