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 the conductor, we can use the formula for current:

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

Where:
- \( I \) = current (in Amperes)
- \( N \) = free electron density (in m\(^{-3}\))
- \( e \) = charge of an electron (in Coulombs)
- \( A \) = cross-sectional area of the conductor (in m\(^2\))
- \( V_d \) = drift velocity (in m/s)

### Step 1: Identify the given values
- Current, \( I = 10 \, \text{A} \)
- Cross-sectional area, \( A = 1 \, \text{cm}^2 = 1 \times 10^{-4} \, \text{m}^2 \)
- Free electron density, \( N = 9 \times 10^{28} \, \text{m}^{-3} \)
- Charge of an electron, \( e = 1.6 \times 10^{-19} \, \text{C} \)

### Step 2: Rearranging the formula to find drift velocity
We need to solve for \( V_d \):

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

### Step 3: Substitute the known values into the formula
Now, we substitute the values into the equation:

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

### Step 4: Calculate the denominator
Calculating the denominator:

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

Calculating step-by-step:

1. \( 9 \times 1.6 = 14.4 \)
2. \( 14.4 \times 10^{28} \times 10^{-19} \times 10^{-4} = 14.4 \times 10^{28 - 19 - 4} = 14.4 \times 10^{5} \)

So, the denominator is:

\[ 14.4 \times 10^{5} \]

### Step 5: Calculate the drift velocity
Now substituting back into the drift velocity formula:

\[ V_d = \frac{10}{14.4 \times 10^{5}} \]

Calculating:

\[ V_d = \frac{10}{14.4} \times 10^{-5} \]

Calculating \( \frac{10}{14.4} \):

\[ \frac{10}{14.4} \approx 0.6944 \]

So:

\[ V_d \approx 0.6944 \times 10^{-5} \]

### Step 6: Final result
Converting to scientific notation:

\[ V_d \approx 6.944 \times 10^{-6} \, \text{m/s} \]

Thus, the drift velocity of free electrons is approximately:

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