A current 0.5 amperes flows in a conductor of cross-sectional area of `10^(-2) m^2`. If the electron density is `0.3 * 10^(28) m^(-3)` , then the drift velocity of free electrons is

Options

A. `2.1 * 10^(-5) ms^(-1)`
B. `2.5 * 10^(-3) ms^(-1)`
C. `1.5 * 10^(-6) ms^(-1)`
D. `1.04 * 10^(-7) ms^(-1)`

Answer

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

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

Where:
- \( I \) = current (in amperes)
- \( N \) = electron density (number of electrons per unit volume, in \( m^{-3} \))
- \( e \) = charge of an electron (approximately \( 1.6 \times 10^{-19} \) coulombs)
- \( A \) = cross-sectional area of the conductor (in \( m^2 \))
- \( v_d \) = drift velocity (in \( m/s \))

Given:
- Current \( I = 0.5 \, \text{A} \)
- Cross-sectional area \( A = 10^{-2} \, m^2 \)
- Electron density \( N = 0.3 \times 10^{28} \, m^{-3} \)

### Step 1: Write down the formula for drift velocity
Rearranging the formula for drift velocity \( v_d \):

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

### Step 2: Substitute the known values into the formula
Substituting the values into the formula:

\[ v_d = \frac{0.5}{(0.3 \times 10^{28}) \cdot (1.6 \times 10^{-19}) \cdot (10^{-2})} \]

### Step 3: Calculate the denominator
Calculating the denominator:

1. Calculate \( N \cdot e \cdot A \):
- \( N = 0.3 \times 10^{28} \)
- \( e = 1.6 \times 10^{-19} \)
- \( A = 10^{-2} \)

\[ N \cdot e \cdot A = (0.3 \times 10^{28}) \cdot (1.6 \times 10^{-19}) \cdot (10^{-2}) \]

\[ = 0.3 \cdot 1.6 \cdot 10^{28} \cdot 10^{-19} \cdot 10^{-2} \]

\[ = 0.48 \cdot 10^{28 - 19 - 2} \]

\[ = 0.48 \cdot 10^{7} \]

### Step 4: Substitute back into the drift velocity formula
Now substituting back:

\[ v_d = \frac{0.5}{0.48 \times 10^{7}} \]

### Step 5: Calculate the drift velocity
Calculating \( v_d \):

\[ v_d = \frac{0.5}{0.48} \times 10^{-7} \]

\[ v_d \approx 1.04167 \times 10^{-7} \, m/s \]

### Step 6: Round off the answer
Rounding off to two decimal places:

\[ v_d \approx 1.04 \times 10^{-7} \, m/s \]

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

\[ v_d \approx 1.04 \times 10^{-7} \, m/s \]

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