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

Options

A. `6.94xx 10^(-6) m//s`
B. `5.94xx 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:

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

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

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

### Step 2: Rearrange the formula to solve for drift velocity
We can rearrange the formula to find the drift velocity \( v_d \):

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

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

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

### Step 4: Calculate the denominator
First, calculate 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. \( 9 \times 10^{28} \cdot 1 \times 10^{-4} = 9 \times 10^{24} \)
2. \( 9 \times 10^{24} \cdot 1.6 \times 10^{-19} = 14.4 \times 10^{5} = 1.44 \times 10^{6} \)

### Step 5: Substitute back to find \( v_d \)
Now substitute back into the equation for \( v_d \):

\[ v_d = \frac{10}{1.44 \times 10^{6}} \approx 6.94 \times 10^{-6} \, m/s \]

### Final Result
Thus, the drift velocity of free electrons is:

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