An Aluminium (Al) rod. With area of cross-section `4 xx 10^(-6) m^(2)` has a current of 5 ampere. Flowing through it. Find the drift velocity of electron in the rod. Density of `AI = 2.7 xx 10^(3) kg//m^3` and Atomic wt.= 27. Assume that each Al atom provides one electron.

Options

A. `8.6 xx 10^(-4) m//s`
B. `1.29 xx 10^(-4) m//s`
C. `2.8 xx 10^(-2) m//s`
D. `3.8 xx 10^(-3) m//s`

Explanation

To find the drift velocity of electrons in an aluminum rod, we can follow these steps:

### Step 1: Write down the known values
- Area of cross-section, \( A = 4 \times 10^{-6} \, \text{m}^2 \)
- Current, \( I = 5 \, \text{A} \)
- Density of Aluminum, \( \rho = 2.7 \times 10^{3} \, \text{kg/m}^3 \)
- Atomic weight of Aluminum, \( M = 27 \, \text{g/mol} = 27 \times 10^{-3} \, \text{kg/mol} \)
- Charge of an electron, \( e = 1.6 \times 10^{-19} \, \text{C} \)

### Step 2: Calculate the number of atoms per unit volume (n)
The number of atoms per unit volume can be calculated using the formula:
\[ n = \frac{\rho}{M} \times N_A \]
where \( N_A \) is Avogadro's number, approximately \( 6.022 \times 10^{23} \, \text{mol}^{-1} \).

1. Calculate the mass per unit volume:
\[ n = \frac{2.7 \times 10^{3} \, \text{kg/m}^3}{27 \times 10^{-3} \, \text{kg/mol}} \times 6.022 \times 10^{23} \, \text{mol}^{-1} \]
\[ n = \frac{2.7 \times 10^{3}}{27 \times 10^{-3}} \times 6.022 \times 10^{23} \]
\[ n = 100 \times 6.022 \times 10^{23} \approx 6.022 \times 10^{25} \, \text{electrons/m}^3 \]

### Step 3: Use the formula for current to find drift velocity (Vd)
The formula for current is given by:
\[ I = n \cdot e \cdot A \cdot V_d \]
Rearranging for drift velocity \( V_d \):
\[ V_d = \frac{I}{n \cdot e \cdot A} \]

### Step 4: Substitute the known values into the equation
1. Substitute \( I = 5 \, \text{A} \), \( n = 6.022 \times 10^{25} \, \text{electrons/m}^3 \), \( e = 1.6 \times 10^{-19} \, \text{C} \), and \( A = 4 \times 10^{-6} \, \text{m}^2 \):
\[ V_d = \frac{5}{(6.022 \times 10^{25}) \cdot (1.6 \times 10^{-19}) \cdot (4 \times 10^{-6})} \]

### Step 5: Calculate the drift velocity
1. Calculate the denominator:
\[ n \cdot e \cdot A = (6.022 \times 10^{25}) \cdot (1.6 \times 10^{-19}) \cdot (4 \times 10^{-6}) \approx 3.8432 \times 10^{1} \approx 38.432 \]
2. Now calculate \( V_d \):
\[ V_d = \frac{5}{38.432} \approx 0.130 \, \text{m/s} \]

Thus, the drift velocity \( V_d \approx 1.3 \times 10^{-1} \, \text{m/s} \).

### Final Answer:
The drift velocity of electrons in the aluminum rod is approximately \( 1.3 \times 10^{-1} \, \text{m/s} \).

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