A current of 1.8 A flows through a wire of cross-sectional area 0.5 `mm^(2)`? Find the current density in the wire. If the number density of conduction electrons in the wire is `8.8 xx 10^(28) m^(-3)`, find the drift speed of electrons.

Answer

To solve the problem step by step, let's break it down into two parts: finding the current density and then calculating the drift speed of the electrons.

### Step 1: Calculate the Current Density (J)

The formula for current density \( J \) is given by:

\[ J = \frac{I}{A} \]

where:
- \( I \) is the current (in Amperes)
- \( A \) is the cross-sectional area (in square meters)

Given:
- \( I = 1.8 \, \text{A} \)
- \( A = 0.5 \, \text{mm}^2 = 0.5 \times 10^{-6} \, \text{m}^2 \) (conversion from mm² to m²)

Now, substituting the values into the formula:

\[ J = \frac{1.8 \, \text{A}}{0.5 \times 10^{-6} \, \text{m}^2} \]

Calculating this gives:

\[ J = 3.6 \times 10^{6} \, \text{A/m}^2 \]

### Step 2: Calculate the Drift Speed (v_d)

The drift speed \( v_d \) can be calculated using the formula:

\[ v_d = \frac{J}{n \cdot e} \]

where:
- \( n \) is the number density of conduction electrons (in \( m^{-3} \))
- \( e \) is the charge of an electron (\( e = 1.6 \times 10^{-19} \, \text{C} \))

Given:
- \( n = 8.8 \times 10^{28} \, \text{m}^{-3} \)

Now substituting the values into the formula:

\[ v_d = \frac{3.6 \times 10^{6} \, \text{A/m}^2}{8.8 \times 10^{28} \, \text{m}^{-3} \cdot 1.6 \times 10^{-19} \, \text{C}} \]

Calculating the denominator:

\[ n \cdot e = 8.8 \times 10^{28} \cdot 1.6 \times 10^{-19} = 1.408 \times 10^{10} \, \text{C/m}^3 \]

Now substituting this back into the equation for \( v_d \):

\[ v_d = \frac{3.6 \times 10^{6}}{1.408 \times 10^{10}} \]

Calculating this gives:

\[ v_d \approx 2.56 \times 10^{-4} \, \text{m/s} = 0.256 \, \text{mm/s} \]

### Final Answers:
- Current Density \( J = 3.6 \times 10^{6} \, \text{A/m}^2 \)
- Drift Speed \( v_d \approx 2.56 \times 10^{-4} \, \text{m/s} \)