At room temperature copper has free electron density of `8.4xx10^(28) per m^(3)` . The copper conductor has a cross-section of `10^(-6)m^(2)` and carries a current of 5.4 A. What is the electron drift velocity in copper?

Options

A. `4ms^(-1)`
B. `0.4ms^(-1)`
C. `"4 cm s"^(-1)`
D. `"0.4 mm s"^(-1)`

Answer

To find the electron drift velocity in copper, we can use the relationship between current (I), electron density (N), charge of an electron (E), cross-sectional area (A), and drift velocity (V_d). The formula we will use is:

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

Where:
- \( I \) = Current (in Amperes)
- \( N \) = Free electron density (in electrons per cubic meter)
- \( A \) = Cross-sectional area (in square meters)
- \( e \) = Charge of an electron (approximately \( 1.6 \times 10^{-19} \) Coulombs)
- \( V_d \) = Drift velocity (in meters per second)

### Step-by-step Solution:

1. **Identify the given values:**
- Free electron density, \( N = 8.4 \times 10^{28} \, \text{m}^{-3} \)
- Cross-sectional area, \( A = 10^{-6} \, \text{m}^{2} \)
- Current, \( I = 5.4 \, \text{A} \)
- Charge of an electron, \( e = 1.6 \times 10^{-19} \, \text{C} \)

2. **Rearrange the formula to solve for drift velocity \( V_d \):**
\[ V_d = \frac{I}{N \cdot A \cdot e} \]

3. **Substitute the known values into the equation:**
\[ V_d = \frac{5.4}{(8.4 \times 10^{28}) \cdot (10^{-6}) \cdot (1.6 \times 10^{-19})} \]

4. **Calculate the denominator:**
- First, calculate \( N \cdot A \cdot e \):
\[ N \cdot A \cdot e = (8.4 \times 10^{28}) \cdot (10^{-6}) \cdot (1.6 \times 10^{-19}) \]
- Calculate \( 8.4 \times 1.6 = 13.44 \)
- Now, \( 13.44 \times 10^{28 - 6 - 19} = 13.44 \times 10^{3} = 1.344 \times 10^{4} \)

5. **Now substitute back into the drift velocity equation:**
\[ V_d = \frac{5.4}{1.344 \times 10^{4}} \approx 4.007 \times 10^{-4} \, \text{m/s} \]

6. **Convert the drift velocity to centimeters per second:**
\[ V_d \approx 4.007 \times 10^{-4} \, \text{m/s} = 0.04007 \, \text{cm/s} \approx 0.04 \, \text{cm/s} \]

7. **Final answer:**
\[ V_d \approx 0.4 \, \text{mm/s} \]

### Final Result:
The electron drift velocity in copper is approximately \( 4 \times 10^{-4} \, \text{m/s} \) or \( 0.04 \, \text{cm/s} \) or \( 0.4 \, \text{mm/s} \).