The diameter of a copper wire is 2mm, if a steady current of 6.25 A is caused by `8.5xx10^(28) //m^(3)` electrons flowing throught it. Calculate the drift velocity of condu ction electrons.

Answer

To calculate the drift velocity of conduction electrons in a copper wire, we can follow these steps:

### Step 1: Identify the given values
- Diameter of the copper wire, \(d = 2 \, \text{mm} = 2 \times 10^{-3} \, \text{m}\)
- Radius of the wire, \(r = \frac{d}{2} = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m}\)
- Current, \(I = 6.25 \, \text{A}\)
- Number density of electrons, \(n = 8.5 \times 10^{28} \, \text{m}^{-3}\)
- Charge of an electron, \(q = 1.6 \times 10^{-19} \, \text{C}\)

### Step 2: Calculate the cross-sectional area of the wire
The cross-sectional area \(A\) of the wire can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (1 \times 10^{-3})^2 = \pi \times 10^{-6} \, \text{m}^2 \]

### Step 3: Use the formula for drift velocity
The drift velocity \(v_d\) can be calculated using the formula:
\[ I = n q A v_d \]
Rearranging the formula to solve for \(v_d\):
\[ v_d = \frac{I}{n q A} \]

### Step 4: Substitute the known values into the equation
Substituting the values we have:
\[ v_d = \frac{6.25}{(8.5 \times 10^{28}) \times (1.6 \times 10^{-19}) \times (\pi \times 10^{-6})} \]

### Step 5: Calculate the denominator
Calculating the denominator:
\[ n q A = (8.5 \times 10^{28}) \times (1.6 \times 10^{-19}) \times (\pi \times 10^{-6}) \]
Calculating \(8.5 \times 1.6 = 13.6\):
\[ n q A \approx 13.6 \times 10^{28} \times 3.14 \times 10^{-6} \approx 42.7 \times 10^{22} = 4.27 \times 10^{23} \]

### Step 6: Calculate the drift velocity
Now substituting back into the drift velocity equation:
\[ v_d = \frac{6.25}{4.27 \times 10^{23}} \approx 1.46 \times 10^{-24} \, \text{m/s} \]

### Step 7: Convert to mm/s
To convert \(v_d\) from m/s to mm/s:
\[ v_d \approx 0.15 \, \text{mm/s} \]

### Final Answer
The drift velocity of conduction electrons is approximately \(0.15 \, \text{mm/s}\).

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