The free electrons of a copper wire of cross-sectional area `10^(-6) m^(2)` acquire a drift velocity of `10^(-4) m//s` when a certain potential difference is applied across the wire. Find the current flowing in the wire if the density of free electrons in copper is `8.5xx10^(28) "electrons"//m^(3)`.

Answer

To find the current flowing in the copper wire, we can use the formula for current in terms of the number density of free electrons, the cross-sectional area of the wire, and the drift velocity of the electrons. The formula is given by:

\[ I = n \cdot A \cdot v_d \]

where:
- \( I \) is the current,
- \( n \) is the density of free electrons (in electrons/m³),
- \( A \) is the cross-sectional area of the wire (in m²),
- \( v_d \) is the drift velocity of the electrons (in m/s).

### Step 1: Identify the given values
- Cross-sectional area, \( A = 10^{-6} \, \text{m}^2 \)
- Drift velocity, \( v_d = 10^{-4} \, \text{m/s} \)
- Density of free electrons, \( n = 8.5 \times 10^{28} \, \text{electrons/m}^3 \)

### Step 2: Substitute the values into the formula
Now we can substitute the known values into the formula for current:

\[ I = n \cdot A \cdot v_d \]

Substituting the values:

\[ I = (8.5 \times 10^{28} \, \text{electrons/m}^3) \cdot (10^{-6} \, \text{m}^2) \cdot (10^{-4} \, \text{m/s}) \]

### Step 3: Calculate the current
Now we perform the multiplication:

\[ I = 8.5 \times 10^{28} \cdot 10^{-6} \cdot 10^{-4} \]

Calculating the powers of ten:

\[ I = 8.5 \times 10^{28 - 6 - 4} = 8.5 \times 10^{18} \, \text{A} \]

### Step 4: Multiply by the charge of an electron
Since the current is in terms of electrons, we need to multiply by the charge of an electron \( e = 1.6 \times 10^{-19} \, \text{C} \):

\[ I = (8.5 \times 10^{18}) \cdot (1.6 \times 10^{-19}) \]

Calculating this gives:

\[ I = 1.36 \, \text{A} \]

### Final Answer:
The current flowing in the wire is approximately \( 1.36 \, \text{A} \).

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