Find the velocity of charge leading to `1A` current which flows in a copper conductor of cross section `1cm^2` and length `10 km`. Free electron density of copper is `8.5 X 10^28//m^3`. How long will it take the electric charge to travel from one end of the conductor to the other?

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Identify the given values
- Current (I) = 1 A
- Cross-sectional area (A) = 1 cm² = \(1 \times 10^{-4}\) m²
- Length of the conductor (L) = 10 km = \(10 \times 10^3\) m
- Free electron density (n) = \(8.5 \times 10^{28}\) electrons/m³
- Charge of an electron (e) = \(1.6 \times 10^{-19}\) C

### Step 2: Use the formula for current
The formula for current in terms of drift velocity (v_d) is given by:
\[ I = n \cdot A \cdot e \cdot v_d \]
We need to rearrange this formula to find the drift velocity (v_d):
\[ v_d = \frac{I}{n \cdot A \cdot e} \]

### Step 3: Substitute the known values into the equation
Substituting the values we have:
\[ v_d = \frac{1}{(8.5 \times 10^{28}) \cdot (1 \times 10^{-4}) \cdot (1.6 \times 10^{-19})} \]

### Step 4: Calculate the denominator
Calculating the denominator:
\[ n \cdot A \cdot e = (8.5 \times 10^{28}) \cdot (1 \times 10^{-4}) \cdot (1.6 \times 10^{-19}) = 8.5 \times 1.6 \times 10^{28 - 4 - 19} = 13.6 \times 10^{5} \]

### Step 5: Calculate the drift velocity
Now substituting back into the equation for v_d:
\[ v_d = \frac{1}{13.6 \times 10^{5}} = 7.35 \times 10^{-7} \text{ m/s} \]

### Step 6: Calculate the time taken to travel the length of the conductor
To find the time (t) taken to travel the entire length of the conductor, we use the formula:
\[ t = \frac{L}{v_d} \]
Substituting the values:
\[ t = \frac{10 \times 10^3}{7.35 \times 10^{-7}} \approx 1.36 \times 10^{10} \text{ seconds} \]

### Final Answers
- Drift velocity \(v_d \approx 7.35 \times 10^{-7} \text{ m/s}\)
- Time taken \(t \approx 1.36 \times 10^{10} \text{ seconds}\)

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