A potential difference of 100 V is applied to the ends of a copper wire one metre long . What is the average drift velocity of electrons ?
(Given, `sigma=5.81xx10^(7)Omega` or `n_(Cu)=8.5xx10^(28)m^(-3)`)

Options

A. `0.43 ms^(-1)`
B. `0.83 ms^(-1)`
C. `0.52 ms^(-1)`
D. `0.95 ms^(-1)`

Explanation

To find the average drift velocity of electrons in a copper wire when a potential difference is applied, we can use the following steps:

### Step 1: Identify the given data
- Potential difference (V) = 100 V
- Length of the wire (L) = 1 m
- Conductivity of copper (σ) = \(5.81 \times 10^7 \, \Omega^{-1} \, \text{m}^{-1}\)
- Carrier density of copper (n) = \(8.5 \times 10^{28} \, \text{m}^{-3}\)
- Charge of an electron (e) = \(1.6 \times 10^{-19} \, \text{C}\)

### Step 2: Calculate the electric field (E)
The electric field (E) in the wire can be calculated using the formula:
\[ E = \frac{V}{L} \]
Substituting the values:
\[ E = \frac{100 \, \text{V}}{1 \, \text{m}} = 100 \, \text{V/m} \]

### Step 3: Use the relationship between current density (J), conductivity (σ), and electric field (E)
The current density (J) can be expressed as:
\[ J = \sigma E \]
Substituting the values we have:
\[ J = (5.81 \times 10^7 \, \Omega^{-1} \, \text{m}^{-1})(100 \, \text{V/m}) = 5.81 \times 10^9 \, \text{A/m}^2 \]

### Step 4: Relate current density (J) to drift velocity (v_d)
The current density can also be expressed in terms of drift velocity (v_d):
\[ J = n e v_d \]
Where:
- n = carrier density
- e = charge of an electron
- v_d = drift velocity

### Step 5: Rearrange the equation to solve for drift velocity (v_d)
We can rearrange the equation to find v_d:
\[ v_d = \frac{J}{n e} \]
Substituting the values:
\[ v_d = \frac{5.81 \times 10^9 \, \text{A/m}^2}{(8.5 \times 10^{28} \, \text{m}^{-3})(1.6 \times 10^{-19} \, \text{C})} \]

### Step 6: Calculate the drift velocity (v_d)
Calculating the denominator:
\[ n e = (8.5 \times 10^{28})(1.6 \times 10^{-19}) = 1.36 \times 10^{10} \]
Now substituting back:
\[ v_d = \frac{5.81 \times 10^9}{1.36 \times 10^{10}} \approx 0.427 \, \text{m/s} \]

### Step 7: Final answer
The average drift velocity of electrons in the copper wire is approximately:
\[ v_d \approx 0.43 \, \text{m/s} \]