A wire made of aluminum having resistivity `rho = 2.8 xx10^(-8) Omega - m ` with a circular cross - section and has a radius of `2xx10^(-3)` m. A current of 5 A flows through the wire . If the voltage difference between the ends is 1 V , the length of the wire in m is

Options

A. 50
B. 60
C. 90
D. 120

Answer

To find the length of the aluminum wire, we will use the relationship between resistance, resistivity, length, and cross-sectional area, along with Ohm's law. Here are the steps to solve the problem:

### Step 1: Identify the given values
- Resistivity of aluminum, \( \rho = 2.8 \times 10^{-8} \, \Omega \cdot m \)
- Radius of the wire, \( r = 2 \times 10^{-3} \, m \)
- Current, \( I = 5 \, A \)
- Voltage difference, \( V = 1 \, V \)

### Step 2: Calculate the cross-sectional area \( A \) of the wire
The cross-sectional area \( A \) of a circular wire is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of the radius:
\[ A = \pi (2 \times 10^{-3})^2 = \pi \times 4 \times 10^{-6} \, m^2 \]

### Step 3: Use Ohm's law to find the resistance \( R \)
Ohm's law states that:
\[ V = I R \]
Rearranging this gives:
\[ R = \frac{V}{I} \]
Substituting the values of \( V \) and \( I \):
\[ R = \frac{1}{5} \, \Omega \]

### Step 4: Relate resistance to resistivity, length, and area
The resistance \( R \) can also be expressed in terms of resistivity \( \rho \), length \( L \), and area \( A \):
\[ R = \frac{\rho L}{A} \]
Substituting the values we have:
\[ \frac{1}{5} = \frac{(2.8 \times 10^{-8}) L}{\pi \times 4 \times 10^{-6}} \]

### Step 5: Rearrange the equation to solve for length \( L \)
Rearranging the equation gives:
\[ L = \frac{(1/5) \times (\pi \times 4 \times 10^{-6})}{2.8 \times 10^{-8}} \]

### Step 6: Calculate the length \( L \)
Now we will compute \( L \):
\[ L = \frac{(0.2 \times \pi \times 4 \times 10^{-6})}{2.8 \times 10^{-8}} \]
Calculating the numerator:
\[ 0.2 \times \pi \times 4 \approx 0.8\pi \approx 2.513 \, (using \, \pi \approx 3.14) \]
Thus,
\[ L \approx \frac{2.513 \times 10^{-6}}{2.8 \times 10^{-8}} \approx 89.4 \, m \]

### Final Answer
The length of the wire is approximately \( 90 \, m \).

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