A resistance of `2Omega` is to be made from a copper wire (specific resistance `1.7xx10^(-8)Omegam`) using a wire of length 50 cm. The radius of the wire is

Options

A. `0.0116 mm`
B. `0.0367 mm`
C. `0.116 mm`
D. `0.367 mm`

Explanation

To find the radius of the copper wire that has a resistance of 2 Ohms, we can use the formula for resistance in terms of resistivity, length, and cross-sectional area:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Resistance (R) = 2 Ohms
- Specific resistance (ρ) = \(1.7 \times 10^{-8} \, \Omega \cdot m\)
- Length (L) = 50 cm = \(50 \times 10^{-2} \, m = 0.5 \, m\)

2. **Use the Formula for Resistance:**
The formula for resistance is given by:
\[ R = \frac{\rho L}{A} \]
where \(A\) is the cross-sectional area of the wire. For a cylindrical wire, the area \(A\) can be expressed in terms of the radius \(r\):
\[ A = \pi r^2 \]
Thus, we can rewrite the resistance formula as:
\[ R = \frac{\rho L}{\pi r^2} \]

3. **Rearranging the Formula to Solve for Radius:**
Rearranging the equation to solve for \(r\):
\[ r^2 = \frac{\rho L}{\pi R} \]
\[ r = \sqrt{\frac{\rho L}{\pi R}} \]

4. **Substituting the Values:**
Now, substitute the known values into the equation:
\[ r = \sqrt{\frac{(1.7 \times 10^{-8} \, \Omega \cdot m)(0.5 \, m)}{\pi (2 \, \Omega)}} \]

5. **Calculating the Values:**
First, calculate the numerator:
\[ \rho L = (1.7 \times 10^{-8})(0.5) = 0.85 \times 10^{-8} \, \Omega \cdot m \]
Now, calculate the denominator:
\[ \pi R = \pi \times 2 \approx 6.2832 \, \Omega \]
Now, substitute these values into the equation for \(r\):
\[ r = \sqrt{\frac{0.85 \times 10^{-8}}{6.2832}} \approx \sqrt{1.352 \times 10^{-9}} \approx 0.0368 \times 10^{-3} \, m \]

6. **Convert to Millimeters:**
To convert meters to millimeters:
\[ r \approx 0.0368 \times 10^{-3} \, m = 0.0368 \, mm \]

### Final Answer:
The radius of the wire is approximately \(0.0368 \, mm\).