Certainly! Let's solve the problem step by step.
Given:
- Resistivity of the material, ρ = [tex]\(1.8 \times 10^{-7}\)[/tex] ohm meters (Ω·m)
- Cross-sectional area of the wire, [tex]\(A = 0.62 \, \text{mm}^2\)[/tex]
- Resistance of the wire, [tex]\(R = 3 \, \Omega\)[/tex]
We need to calculate the length of the wire, [tex]\(L\)[/tex].
### Step-by-Step Solution:
1. Convert the cross-sectional area from square millimeters to square meters:
[tex]\[
0.62 \, \text{mm}^2 = 0.62 \times 10^{-6} \, \text{m}^2
\][/tex]
2. Use the formula for resistance in terms of resistivity, length, and cross-sectional area:
[tex]\[
R = \rho \left(\frac{L}{A}\right)
\][/tex]
Rearrange this formula to solve for the length [tex]\(L\)[/tex]:
[tex]\[
L = \frac{R \cdot A}{\rho}
\][/tex]
3. Substitute the given values into the formula:
[tex]\[
L = \frac{3 \, \Omega \times 0.62 \times 10^{-6} \, \text{m}^2}{1.8 \times 10^{-7} \, \Omega \cdot \text{m}}
\][/tex]
4. Perform the multiplication in the numerator:
[tex]\[
3 \times 0.62 \times 10^{-6} = 1.86 \times 10^{-6}
\][/tex]
5. Divide by the resistivity in the denominator:
[tex]\[
L = \frac{1.86 \times 10^{-6}}{1.8 \times 10^{-7}}
\][/tex]
6. Carry out the division:
[tex]\[
L = 10.333333333333334 \, \text{m}
\][/tex]
Therefore, the length of the wire that will have a resistance of 3 ohms is approximately [tex]\(10.33 \, \text{meters}\)[/tex].
So, the detailed solution provides us with a wire length of approximately 10.33 meters.
