Certainly! Let's go through the steps to determine the length of the copper wire given the diameter, resistivity, and resistance.
### Step 1: Convert Diameter to Radius
First, convert the diameter of the wire into its radius. The given diameter is [tex]\(1.58 \, \text{mm}\)[/tex].
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{1.58 \, \text{mm}}{2} = 0.79 \, \text{mm}
\][/tex]
Since the resistivity is given in meters, we need to convert the radius from millimeters to meters.
[tex]\[
0.79 \, \text{mm} = 0.79 \times 10^{-3} \, \text{m} = 0.00079 \, \text{m}
\][/tex]
### Step 2: Compute the Cross-Sectional Area
Next, calculate the cross-sectional area (A) of the wire. The cross-sectional area of a wire with a circular cross-section is given by the formula:
[tex]\[
A = \pi r^2
\][/tex]
Substitute the radius into the formula:
[tex]\[
A = \pi (0.00079 \, \text{m})^2 \approx 1.9606679751053896 \times 10^{-6} \, \text{m}^2
\][/tex]
### Step 3: Use the Resistivity Formula to Find Length
The relationship between resistance (R), resistivity ([tex]\(\rho\)[/tex]), length (L), and cross-sectional area (A) of a wire is given by the formula:
[tex]\[
R = \rho \frac{L}{A}
\][/tex]
Rearrange the formula to solve for length (L):
[tex]\[
L = \frac{R \cdot A}{\rho}
\][/tex]
Substitute the known values into the equation. Here, [tex]\(R = 0.50 \, \Omega\)[/tex], [tex]\(\rho = 1.7 \times 10^{-8} \, \Omega \cdot \text{m}\)[/tex], and [tex]\(A = 1.9606679751053896 \times 10^{-6} \, \text{m}^2\)[/tex]:
[tex]\[
L = \frac{0.50 \, \Omega \times 1.9606679751053896 \times 10^{-6} \, \text{m}^2}{1.7 \times 10^{-8} \, \Omega \cdot \text{m}}
\][/tex]
Perform the calculation:
[tex]\[
L \approx 57.66670515015852 \, \text{m}
\][/tex]
### Conclusion
The length of the copper wire is approximately [tex]\(57.67 \, \text{m}\)[/tex].
