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Calculate the length of a wire with a cross-sectional area of [tex]$0.62 \, \text{mm}^2$[/tex] and a resistivity of [tex]$1.8 \times 10^{-7} \, \Omega \cdot \text{m}$[/tex] that will have a resistance of [tex]3 \, \Omega[/tex].

Use the formula:
[tex]\[ R = \rho \frac{L}{A} \][/tex]

Where:
- [tex]\( R \)[/tex] is the resistance,
- [tex]\( \rho \)[/tex] is the resistivity,
- [tex]\( L \)[/tex] is the length of the wire,
- [tex]\( A \)[/tex] is the cross-sectional area.


Sagot :

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.
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