The resistivity of a wire is 1.69x10^-8 ohm-m. if it has a diameter of .1 cm, what is its length if the total resistance is 10 ohms?
To find the length of the wire given its resistivity, diameter, and total resistance, we can use the formula for the resistance of a cylindrical conductor:
[ R = \rho \frac{L}{A} ]
where:
First, we need to find the cross-sectional area ( A ) of the wire. The wire has a circular cross-section, so the area can be calculated using the formula for the area of a circle:
[ A = \pi r^2 ]
The diameter of the wire is given as 0.1 cm, which we need to convert to meters:
[ \text{Diameter} = 0.1 , \text{cm} = 0.1 \times 10^{-2} , \text{m} = 0.001 , \text{m} ]
The radius ( r ) is half of the diameter:
[ r = \frac{\text{Diameter}}{2} = \frac{0.001 , \text{m}}{2} = 0.0005 , \text{m} ]
Now, we can calculate the cross-sectional area ( A ):
[ A = \pi (0.0005 , \text{m})^2 ] [ A = \pi (0.0005)^2 , \text{m}^2 ] [ A = \pi \times 0.00000025 , \text{m}^2 ] [ A \approx 3.14159 \times 0.00000025 , \text{m}^2 ] [ A \approx 7.854 \times 10^{-7} , \text{m}^2 ]
Next, we can rearrange the resistance formula to solve for the length ( L ):
[ L = \frac{R \cdot A}{\rho} ]
Substitute the known values into the equation:
[ L = \frac{10 , \text{ohms} \times 7.854 \times 10^{-7} , \text{m}^2}{1.69 \times 10^{-8} , \text{ohm-m}} ]
[ L = \frac{7.854 \times 10^{-6} , \text{ohm-m}^2}{1.69 \times 10^{-8} , \text{ohm-m}} ]
[ L = \frac{7.854}{1.69} \times 10^{-6 + 8} , \text{m} ]
[ L = \frac{7.854}{1.69} \times 10^2 , \text{m} ]
[ L \approx 4.65 \times 10^2 , \text{m} ]
[ L \approx 465 , \text{m} ]
So, the length of the wire is approximately 465 meters.