You've made the finals of the Science Olympics! As one of your tasks, you're given 1.1 g of copper and asked to make a wire, using all the metal, with a resistance of 2.0 (ohms). Copper has a density of 8900 kg/m^3.
What length will you choose for your wire?
What diameter will you choose for your wire?
Please help me out. I got 3600 m for the length and .0062mm in diameter. It's wrong but I have no idea what I did wrong. Please help with some details. it would be much appricated!
Thank you in advance!
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The length is 3.8m, the diameter is 0.19
To determine the length and diameter of the copper wire, we can use the following equations and information:
1. Resistance (R) = Resistivity (ρ) x Length (L) / Cross-sectional Area (A)
2. Resistivity (ρ) for copper = 1.7 x 10^-8 (ohm-m)
3. Density (ρ') of copper = 8900 kg/m^3
4. Mass (m) of copper wire = 1.1 g
First, let's find the volume (V) of the copper wire:
V = m / ρ'
= 1.1 g / 8900 kg/m^3
= 1.1 x 10^-3 kg / 8900 kg/m^3
≈ 1.24 x 10^-7 m^3
Now, let's find the length (L) of the wire using its volume:
V = πr^2 x L
1.24 x 10^-7 m^3 = πr^2 x L
Since we have been given the resistance (R = 2.0 ohms), and resistivity (ρ = 1.7 x 10^-8 ohm-m) for copper, we can rearrange the resistance equation to solve for cross-sectional area (A):
R = ρ x L / A
2.0 ohms = 1.7 x 10^-8 ohm-m x L / A
Now, if we substitute the value of L obtained earlier, we get:
2.0 ohms = 1.7 x 10^-8 ohm-m x [1.24 x 10^-7 m^3 / (πr^2)]
Now, we can manipulate the equation to solve for the diameter (d) of the wire:
2.0 ohms = 1.7 x 10^-8 ohm-m x [1.24 x 10^-7 m^3 / (πr^2)]
Rearrange the equation:
(πr^2) / (1.24 x 10^-7 m^3) = 1.7 x 10^-8 ohm-m / 2.0 ohms
Simplify:
πr^2 = (1.7 x 10^-8 ohm-m / 2.0 ohms) x (1.24 x 10^-7 m^3)
Now, solve for the radius (r):
r^2 = [(1.7 x 10^-8 ohm-m / 2.0 ohms) x (1.24 x 10^-7 m^3)] / π
Take the square root to find the radius (r):
r = √([(1.7 x 10^-8 ohm-m / 2.0 ohms) x (1.24 x 10^-7 m^3)] / π)
Finally, to find the diameter (d), multiply the radius (r) by 2:
d = 2r = 2 x √([(1.7 x 10^-8 ohm-m / 2.0 ohms) x (1.24 x 10^-7 m^3)] / π)
Using these equations and calculations, you can determine the correct length and diameter of the copper wire.