You've made the finals of the Science Olympics! As one of your tasks, you're given 1.1g of copper and asked to make a cylindrical wire, using all the metal, with a resistance of 1.7 Ohm.
What length l will you choose for your wire?
What diameter d will you choose for your wire?
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To find the length (l) and diameter (d) of the wire, we need to use the formula that relates the resistance, length, cross-sectional area, and resistivity of the wire:
Resistance (R) = (Resistivity * Length) / Cross-sectional area
In this case, resistance is given as 1.7 Ohms, and we need to find the length and diameter. We can make use of the fact that the wire is in the shape of a cylinder - the cross-sectional area will be given by the formula:
Cross-sectional area = π * (diameter/2)^2
First, let's find the resistivity of copper. The resistivity of copper is a known constant, which you can look up online or find in a physics reference book. Let's say it is given as ρ (rho).
Now we can rearrange the resistance formula to solve for the length:
Length = (Resistance * Cross-sectional area) / Resistivity
We already have the resistance and resistivity, so we need to find the cross-sectional area. Substituting the formula for cross-sectional area into the length equation:
Length = (Resistance * (π * (diameter/2)^2)) / Resistivity
Now, we need to substitute in the given values. The resistance is 1.7 Ohms, and the diameter is yet to be determined. However, we know the mass of copper is 1.1g. To find the diameter, we can use the density of copper and the mass of the wire:
Density = Mass / Volume
The volume of a cylinder is given by the formula:
Volume = π * (diameter/2)^2 * length
Rearranging the formula for volume to solve for diameter:
Diameter = √(4 * Mass / (π * length * Density))
Now, let's substitute the values and solve the equation for diameter:
Diameter = √(4 * 1.1g / (π * length * Density))
(Note: We need to know the density of copper, which is approximately 8.96 g/cm³.)
Once we have the diameter, we can substitute it back into the equation for length to find the final length:
Length = (Resistance * (π * ((√(4 * 1.1g) / (π * length * Density))/2)^2)) / Resistivity
We can iteratively solve these equations by approximating the values for diameter and length until they converge to a stable solution.