A 50.0-g sample of a conducting material is all that is available. The
resistivity of the material is measured to be 11�~ 10-8 ƒ¶m and the density is 7.86 g/cm3. The
material is to be shaped into a solid cylindrical wire that has a total resistance of 1.5 Ħ.
a) What length of wire is required?
b) What must be the diameter of the wire?
To solve this problem, we will use the formulas that relate resistivity, resistance, length, and cross-sectional area of a wire.
a) To determine the length of the wire required, we can use the formula:
Resistance = Resistivity x (Length / Cross-sectional Area)
We are given the resistance (1.5 Ω), the resistivity (11 x 10^-8 Ωm), and the density (7.86 g/cm^3). However, we need to find the cross-sectional area before we can calculate the length.
b) To find the diameter of the wire, we can use the formula:
Cross-sectional Area = (π x (Diameter/2)^2)
To calculate the diameter of the wire, we can rearrange the formula:
Diameter = 2 x sqrt(Cross-sectional Area / π)
Now let's calculate!
a) To find the length of the wire:
Since we are given the resistivity, resistance, and density of the conducting material, we can calculate the cross-sectional area using the formula for mass:
Mass = Volume x Density
Volume = (π x (Diameter/2)^2) x Length
The mass of the wire is given as 50.0 g, so we can calculate the volume:
50.0 g = (π x (Diameter/2)^2) x Length
Now, we need to calculate the cross-sectional area using the formula for resistance:
1.5 Ω = (11 x 10^-8 Ωm) x (Length / Cross-sectional Area)
Let's solve these two equations simultaneously to find the length.
b) To find the diameter of the wire:
We can now rearrange the formula for cross-sectional area to solve for the diameter:
Diameter = 2 x sqrt(Cross-sectional Area / π)
Now that we know the length, we can plug it into the equation and calculate the diameter.
Keep in mind that we need to convert the density from g/cm^3 to kg/m^3 and the diameter from meters to centimeters for the units to match.
Once we have the diameter, we can use it to calculate the length of the wire using the formula for mass.
That's it! Using the formulas and given values, we can find both the length and diameter of the wire.