As part of a class project you are given 0.900 g of nichrome and asked to fabricate a wire with uniform cross-section. You use up 95% of the nichrome and make a wire with a resistance of 0.840 Ω. The resistivity of nichrome is 1.00 10-6 Ω · m and its density is 8.31 103 kg/m3.

(a) What length of wire do you end up with?

(b) What is the diameter of the wire?

To find the length of the wire (a), we can start by determining the mass of the wire based on the percentage used and the initial mass of the nichrome.

(a) Length of the wire:
1. Calculate the mass of nichrome used:

Mass used = Initial mass of nichrome × Percentage used
Mass used = 0.900 g × 0.95
Mass used = 0.855 g

2. Convert the mass used to the volume of the wire using the density of nichrome:

Volume of wire = Mass used / Density of nichrome
Volume of wire = 0.855 g / 8.31 × 10^3 kg/m^3

3. Calculate the length of the wire using the formula for the volume of a cylinder:

Volume of wire = Cross-sectional area × Length of wire
Length of wire = Volume of wire / Cross-sectional area

The cross-sectional area can be calculated using the formula for the resistivity of a material:

Cross-sectional area = Resistance × Length of wire / Resistivity

Now, let's proceed to find the length of the wire.

(a) Length of the wire:
1. Calculate the cross-sectional area:

Cross-sectional area = Resistance × Length of wire / Resistivity
0.840 Ω = (1.00 × 10^-6 Ω · m) × Length of wire / Cross-sectional area

2. Rearrange the equation to solve for the cross-sectional area:

Cross-sectional area = (0.840 Ω) × (1.00 × 10^-6 Ω · m) / Length of wire

3. Substitute the known values into the equation:

Cross-sectional area = (0.840 Ω) × (1.00 × 10^-6 Ω · m) / Length of wire

4. Solve for the length of the wire:

Length of wire = (0.840 Ω) × (1.00 × 10^-6 Ω · m) / Cross-sectional area

(b) Diameter of the wire:
To find the diameter of the wire, we will use the cross-sectional area.

1. Calculate the radius of the wire:

Radius = √(Cross-sectional area / π)

2. Calculate the diameter of the wire:

Diameter = 2 × Radius

Now, we have the process for finding the length of the wire (a) and the diameter of the wire (b).