Question

You want to produce three 1.00-mm-diameter cylindrical wires, each with a resistance of 5.00 Ω at room temperature. One wire is gold, one is copper, and one is aluminum. Refer to Table 25.1 in the textbook for the resistivity values.

Part A

What will be the length of the gold wire?

Part B

What will be the length of the copper wire?

Part C

What will be the length of the aluminum wire?

Part D

Gold has a density of 1.93 × 104 kg/m3. What will be the mass of the gold wire?

Part E

If gold is currently worth $40 per gram, what is the cost of the gold wire?

Answer 1

We know that resistance of a cylindrical wire is given by:

R = rho*L/A

rho = resistivity of wire

L = length of wire & A = Cross-sectional area of wire

Part A

If value of resistivity are different than used, then let me know.

Resistivty of gold = 2.44*10^-8 ohm.m

A = Cross-sectional area = pi*d^2/4

R = resistance of wire = 5.00 ohm

d = diameter = 1.00 mm = 1.00*10^-3 m

So,

L = R*A/rho

L = 5.00*pi*(1.00*10^-3)^2/(4*2.44*10^-8)

L = 160.94 m = 161 m = length of gold wire

Part B.

Resistivty of copper = 1.72*10^-8 ohm.m

A = Cross-sectional area = pi*d^2/4

R = resistance of wire = 5.00 ohm

d = diameter = 1.00 mm = 1.00*10^-3 m

So,

L = R*A/rho

L = 5.00*pi*(1.00*10^-3)^2/(4*1.72*10^-8)

L = 228.31 m = 228 m = length of copper wire

Part C.

Resistivty of aluminum = 2.75*10^-8 ohm.m

A = Cross-sectional area = pi*d^2/4

R = resistance of wire = 5.00 ohm

d = diameter = 1.00 mm = 1.00*10^-3 m

So,

L = R*A/rho

L = 5.00*pi*(1.00*10^-3)^2/(4*2.75*10^-8)

L = 142.80 m = 143 m = length of aluminum wire

Part D.

Mass is given by:

Mass = Volume*density

Volume of wire = length*cross-sectional area = L*A = pi*d^2*L/4

density of gold = 1.93*10^4 kg/m^3

Mass = density*pi*d^2*L/4

Using given values:

Mass = 1.93*10^4*pi*(1.00*10^-3)^2*160.94/4

Mass = 2.43956 kg

Mass of gold wire = 2.44 kg

Part E.

Mass of 1 gram gold = $40

Mass of 1 kg gold = $40000

So mass of 2.44 kg gold = $40000*2.44 = $97600

Let me know if you've any query.