Diameter of copper wire to have same resistance as equal length aluminum wire

In summary, the copper wire must have a diameter of 3.077059 mm to have the same resistance as the aluminum wire with a diameter of 3.94mm.
  • #1
Northbysouth
249
2

Homework Statement


What diameter must a copper wire have if its resistance is to be the same as that of an equal length of aluminum wire with diameter 3.94mm?

Give your answer in mm

Homework Equations



R = ρL/A

where A is the cross sectional area, L is the length and ρ resistance of the material

ρCopper = 1.72x10-8 Ωm
ρAluminum = 2.82x10-8 Ωm

The Attempt at a Solution



So I knew that Rcopper = RAluminum

(1.72x10-8Ωm)(L)/Acopper = (2.82x10-8Ωm)(L)/(0.00394m)

A = 0.00240 m
A = πr2

r = 0.027657 m

diameter = 55.31499 mm

But the system says its wrong and I don't see where I've gone wrong.
 
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  • #2
Northbysouth said:

Homework Statement


What diameter must a copper wire have if its resistance is to be the same as that of an equal length of aluminum wire with diameter 3.94mm?

Give your answer in mm

Homework Equations



R = ρL/A

where A is the cross sectional area, L is the length and ρ resistance of the material

ρCopper = 1.72x10-8 Ωm
ρAluminum = 2.82x10-8 Ωm

The Attempt at a Solution



So I knew that Rcopper = RAluminum

(1.72x10-8Ωm)(L)/Acopper = (2.82x10-8Ωm)(L)/(0.00394m)
Start with what I've highlighted above in red.

  • What are units for area: meters or square meters?
  • Before calculating the cross section area of the copper wire, first ask, what is the cross sectional area of the aluminum wire? :wink:

[Edit: On a much lesser note, and I'm not sure if this matters, according to some sources I've looked at on the Internet, 1.72x10-8 Ωm is the resistivity of annealed copper. Plain old copper has a very similar resistivity, but slightly different. Use whatever your course material advises. And again, the difference is small and I'm not sure if copper vs. annealed copper matters.]
 
Last edited:
  • #3
I see what you're saying.

The cross section area od aluminum is π(0.00197m)2

3.8809x10-6 m2

Then solving for the cross sectional area of copper I get

Acopper = 7.4364x10-6 m2

rcopper wire = sqrt(7.4363x102/π)

Diameter of copper wire = 2r = 3.077059 mm

Thanks for your help
 

1. What is the significance of using copper wire instead of aluminum wire for the same resistance?

The choice of using copper wire over aluminum wire for the same resistance is mainly based on the conductivity of the material. Copper is a better conductor of electricity compared to aluminum, meaning it can transmit electricity more efficiently. This results in less energy loss and potentially lower electricity bills.

2. How does the diameter of the copper wire affect the resistance compared to an equal length of aluminum wire?

The diameter of a wire is directly proportional to its resistance. This means that a thicker copper wire will have less resistance compared to a thinner aluminum wire of the same length. This is because a thicker wire has a larger cross-sectional area, allowing for more flow of electrons and thus lower resistance.

3. What is the formula for calculating the diameter of copper wire to have the same resistance as an equal length of aluminum wire?

The formula for calculating the diameter of copper wire to have the same resistance as an equal length of aluminum wire is: dCu = (dAl * ρAl * LAl) / (ρCu * LCu), where d is the diameter, ρ is the resistivity, and L is the length of the wire.

4. How does the resistivity of copper and aluminum affect the diameter of the wire for the same resistance?

The resistivity of a material is a measure of how strongly it resists the flow of electricity. Copper has a lower resistivity compared to aluminum, which means that a copper wire of the same diameter as an aluminum wire will have a lower resistance. In order to have the same resistance, the diameter of the copper wire needs to be larger than the aluminum wire.

5. Are there any other factors to consider when choosing between copper and aluminum wire for the same resistance?

Aside from their resistivity and diameter, there are other factors to consider when choosing between copper and aluminum wire for the same resistance. These include the cost, availability, and the application of the wire. Copper wire is more expensive and heavier compared to aluminum wire, but it is also more durable and suitable for high-power applications. On the other hand, aluminum wire is lighter and cheaper, making it more suitable for low-power applications.

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