# Resistance of a Composite Wire

#### SeannyBoi71

1. The problem statement, all variables and given/known data
A transmission wire is made of a cylindrical core of copper surrounded by a sheath of steel. The copper core is 1 cm in diameter, and the sheath has an outer diameter of 2 cm. What is the resistance of 1 km of wire?

2. Relevant equations
R=ρl/A

3. The attempt at a solution
I tried treating the two layers of wire as resistors in parallel, so I found the resistance of each one and added them that way (R-1=R1-1+R2-1). For the area of the outer sheath, I used the area of the total wire minus the area of the inner copper part.

Resistance of copper = (1.72e-8 * 1000) / 7.85e-5 = 0.219 ohm
Area of steel part: 7.07e-4 - 7.85e-5 = 6.28e-4 m2
Resistance of steel = (29e-8 * 1000) / 6.28e-4 = 0.46 ohm

Equivalent resistance = (Rcopper-1 + Rsteel-1)-1 = 0.15. But the answer key says it's 0.19. I tried to keep as many decimal places all the way through, and used values for resistivity straight out of the textbook. Any help please?

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#### technician

I think you have mis-calculated the cross sectional area of the steel.
The outside diameter of the steel is 2cm and the inside diameter is 1cm
I got the area to be 3.142 x 10^-4 - 7.86 x 10^-5
This gave a total resistance of 0.186 (0.19)

#### SeannyBoi71

Alright, I see how you did that. but how come you used 1 (half of 2) instead of 1.5 (half of 3, total diameter of the wire) for the first radius?

#### technician

The wire has a central core of diameter 1cm. The steel coating has a diameter of 2cm.....it is not 3cm.
Imagine a Doughnut.... the hole in the middle is the copper (1cm diameter) the best bit of the doughnut is the steel.... diameter 2cm

#### SeannyBoi71

Oh ok silly mistake on my part. Thank you very much.

#### technician

glad you got it.... enjoy the doughnut

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