Resistance of a Composite Wire

[SeannyBoi71]

Homework Statement

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?

Homework Equations

R=ρl/A

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 = (R_copper^-1 + R_steel^-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?

 

[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