Resistance of a Copper Wire with initial and final radius

[jawhnay]

Homework Statement

A Copper wire has a shape given by a radius that increases as R(x)= ae^x+ b. Its initial radius is .45 mm and final radius is 9.67 mm and its horizontal length is 38 cm. Find its resistance.

Homework Equations

R = pL/A where p = resistivity of copper
A = ∏r²
L = length of copper wire

resistivity of copper = 1.7 x 10^-8

1/R_total = 1/R_initial + 1/R_final

The Attempt at a Solution

My attempt at the solution was finding the resistance of the wire when the radius is initially at .45 mm and then finding the resistance of the wire at 9.67 mm using the formula R = pL/A. Then I tried adding those resistances together by using the Resistance formula for parallel circuits which is 1/R_total = 1/R_initial + 1/R_final

 

[kontejnjer]

Think of a small wire element of length $dl$ and find its resistance $dR$, then integrate over the length to find the total resistance.

 

[jawhnay]

so you're saying R = pL/A ... dR = pdr/A what would my area be? Would it be A = ∏r² or A = 2∏rL How exactly do I go about doing this?

 

[jawhnay]

Can anyone else offer any insight on how to do this problem?

 

[milesyoung]

So you're given:
r(x) = a*e^x + b
r(0 m) = 0.45e-3 m
r(38e-2 m) = 9.67e-3 m

A(x) = π*r(x)^2

You could imagine summing up the resistances of a bunch of thin slices of wire with width dx as a heuristic to help you set up the integral:
$$R(x_e) = \int_0^{x_e} \frac{\rho}{A(x)} \mathrm{d}x$$

 

[jawhnay]

where are you getting r(0 m) = 0.45e-3 m and r(38e-2 m) = 9.67e-3 m ?

 

[milesyoung]

Let x = 0 cm and x = 38 cm represent the endpoints with the small and large radius, respectively. You're then given the values that r(x) should take at these endpoints.

 

[jawhnay]

How do you know that r(0) = .45 x 10^-3 and r(.38) = 9.67 x 10^-3 ? I'm kind of confused... Can anyone make this clearer for me?

 

[milesyoung]

Imagine putting a coordinate axis down the length of the wire. You could call this the x-coordinate axis and x would represent a specific position along the length of the wire.

You know the length of the wire is 38 cm and the radii of the thin and thick ends are 0.45 mm and 9.67 mm, respectively. You also know how the radius of the wire changes from one end to the other.

If r(x) is the radius at position x, you could let x = 0 represent the thin end of the wire and x = 0.38 could then be the thick end (it's usually good practice to convert everything to SI base units, meter [m] for length).

Does that help?