Calculate Resistance of Frustrum Wire

[pizzasky]

How do I calculate the resistance of a wire shaped like a frustrum (in layman's terms, the solid that remains when a cone is removed from the top of a bigger cone)?

Let the radius of the smaller upper plane of the frustrum be a, the radius of the larger lower plane of the frustrum be b, the vertical length of the frustrum be l, and the resistivity of the material be r.

I believe the equation
Resistance = Resistivity * Length / Cross-sectional Area
can be applied. Some integration and calculus will also be needed in solving this problem, but I do not know how to begin.

Should I take an elemental cross-sectional area of the wire and integrate?

 

[Tide]

Yes. You are effectively adding all the elemental resistors in series.

 

[pizzasky]

How do I obtain the general expression to integrate?
What is the expression for the cross-sectional area I should use?

dR = r (dl) / (Expression for cross-sectional area?),
where d represents a small quantity

 

[Tide]

Use simple proportions for the radius of the cross section: R = b + \frac {a-b}{L} z where z goes from 0 to L. The cross-sectional area is just \pi R^2.

 

[pizzasky]

All right, I finally understand how to solve the problem.

Thank you for your help!

 

[Tide]

You are most welcome!