Calculate Resistance of Frustrum Wire

In summary, To calculate the resistance of a wire shaped like a frustrum, the equation Resistance = Resistivity * Length / Cross-sectional Area can be used. Integration and calculus are needed, and the general expression for integration is dR = r (dl) / (Expression for cross-sectional area?). The cross-sectional area can be found using simple proportions for the radius of the cross section, R = b + (a-b)/L z where z goes from 0 to L, and the cross-sectional area is pi * R^2.
  • #1
pizzasky
172
2
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?
 
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  • #2
Yes. You are effectively adding all the elemental resistors in series.
 
  • #3
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
 
  • #4
Use simple proportions for the radius of the cross section: [itex]R = b + \frac {a-b}{L} z[/itex] where z goes from 0 to L. The cross-sectional area is just [itex]\pi R^2[/itex].
 
  • #5
All right, I finally understand how to solve the problem.

Thank you for your help!
 
  • #6
You are most welcome!
 

What is a frustrum wire?

A frustrum wire is a type of wire with a tapered shape, meaning it has a larger diameter at one end and a smaller diameter at the other end. It is commonly used in electrical circuits and is typically made of a conductive material such as copper or aluminum.

Why is it important to calculate the resistance of a frustrum wire?

Calculating the resistance of a frustrum wire is important because it allows us to understand how the wire will affect the flow of electricity in a circuit. Resistance is a measure of how difficult it is for electrons to flow through a material, and it can impact the efficiency and performance of electronic devices.

What factors affect the resistance of a frustrum wire?

The resistance of a frustrum wire is affected by several factors, including the material it is made of, its length, its cross-sectional area, and the temperature of the wire. Additionally, the resistance may also be influenced by the type of circuit the wire is used in and any external factors such as humidity.

How do you calculate the resistance of a frustrum wire?

To calculate the resistance of a frustrum wire, you can use the formula R = ρL/A, where R is resistance, ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area. You can also use online calculators or consult a reference table for the resistivity of different materials.

What are some practical applications for calculating the resistance of a frustrum wire?

Calculating the resistance of a frustrum wire is useful in a variety of applications, including designing and building electrical circuits, determining the appropriate wire size for a specific circuit, and troubleshooting issues with electronic devices. It is also important in industries such as telecommunications, aerospace, and automotive engineering.

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