Electrical resistance of a paraboloid.

In summary, the conversation discusses the calculation of electrical resistance for a paraboloid from the points y=0 to L. The solution involves breaking the paraboloid into a finite sum of discs and using the equation R = \rho \frac{L}{A}. The integral for the calculation resolves to an undefined numerical answer, indicating infinite resistance. The validity of the solution is also questioned.
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Homework Statement


What would the electrical resistance of a paraboloid from y = 0 to L be?


Homework Equations


[tex]R = \rho \frac{L}{A}[/tex]


The Attempt at a Solution


Okay, so I'll put the parabola (that would rotate into the paraboloid) into the form [tex] y = \sqrt{x}[/tex]

The function A(x) is just the area of the circle, at distance x.

A = [tex]\pi y^{2} = \pi x[/tex]

I'll break the paraboloid up first into a finite sum of discs, from 0 to L.

[tex]R = \Sigma \rho \frac{\Delta x}{A(x)}[/tex]

==>

[tex]R = \int^{L}_{0} \rho \frac{dx}{\pi x}[/tex]

==>

[tex]R = \frac{\rho}{\pi} \int^{L}_{0}\frac{1}{x} dx[/tex]

This integral resolves to:

[tex]R = [\frac{\rho}{\pi}ln(x)]^{L}_{0}[/tex]

Natural log of 0 is undefined, so this would resolve to a numerical answer, as at the limit of x ==> 0, the area approaches zero and this means infinite resistance. But is the maths correct? Or can someone suggest a better way to actually find an answer.
 
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bump. <_<
 

1. What is the electrical resistance of a paraboloid?

The electrical resistance of a paraboloid is a measure of how difficult it is for electricity to flow through the paraboloid. It is influenced by the material, size, and shape of the paraboloid.

2. How is the electrical resistance of a paraboloid calculated?

The electrical resistance of a paraboloid can be calculated using the formula R = ρL/A, where R is the resistance, ρ is the resistivity of the material, L is the length of the paraboloid, and A is the cross-sectional area.

3. What factors affect the electrical resistance of a paraboloid?

The factors that affect the electrical resistance of a paraboloid include the material it is made of, its dimensions, and its temperature. Different materials have different resistivities, and longer and thinner paraboloids have higher resistance. Temperature also affects resistance, with higher temperatures leading to higher resistance.

4. How does the shape of a paraboloid affect its electrical resistance?

The shape of a paraboloid can affect its electrical resistance by changing its cross-sectional area. A paraboloid with a larger cross-sectional area will have lower resistance compared to one with a smaller cross-sectional area, assuming all other factors are constant.

5. What is the practical application of understanding the electrical resistance of a paraboloid?

Understanding the electrical resistance of a paraboloid is important in various engineering and scientific fields. It is used in designing and building electrical circuits, determining the efficiency of electrical components, and analyzing the performance of materials in different environments. It also plays a crucial role in the design and operation of electrical systems, such as power grids and electronic devices.

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