- #1

BlackWyvern

- 105

- 0

## 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.