What Determines the Proximity of a Proton to an Infinitely Charged Line?

[Alexander2357]

Homework Statement

An infinitely long line of charge has a linear charge density of λ C/m. A proton is at distance d m from the line and is moving directly toward the line with speed v m/s.
How close does the proton get to the line of charge?

Homework Equations

$$\Delta KE = W = -\Delta U$$
$$\frac{1}{2}m(v_{2})^{2}-\frac{1}{2}m(v_{1})^{2}=\frac{q_{1}q_{2}}{4\pi \epsilon _{0}r_{2}}-\frac{q_{1}q_{2}}{4\pi \epsilon _{0}r_{1}}$$

Potential difference at distance d from an infinite line of charge: $$V=\frac{\lambda }{4\pi \epsilon _{0}}\int_{0}^{\infty }\frac{dx}{\sqrt{x^{2}+d^{2}}}$$

Distance from the infinite wire that the electron can reach before being stopped:

$$r=d\times e^{\frac{-m(v)^{2}4\pi \epsilon _{0}}{\lambda q}}$$

Where e is Euler's Number.

The Attempt at a Solution

The equation is correct as when I substitutes numbers into it I got the correct answer but how is it derived?

 

[haruspex]

Where did you get the final equation from? If it's the book answer, you are asking us to do your homework for you. You need to show some attempt.

 

[Alexander2357]

Where did you get the final equation from? If it's the book answer, you are asking us to do your homework for you. You need to show some attempt.

The question does not ask me to derive the equation that I got there, it asks me to find the distance that the proton can reach before being repelled and I found it using that equation. I found the equation in the lecture notes but the derivation isn't shown so I would like to at least know where to start to derive it.

I know it has to do with equating the kinetic energy and electric potential energy but what do I have to do after that? Where do I get Euler's Number from?

 

[haruspex]

The integral you quote for the potential from a line of charge can be solved. Put x = d tan(θ), leading to the integral of sec(θ). That integral has a ln() function.
Euler's constant is the base of natural logarithms. The equation y = ln(x) can be rewritten x = e^y.
Does that help, or do you want the full derivation?

 

[Alexander2357]

I get infinity when I evaluate that integral though.

 

[haruspex]

I get infinity when I evaluate that integral though.

The infinity can be avoided by taking limits in the right order, and bearing in mind that what you want in the end is the potential difference between two distances from the wire. See http://www.math.oregonstate.edu/BridgeBook/book/physics/vlineinf