# How is the equation derived?

1. Aug 24, 2014

### Alexander2357

1. The problem statement, all variables and given/known data

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?

2. Relevant 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.

3. 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?

2. Aug 24, 2014

### 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.

3. Aug 24, 2014

### Alexander2357

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?

4. Aug 24, 2014

### 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 = ey.
Does that help, or do you want the full derivation?

5. Aug 25, 2014

### Alexander2357

I get infinity when I evaluate that integral though.

6. Aug 25, 2014

### haruspex

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted