# Deriving Rutherford's Formula

1. Apr 27, 2010

### jameson2

1. The problem statement, all variables and given/known data
2. Relevant equations
This is in my mechanics book. It gives the final formula, only saying "carrying out the elementary integration..."
Derive Rutherford's Formula:
$$\phi_0=cos^{-1}\frac{a}{\sqrt{1+a^2}}$$
where
$$a=\frac{\alpha}{mv_\infty^2 \rho}$$

From the equation
$$\phi_0 = \int_{r_{min}}^\infty \frac{\frac{\rho}{r^2}dr}{\sqrt{1-\frac{\rho^2}{r^2}-\frac{2U}{mv_\infty^2}}}$$

Using $$U=\frac{\alpha}{r}$$

3. The attempt at a solution
This question comes up regularly on exams, and the hint that is given is that

$$\int_{x_+}^\infty \frac{1}{x\sqrt{(x-x_+)(x-x_-}} = arccos \frac{a}{\sqrt{1+a^2}}$$

where $$x_+=a+\sqrt{1+a^2}$$ and $$x_-=a-\sqrt{1+a^2}$$

I don't see how this helps, since the integral is from r min, which corresponds to $$x_-$$ I think. Also, I have $$\rho$$ instead of 1 in the expressions for $$x_-$$ and $$x_+$$
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution