A proton with kinetic energy `T= 10 MeV` flies past a stationary free electron at a distacne `b= 10 p m`. Find the energy acquired by the electron. Assuming the proton's trajectory to be rectilinear and the electron to paractically motionless as the proton flies by.

Explanation

The proton moving by the first accelerates and then decelerates and it not easy to calculate the energy lost by the proton so energy conservation does not do the trick. Rather we must directly calcuate the momentum acquire by the electron. By symmetry that momentum is along `Ovec(A)` and its magnitude is `P_(d)~= intF_(_|_)dt`
where `F_(_|_)` is the component along `OA` of the force on electron. Thus `p_(d)= int_(-oo)^(oo)(e^(2))/(4 pi epsilon_(0))(b)/(sqrt(b^(2)+v^(2)t^(2))).(1)/(b^(2)+v^(2)t^(2))dt`
`=(e^(2)b)/(4pi epsilon_(0)v) int_(-oo)^(oo) (dx)/((b^(2)+x^(2))3//2)`
Evauate the integral by substituting
`x= b tan theta`
Then `P_(e )-(2e^(2))/((4pi epsilon_(0))vb).`
Then `T_(e )=(P_(e )^(2))/(2m_(e))=(m_(p)e^(4))/((4pi epsilon_(0))^(2)Tb^(2)m_(e))`
In Gaussian units there is no factor `(4 pi epsilon_(0))^(2)`. Substituting the values we get
`T_(e )= 3.82V`