# Relativistic motion in a particle accelerator: Rate of Energy Loss

## Homework Statement

(a) Consider a 10-Mev proton in a cyclotron of radius .5m. Use the formula (F1) to calculate the rate of energy loss in eV/s due to radiation.
(b) Suppose that we tried to produce electrons with the same kinetic energy in a circular machine of the same radius. In this case the motion would be relativistic and formula (F1) is modified by an extra factor of $\gamma$$^{4}$. Find the rate of energy loss of the electron and com¬pare with that for a proton.

## Homework Equations

(F1):
P = $\frac{2kq^{2}a^{2}}{3c^{3}}$

(F1): (modified to have the "extra factor")
P = $\frac{2kq^{2}a^{2}\gamma^{4}}{3c^{3}}$

$\gamma$ = $\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$

a = $\frac{v^{2}}{r}$

KE = .5mv$^{2}$

## The Attempt at a Solution

I did part (a) as best I could. I set kq$^{2}$ = (2.307•10$^{28}$ J•m), solved for v by using the Kinetic Energy formula and converting the 10-MeV value into Joules, doubling it, divide by the proton's mass, and take the square root (my velocity was 43738998.62 m/s). Plug it all in to the equation, the answer comes out in J/s. Convert to eV/s and I get 5.21$^{-4}$ eV/s. The book's answer is P = 5.23$^{-4}$ eV/s. I disregarded the small error as due to rounding numbers throughout the equation.

Part (b) is what's grinding my gears. I do the same thing I did to get v as before. 10-MeV into Joules, double, divide by the electron's mass, and square root:

v = $\sqrt{\frac{2(1.6•10^{-12}J)}{9.11•10^{-31}kg}}$

I get 1.874•10$^{9}$ m/s. That isn't possible at all. But I run with it, solve for gamma (I got an imaginary number given that v was bigger than c), then solve the equation, convert the answer from J/s to eV/s and I get 1.2eV/s. The answer should be 2.05•10$^{5}$eV/s. The equation I'm plugging all my numbers into looks like this:

P = $\frac{2kq^{2}v^{4}\gamma^{4}}{3c^{3}r^{2}}$

I understand it may appear that I only tried once, and in laziness decided to post the question on here, but I assure you all that I have tried the problem many times. I'm sure the velocity is wrong, but I don't know how it is wrong. Any help is much appreciated.

fzero
Homework Helper
Gold Member
For relativistic motion, the energy is

$$E = \gamma m c^2,$$

so the kinetic energy is

$$T = \gamma m c^2 - mc^2.$$

The fact you found $v>c$ was a hint that you were using an invalid nonrelativistic expression. As an aside, the equation above lets you determine $\gamma$ directly.

Mother of mercy, that's it! Solving for Kinetic Energy to get gamma, then solving for v using the equation I got, then plugging it all into the last equation got me my answer. It also helped me out on the two questions after the one I posted, where my velocities were all just a hair below the speed of light.
Thank you very much. I recognized the equations you posted and was able to get the answers right away. Perhaps I should pay better attention to my textbook, huh?