Relativistic motion in a particle accelerator: Rate of Energy Loss

[JackFlash]

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 into 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]

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 let's you determine \gamma directly.

 

[JackFlash]

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?