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## 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 [itex]\gamma[/itex][itex]^{4}[/itex]. Find the rate of energy loss of the electron and com¬pare with that for a proton.

## Homework Equations

(F1):

P = [itex]\frac{2kq^{2}a^{2}}{3c^{3}}[/itex]

(F1): (modified to have the "extra factor")

P = [itex]\frac{2kq^{2}a^{2}\gamma^{4}}{3c^{3}}[/itex]

[itex]\gamma[/itex] = [itex]\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}[/itex]

a = [itex]\frac{v^{2}}{r}[/itex]

KE = .5mv[itex]^{2}[/itex]

## The Attempt at a Solution

I did part (a) as best I could. I set kq[itex]^{2}[/itex] = (2.307•10[itex]^{28}[/itex] 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[itex]^{-4}[/itex] eV/s. The book's answer is P = 5.23[itex]^{-4}[/itex] 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 = [itex]\sqrt{\frac{2(1.6•10^{-12}J)}{9.11•10^{-31}kg}}[/itex]

I get 1.874•10[itex]^{9}[/itex] 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[itex]^{5}[/itex]eV/s. The equation I'm plugging all my numbers into looks like this:

P = [itex]\frac{2kq^{2}v^{4}\gamma^{4}}{3c^{3}r^{2}}[/itex]

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.