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silmaril89
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Homework Statement
I'm confused about part (c) and on in the following questions:
Consider a betatron with these parameters: the radius of the electron orbit is 0.5 m; the kinetic energy of an electron injected into the accelerator is 2.0 MeV; and the rate of increase of the magnetic flux through the area of the toroidal vacuum chamber is 25 Wb/s. The electrons are ejected after 4 ms of acceleration.
(a) Compute the magnitude of the induced electric field
(b) Compute the work done on an electron per revolution around the orbit.
(c) Compute the number or revolutions completed by an electron before ejection. (Approximate the electron speed by c.)
(d) Compute the final kinetic energy of the electron.
(d) In order to keep the radius r of the electron path constant, B at r must be one half of the average of B over the area enclosed by the circular orbit. What is dB/dt at r during the acceleration?
Homework Equations
curl E = -dB/dt
B(r = electron orbit radius, t) = 1/2 B(t) (average field inside electron orbit)
dW = F . dl
The Attempt at a Solution
I solved (a) using ampere's law, and found the electric field to be:
E = (-r/2) (dB/dt) in the phi direction, and B is the average field
I solved (b) by using the equation for W above:
W = pi * r^2 dB/dt, where B is again the average field inside the orbital radius of the electron
Now, I'm stuck on (c), I don't get why we would be approximating the electron speed to be c. If it was c, and we know it is in the betatron for 4ms, the calculation is trivial. But, the electrons would be accelerating the whole time, they couldn't just have one speed?
In part (d), I'm really not sure how to calculate the final kinetic energy. Do I have to take into account relativistic effects?
Thanks to anyone who replies.