Lorentz transformations combined with force

[daselocution]

Homework Statement

First part of the problem:
Newton’s second law is given by F=dp/dt. If the force
is always perpendicular to the velocity, show that F=gamma*m*a, where a is the acceleration.

Second part of the problem: Use the result of the previous problem to show that
the radius of a particle’s circular path having charge q traveling with speed v in a magnetic field perpendicular to the particle’s path is r = p/qB. What happens to the radius as the speed increases as in a cyclotron?

Homework Equations

p=gamma*m*v

F_{magnetic field} = qv x B = (in this case b/c of θ=90º) qvB

The Attempt at a Solution

The first part: I am thinking that since the force is perpendicular to the path of motion, that the speed of the particle will not change, only it's direction--is this logical? If this were indeed the case, then I would solve as follows (and get the answer as directed):

F=dp/dt=d(m*gamma*v)/dt = m*gamma*d(v)/dt
=m*gamma*a

where dv/dt=a and where speed is unchanging so gamma should be constant

Second part of the problem:

F=m*gamma*acceleration=qvB

At this point I'm entirely unsure of how to proceed. I remember that in classical physics a=v²/r, but I don't know if that applies here.

If it does indeed apply here, then the answer seems to be straightforward:

F=m*gamma*acceleration=m*gamma*v²/r=qvB

r=m*v*gamma/(qb)=p/(qB)

Thus, as the speed increases in a cyclotron, the radius should increase as well

 

[apelling]

I would agree with what you have said. However you are perhaps expected to say something more about the cyclotron.

Even at non-relativistic speeds r increases with v. However r is directly proportional to v and since time for 1 rev=2∏r/v the frequency of revolution is a constant which makes accelerating particles easy while they are only traveling at about 0.1c or less. (apply an alternating electric field of the expected frequency)

However once relativistic speeds are approached r increases faster than v and the frequency of rotation reduces giving synchronization problems. Thus cyclotrons are limited in how much energy they can give particles especially light ones that go relativistic easily such as the electron. (There have been attempts to overcome this problem such as the synchrocyclotron)

 

[daselocution]

Thank you very much--I went to office hours today to ask my teacher and he echoed more or less what you said about the speed of the cyclotron