Velocity of a relativistic particle in a uniform magnetic field

[B Rylen]

Homework Statement

I am trying to solve for the velocity of a relativistic particle in a magnetic field using magnetic force (F = q(v x B)) and F = dp/dt (where p = ɣmv) and equating the two equations to each other. However, I stuck on how to isolate v in this setup.

Relevant Equations

F = dp/dt = d(ɣmv)/dt
F = q(v x B)
ɣ = 1/sqrt(1-(v/c)^2)

d(ɣmv)/dt = qvB
(dɣ/dt)mv + ɣm(dv/dt) = qvB
Substituting gamma in and using the chain rule, it ends up simplifying to the following:
ɣ^3*m(dv/dt) = qvB

Now, I am confused on how to solve for v.

 

[Orodruin]

(dɣ/dt)mv + ɣm(dv/dt) = qvB

Your problem is one of using scalars instead of vectors. The force is orthogonal to velocity so the speed is constant. Hence (dɣ/dt) = 0.

 

[Orodruin]

It should also be mentioned that dv/dt (where v is the speed and not velocity) is also equal to zero.

 

[vanhees71]

Another possibility is to solve the manifestly covariant equations using the proper time $\tau$ as the parameter of the trajectory/world line. This usually simplifies such calculations!

 

[TSny]

Homework Statement:: I am trying to solve for the velocity of a relativistic particle in a magnetic field using magnetic force (F = q(v x B)) and F = dp/dt (where p = ɣmv) and equating the two equations to each other.

Are you assuming that the particle is orbiting in a plane perpendicular to the field? If so, you will need to choose which parameters you want to use for expressing the speed v. Besides q, B, and m there is the radius R of the orbit, the period T of the orbit, and the frequency f (or angular frequency ω) of the orbit.