Cosmic Rays passing through a magnetic field

[Gogsey]

Assume that it hits the Galactic magnetic field at an angle of 45o, i.e. such that
the components of its momentum parallel and perpendicular to the magnetic field
are equal. What is its cyclotron radius? How long does it take to execute one
cyclotron orbit?

W ealso, know velocity From part a0. which I didn't post), magnetic field strength, and the Lorentz factor, since were accounting for relativistic objects?

Mainly I'm not sure about the whole 45 degrre angle and equal components of momentum and how they relate to cyclotron motion.

 

[Brian_C]

You need to set up your equations of motion. Start with the Lorentz force on the electron, making sure to use the relativistic momentum.

 

[Gogsey]

Ok, so we have:

P(x)=gamma*mV(x)cos 45
P(y)=gamma*mV(y)sin 45

So this means the velocity in the y and x directions are not equal, but the momentums are.

Therefore we are left with V(x)cos 45 = V(y)sin 45.

Now which velocity are we interested in? Is it the combined velocity? Do we have to write one in terms of the other, and then solve for this value, then use this to find the other and find the total velocity? How do we do this using only one equation?

 

[Brian_C]

You did not write the components of momentum correctly. Write it down as a vector before trying to decompose it into components.

To calculate a trajectory, you first need to set up an equation of motion. In this case, you will need Newton's Second Law ($$\vec{F}$$ = d$$\vec{p}$$/dt). The relativistic momentum is $$\vec{p}$$ = $$\gamma$$m$$\vec{v}$$. The vector force is q$$\vec{v}$$x$$\vec{B}$$.