Newton's 2nd Law and Orbital Motion

[runevxii]

Here's the problem...unfortunately I don't remember much about orbital motion. I'm a bit stuck on where to begin. If somebody could give me a little advice on how to tackle this problem I would appreciate it.

Recall that the magnetic force on a charge q moving with velocity v in a magnetic field B is equal to qvXB. If a charged particle moves in a circular orbit with a fixed speed v in the presence of a constant magnetic field, use the relativistic form of Newton's 2nd law to show that the frequency of its orbital motion is

f=((qB)/(2pim))(1-(v^2/c^2))^(1/2)

 

[Tide]

If the speed is constant then

$$\frac {d \vec v}{dt} = \vec v \times \vec \Omega$$

where $\vec \Omega = q \vec B / m_0$. There are a number of ways to proceed from here but it should be apparent that the same analysis you did in the classical case will work except that B is replaced by $B / \gamma$ from which your result follows.

 

[runevxii]

still stuck

Still stuck since I don't really remember the classical case.

 

[Tide]

In that case, consider ...

$$\frac {d v_x} {dt} = \Omega v_y$$

and

$$\frac {d v_y} {dt } = - \Omega v_x$$

Differentiate, say, the first and substitute the second into the first:

$$\frac {d^2 v_x} {dt^2} = - \Omega^2 v_x$$

from which it should be evident that the motion is sinusoidal with frequency $\Omega$.