A charged particle would continue to move with a constant velocity in a region wherein,
Verify that the cyclotron frequency $\omega=e B / m$ has the correct dimensions of $[T]^{-1}$.
For a charge particle moving perpendicular to the magnetic field, the magnetic Lorentz forces provides necessary centripetal force for revolution.
$$\frac{m v^2}{R}=q v B$$
On simplifying the terms, we have
$\therefore \quad\frac{q B}{m}=\frac{v}{R}=\omega$
Finding the dimensional formula of angular frequency
$$\therefore\quad[\omega]=\left[\frac{q B}{m}\right]=\left[\frac{v}{R}\right]=[T]^{-1}$$
Show that a force that does no work must be a velocity dependent force.
Let no work is done by a force, so we have
$$\begin{aligned} &\begin{aligned} \mathrm{dW} & =\mathbf{F} \cdot \mathrm{dl}=0 \\ \Rightarrow\quad\text { F. } \mathbf{v} d t & =0 \quad\text { (Since, } \mathbf{d l}=\mathbf{v} d t \text { and } \mathrm{dt} \neq 0 \text { ) }\\ \Rightarrow\quad\text { F. } \mathbf{v} & =0 \end{aligned}\\ \end{aligned}$$
Thus, $F$ must be velocity dependent which implies that angle between $F$ and $v$ is $90^{\circ}$. If $v$ changes (direction), then (directions) F should also change so that above condition is satisfied,
The magnetic force depends on $v$ which depends on the inertial frame of reference. Does then the magnetic force differ from inertial frame to frame? Is it reasonable that the net acceleration has a different value in different frames of reference?
Yes, the magnetic force differ from inertial frame to frame. The magnetic force is frame dependent.
The net acceleration which comes into existing out of this is however, frame independent (non -relativistic physics) for inertial frames.
Describe the motion of a charged particle in a cyclotron if the frequency of the radio frequency (rf) field were doubled.
Here, the condition of magnetic resonance is violated.
When the frequency of the radio frequency (rf) field were doubled, the time period of the radio frequency ( rf ) field were halved. Therefore, the duration in which particle completes half revolution inside the dees, radio frequency completes the cycle.
Hence, particle will accelerate and decelerate alternatively. So, the radius of path in the dees will remain same.