The magnetic force for charge particle (with charge q) moving with velocity

The magnetic force is perpendicular to both the velocity and magnetic field ,

The power done by the magnetic force
So the magnetic force will not change the kinetic energy of the charge particle,

It only change the direction of the velocity for the charge particle.

If the velocity is in the same direction of the magnetic field,
then there is no magnetic force acting on the charge particle.

For charge particle moving in a uniform magnetic field ,
The velocity can be represented as ,
is a component of velocity
along B, which will not change.

The particle advances along while it moves in a circle in the plane formed by and.

The resulting trajectory forms a spiral with its axis along .

So we have momentum
And the angular velocity of the circular motion
Which is independent of the velocity of the charge particle!( is the cyclotron frequency)

The constancy of the cyclotron frequency led to a device called cyclotron.
This java applet let you play with cyclotron.

This is a top view of the region of a cyclotron in which the particle circulate.

is a picture of a real cyclotron.

The two hollow D-shaped objects (open on their straight edges) are made of copper sheet.
These dees, as they are called, form part of an electrical oscillator, which establishes an alternating potential difference across the gap between them.

The dees are immersed in a magnetic field whose direction is into the plane of the screen,

Suppose that a proton, starts from the blue dot near the center of the cyclotron, initially moves toward a negatively dee.
It will accelerate toward this dee and will enter it.

Once inside, it is "screened" from electric fields by the copper walls of the dee.
The magnetic field is not screen by the (nonmagnetic) copper dee, so the proton moves in the circle path.

Assume that at the instant the proton emerges into the center gap (again) from first dee,
The accelerating potential difference has changed sign.
Thus the proton again faces a negatively charged dee and is again accelerated.

This process continues, the circulating proton always being in step with the oscillations of the dee potential, until the proton spiral out of the edge of the dee system.
The frequency of the electric oscillator must match the cyclotron frequency.

If the gap between the dees is very small , the two frequency is the same.
What if the gap is not small? You will need to adjust the frequency of the electric oscillator.

Enter the value of the frequency ratio (oscillator frequency/cyclotron frequency) into the textfield.

Click the red dot near the electric oscillator and drag it up/down to change the voltage of the oscillator.
Click Start to start the animation,
Click right mouse button to pause, click right mouse button again to resume.

The velocity of the charge particle is represented by yellow line.
The force acting on the particle is represented by red line.
Vy-Vx (velocity) of the charge particle is also shown to the right.
Click Clear to erase the trace of the trajectory.
Click Reset for default values.

For an electron : the charge , mass ,
If the energy of the electron is 1eV (accelerated under 1V biased voltage) and the magnetic field is 1T(Tesla),
The cyclotron frequency (It is so big!)

The velocity of the electron (it is moving so fast!)

If the velocity is perpendicular the magnetic field, the radius of the circular motion
( it is so small!)
If the velocity has a small component along the magnetic (say 1%),
The electron will move along the magnetic field line with velocity
Image how the electron moves under the above conditions!