Why a magnet can move the particles in a cathode rays tube (CRT)? I have been looking experiments on cathode rays tubes, and I looked that a near magnet can move the particles of the ray.
Why?
How does the relationship between the magnetic field and the movement of particles work? Has it about to be with the electric charge of the particles on the CRT?
I can't find any equation or law to link it.
 A: The particles inside catode ray tubes (CRT) are electrons. So, they are obviously charged. And as any charged particles they experience the Lorentz force
$$\vec{F} = q \vec{v} \times \vec{B}$$
A: If you don't know about it yet, you may want to have a look on Lorentz's force. Basically, a charged particle moving under the influence of an electromagnetic field is exerted a force proportional to the fields, its charge and the velocity it has:
$$\vec{F}=q\vec{E}+q\vec{v}\times\vec{B}$$
We want to focus on the second term of this equation, dependant on the magnetic field $\vec{B}$. As you can see, the cross product  makes the force perpendicular both to the field and the velocity vector  of the particle. If these two vectors are parallel (which would give a null cross product), the explanation is a bit more extensive.
A: The answers given so far tell you that the experimental fact was put into an equation by Lorentz. And - by the way - the connection between the magnetic field and the resulting force (deflection of the electrons) was established. If one had been aware of the relationship between the electric field of the electrons and its magnetic field, the equation would contain a constant value.

How does the relationship between the magnetic field and the movement of particles work? Has it about to be with the electric charge of the particles on the CRT?

We should review a few experimental facts related to the deflection of electrons by an external magnetic field.

*

*Deflection is a kind of acceleration. And every acceleration is accompanied by the emission of photons from the deflected electron.


*The electron has a kinetic energy content as long as it is moving. The emitted photons repel the electron, and - viewed from the other side - the electron loses kinetic energy in the form of photons.
This point is observable, and indeed an electron in a magnetic field exhausts its kinetic energy, moves along a spiral path (or a helical path if the direction of movement and the magnetic field are not at 90° to each other) and comes to a standstill after a while.
And - again by the way - the recoil by the photons happens in portions and the spiral path is in detail a spiral of mandarin tangerine slices.


*In the electrostatic case, a magnetic field never interacts with an electric field. It needs a relative movement or a change in the value of the magnetic field to obtain the deflection.
The last point makes me think a little bit about the edge of my plate (über den Tellerrand schauen, a German saying and I’m not sure, it is translated right.)
Beside the electric field the electron has its own magnetic field. Which is a dipole, means, it has an orientation in space. Well, randomly distributed if the electrons are not polarized. The external magnetic field will do the alignment.
What if this alignment is sufficient for the emission of photons? The recoil of the photon re-aligns the magnetic dipole of the electron and the electron is deflected laterally - similar to the gyroscopic effect? This is the only explanation that exists. Search the web.
I’ve answered your question because you asked how it happens and this characterizes the question as unique and deserves the attempt of an answer.
