Chapter 6

# Motions of charged particles inside electromagnetic fields

### Fondamental : Electric field

Let us consider a punctual charged particle M ( ) of mass in motion in a uniform electrostatic field .

The frame of study is the one of the laboratory. It is supposed to be inertial.

The second law of motion gives the equation of the position of M :

We can see an analogy with a material point placed in a uniform gravity field :

Thus the motion of the particle will either be a straight line or a parabola.

### Fondamental : Magnetic field

Power of the magnetic force :

Let us consider a punctual charged particle M ( ) of mass in motion in a uniform magnetostatic field .

The frame of study is the one of the laboratory, it is supposed to be inertial.

The second law of motion gives the equation of the position of M :

The power of the magnetic force is zero ( ).

By using the work energy theorem :

A magnetic field does not change the norm of the velocity vector : it only changes its direction.

Circular motion :

Let us consider a punctual charged particle M ( ) of mass in motion in a uniform magnetostatic field .

The initial velocity of the particle is perpendicular to the field.

It is, for instance, along the (Ox) axis :

The second law of motion will give the radius of the circle (we suppose ) :

So :

The particle moves in a circular orbit with a constant angular velocity (called cyclotron angular velocity) :

Circular motion in a magnetic field

Helical motion :

Let us consider a punctual charged particle M ( ) of mass in motion in a uniform magnetostatic field .

The initial velocity of the particle is given. By picking the axises right, we write it :

The trajectory contained in the plane perpendicular to the (Oz) axis is a circle of radius :

which the particle follows with a constant angular velocity .

The trajectory with respect to the (Oz) axis is rectilinear and uniform.

The particle speed is : .

The trajectory is a helix with constant pitch.

This pitch (the width the particle travels after one period of the circular motion) is :

Helical motion

Some videos about motions in magnetic field (http://physics-animations.com/) :

Séparation isotopique par champ magnétique
Illustration d'une "lentille magnétique"

### Fondamental : Electric and magnetic fields

Helix with variable pitch :

Let us consider a punctual charged particle M ( ) of mass in motion in a uniform magnetostatic field and a uniform elctrostatic field .

The initial velocity of the particle is given. By picking the axises right, we write it :

With respect to the axis (Oz), the motion is accelerated :

Helix with variable pitch

Cycloid :

Let us consider a punctual charged particle M ( ) of mass in motion in a uniform magnetostatic field and a uniform elctrostatic field .

The particle is initially at the origin O. Its initial velocity is zero.

We note :

We can show that the parametric equations of the trajectory are :

Cycloid

Une animation Java sur un filtre de vitesse et la mesure du rapport e / m :

Animation java (Jean-Jacques Rousseau, Université du Mans)

Une vidéo illustrant le mouvement d'une particule dans des champs électrique et magnétique croisés (http://physics-animations.com/) :

Mouvements de particules dans des champs électrique et magnétique croisés