1. LECTURE I: SINGLE-PARTICLE MOTIONS IN ELECTRIC AND MAGNETIC FIELDS
Univerzitet Crne Gore
LECTURE I: SINGLE-PARTICLE MOTIONS IN ELECTRIC AND MAGNETIC FIELDS
4/22/2019
Dr Slavoljub Mijović
University of Montenegro
Faculty of Natural Sciences and
Mathematics, Podgorica, MONTENEGRO
4/22/2019

2. Why Study Particle Motion?
Presence of free mobile charged particles differentiates plasma from usual gas
Motion of charged particles gives plasma its special characteristics:
Conductivity
Interaction with E-M waves
Usefulness in Applications:
Light sources
Plasma processing
Etc.
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3. Lorentz Force For single particle, with no radiation losses and
Univerzitet Crne Gore
Lorentz Force
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For single particle, with no radiation losses and
no space charge effects:
There are many possibilities, depending on space and time-dependence of electric and magnetic fields!!

4. Single Charged Particle Motion in a Static Electric Field
In General: x y E
Consider Static E only: +
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5. Example 1.
Describe the motion of a charged particle in a permanent electric field.
Solution: The equation of motion of a charged particle in such field can be described as
(1)
The change of particle's kinetic energy in this field is equal to the work of the electric field
(2)
Here is the integral taken along the trajectory of the paricle. If the electric field has the potential not depending of time (this case!) than the right hand of the eq. 2 will be

6. Example 1-cont’d
For a particle which was at rest in the initial state, the formula (2) becomes:
Thus, during a motion of a charged particle in a static potential field, the kinetic energy of a charged particle is determined with the potential difference, prevailed by the particle.

7. Motion in a Static Magnetic Field
No Motion  No Force
Motion parallel to B
(i.e. in z direction):
No magnetic force – no influence
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8. Charged particle in an uniform magnetic field-cont’d
Magnetic field into board
No Motion  No Motion +
Note speed never changes + +
but direction does + + v FB +
Force is always  to v + v
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9. Example 2. Motion  to B Solution: The equation of motion
Verify that a charged particle which entered into a region with a homogenious magnetic field and has the velocity
 to B, makes a circular motion.
Solution: The equation of motion
Or in scalar form
These eq’s are in a form as for a harmonic oscillator at the cyclotron-
frequency The solution of these eqs are then
where ± taking into account charge sign.
The phase  can be chosen so than
Then

10. Example 2. Motion  to B-con’d
Integrating one more, we got
substitute with
and taking real part from the above eq-s
Larmor’s radius
The equation describes a circular orbit around the field with gyroradius, and gyrofrequency, The orbit‘s center (x0,y0) is called the guiding center. The gyration represents a microcurrent, which creates a field opposite to the background one. This behaviour is called diamagnetic effect.

11. Example 3. Motion under an angle  to B-con’d
A charged particle is in a homogenious magnetic field and has the velocity . Show that the modul and projections along and normal to the field of its velocity stay constant.
Solution: The equation of motion for a charged particle in a magnetic field is :
(1)
For the component of the velocity along the magnetic field the derivative
and thus
Put now in the equation (1)
and make scalar multiplication of the
both sides of the equation with
The result is
thus
and
The modul of the velocity of the charged particle is

12. Example 3. Motion under an angle  to B-con’d
Helicoidal ion orbit in a uniform magnetic field
If one includes a constant speed parallel to the field, the particle motion is three-dimensional and looks like a helix. The pitch angle of the helix or particle velocity with respect to the field depends on the ratio of perpendicular to parallel velocity components.

15. Charged particle in an inhomogeneous magnetic field
Magnetic field into board
No Motion  No Motion
Gradient
Note speed never changes
but direction does
Divergence

16. Example 4.
Analyse a motion of a charged particle in a magnetic field, which is continually changing along the trajectory of the particle. The curves of the magnetic lines are negligible. The radius of the curve of the particle trajectory is small in compare to the characteristic length, i.e., to the distance where the change of the magnitude of the magnetic field is significant,
Solution:
Note speed never changes
During a motion of a particle in a magnetic field, which increases, the radius and the pace of the spiral are decreasing. This is due to a force occurs in inhomogeneous magnetic field which slow-down the motion along the lines. This force is a component of Lorenz’s force, whose direction is normal at Larmor’s circle and influences the particle in a such manner to reject it from the region where the field is stronger to the region where the magnetic field is weaker.
but direction does z

17. Example 5.
Find an expression for the force analysed in the previous example
Solution: Due to for an axially symmetrical field is as
(1)
(2)
Integrating (1) along r, follows
follows
is not depend of r, than
and for If
Then, the equation of motion along the z, is
The right hand of the eq. can be transferred as z
the magnetic momentum

18. Example 6.
Prove that during a motion of a charged particle in the above described inhomogeneous magnetic field, its magnetic momentum stays constant.
Solution: A charged particle which moves in a loop  to B makes a current. The magnetic momentum of this current is:
Here S-the surface of the circle with Larmor’s radius.
(1)
The equation of motion in the field direction is
(2)
Multiplying the eq. (1) with
(the conservation energy law) and
from (2)

19. Example 7.- Magnetic mirror
Find a condition for reflecting a charged particle in the above described magnetic field from the region where the magnetic field is higher to the region where it is weaker.
Solution: From the condition determined in the previous example
where
follows
an angle of the velocity to B. It is clear that with increasing B increases the angle because of
The above condition is achieved only if the magnitude of the field is not more than
determined by the given direction of motion. This critical magnitude of B is determined as

20. AN INHOMOGENEITY OF A MAGNETIC FIELD  TO THE MAGNETIC LINESB - + x x v
Magnetic drifts I
Inhomogeneity will lead to a drift, perpendicular to both, the field and its gradient + B

21. MOTION IN COMBINED ELECTROSTATIC AND MAGNETOSTATIC FIELDS
E and B in same direction (e.g. y):
If no initial motion in x or z directions:
B has no effect
Acceleration in y direction per previous electrostatic solution

22. Motion in Combined Electrostatic and Magnetostatic Fields-cont’d
E and B in same direction (e.g. y) but with initial motion in x or z direction:
Independent solutions for y-direction and for x-z plane
Acceleration by E in y-directions, identical to previous solution
Circular motion in x-z plane, as in previous magnetostatic case
Superimpose – spiral motion in y direction, with increasing pitch

23. Motion in Crossed Electrostatic and Magnetostatic Fields
Initial motion in x-y plane
Taking a new referent system E y B x z +
If we choose u to be a constant -

24. General Motion in Combined, Uniform, Static E, B Fields
Resolve E into two components, Eand E (with respect to B direction).
In || direction, no influence of B – usual electrostatic acceleration
In  direction, crossed-field motion
Drift velocity, independent of charge,
Superimpose  and || components

25. General Motion in Combined Permanent Force  to B Fields
F=mg

26. Motion a charged particle in an increasing in time B Fieldv + +

27. Application

28. Application-con’d

29. Application-con’d

30. Application-con’d

31. Thank you
for your attention !
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