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 E-mail: slavom@rc.pmf.cg.ac.yu 4/22/2019

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. 4/22/2019

Lorentz Force For single particle, with no radiation losses and Univerzitet Crne Gore Lorentz Force 4/22/2019 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!!

Single Charged Particle Motion in a Static Electric Field In General: x y E Consider Static E only: + 4/22/2019

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

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.

Motion in a Static Magnetic Field No Motion  No Force Motion parallel to B (i.e. in z direction): No magnetic force – no influence 4/22/2019

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 4/22/2019

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

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.

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

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.

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

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

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

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)

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

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

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

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

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 -

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

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

Motion a charged particle in an increasing in time B Field v + +

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Thank you for your attention ! 4/22/2019