A magnetic mirror is a magnetic field configuration that can confine charged particles within a certain region of space. It consists of a magnetic field that has a varying strength along the direction of particle motion. The magnetic mirror effect arises from the interplay between the magnetic field and the particle's gyromotion.

In a magnetic mirror configuration, the magnetic field strength increases as particles move away from the center of the magnetic field region. This variation in the magnetic field strength creates a potential well that can trap and reflect charged particles.

When a charged particle moves along magnetic field lines, it experiences the Lorentz force, which causes it to undergo circular or helical gyromotion around the magnetic field lines. In a uniform magnetic field, the gyromotion of the particle is unrestricted. However, in a magnetic mirror configuration, the increasing magnetic field strength towards the mirror region leads to a stronger force that opposes the particle's motion.

As a result, particles with insufficient energy or pitch angle (angle between the particle's velocity vector and the magnetic field vector) cannot overcome the increasing magnetic field and are reflected back towards the region of weaker magnetic field strength. This reflection effectively confines the particles within the magnetic mirror region.

The region of confinement in a magnetic mirror configuration is often referred to as the "mirror cell." The strength and shape of the magnetic field determine the confinement properties and the dimensions of the mirror cell.

Magnetic mirrors have been used in various plasma physics experiments and magnetic confinement devices, such as mirror machines and tandem mirrors, to confine and study plasma particles. The mirror effect allows for enhanced particle confinement and can be utilized in the design of magnetic fusion devices for controlling plasma stability and confinement.

Consider magnetic field directed in a direction and its magnitude varies in z direction as shown in figure.

Let the field be axis symmetric with B_𝜃 = 0 and ∂B_𝜃/ ∂𝜃. Since lines of force converge and diverse, there is necessarily a component Br. We wish to show that this gives rise to a force which can trap a particle in a magnetic field.

Now, homogeneous Maxwell's equation is

If ∂Bz/∂z is given at r=0 and doesn't vary much with r, we have

The variation of lBl with r causes grad B drift of guiding centre about the axis of symmetry but there is radial grad B drift because ∂B/∂𝜃 = 0. Now Lorentz force is

The component of Lorentz force are

The term a and b gives Larmor gyration and c vanishes at the axis, when it doesn't vanish this azimuthal forces causes a drift in the radial direction. Now, z component is

using equation 2 in equation 3 we get

Let us consider a particle whose guiding centre lies on the z axis. V_𝜃 is the direction ie clockwise and perpendicular to z direction. The particle gyrates such that it opposes magnetic field causing gyration. hence a positive charge gyrates opposite to v⟂ direction

Also r = rL such that

The defining of 𝜇 is same as usual defination for the magnetic moment. In case of single charged ion. I is generated by (e) moving around wc/2π times a second

I = e. wc/ 2

Area = π r²

To see variation of 𝜇, consider equation of motion along B from 6

But particle energy should be conserved i.e dE/dt =0

When a particle transitions from a weak field to a strong field, it experiences an increase in B and v⟂ must increase in order to keep 𝜇 constant. Since the total energy must be constant. Vll must inevitably decline. If B is high enough Vll drops to zero and the particle is reflected back to the weak field zone if B is high enough. This serves as the foundation for magnetic mirror particle entrapment.

This note is a part of the Physics Repository.

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