Magnetic Bottles.

Okay, firstly, sorry for not having posted in the last month or so. I'm just sending this apology into the vast expenses of the world wide web as i don't really know who follows my blog and who doesn't. Y'all need to comment more and give me feedback! And I promised myself last summer that the only articles on this blog will be ones on science or at the very least related to the pursuit science and technology. And lately, I have been a bit involved with my course work and the like. So, I was looking for the right topic to blog on. And whatdoyaknow, I stopped looking for the right topic and started making a topic the right one! So, here is something interesting I learnt a couple of weeks back.

As part of a course on Radiative Processes in Astrophysics here at IIT Madras, I was studying the behavior of particles in uniform Electric and Magnetic fields. Something most of us studied during our 12th and maybe in a bit more detail in college, if you pursued science i.e. Anyway, we've all studied how particles move in straight lines or conic sections in the presence of Electric fields and how particles move in helixes in the presence of constant Magnetic fields. Here is where the catch comes.

There is such a thing defined as the Adiabatic Invariant for particle motion and using the adiabatic invariant, we can study how the motion of a particle is in the presence of a slowly varying magnetic field! Look at the picture for a better idea of what i'm referring to -

So, as you can see, the strength of the magnetic field is slowly varying along the horizontal axis! Now, if we look at the motion of a particle in such a field - the motion will be a helix but one whose radius keeps getting smaller and smaller, as is obvious from the picture!

We can derive the change in longitudinal momentum of the particle, which as it turns out is dependent on the square root of the strength of the magnetic field. And we already know that the magnetic field does not do any work on the particles i.e it does not impart energy to the particles i.e the sum of the squares of the longitudinal and transverse momenta should be constant! 

$$P_0^2 = P_t^2 + P_l^2  = constant$$
$$P_t^2 \alpha \sqrt{H}$$

Using these two results, we can infer that, for the energy of the particle to remain constant while the particle is moving to regions of stronger and stronger magnetic fields, the longitudinal momentum of the particle will increase and the transverse momentum will decrease.

$$P_l^2 \geq 0$$

And since the square of the transverse momentum has to be greater than or equal zero, we can infer that there exists an invisible wall where the magnetic field is so strong that all of the transverse momentum of the particle is lost before it reaches this wall and the particle will then start moving backwards into regions of lower magnetic field. Which is why it is an invisible wall, a barrier the magnetic field is creating which particles cannot cross! This is what is referred to as a magnetic bottle, which can and is used to confine plasma! This can also be used to filter particles based on their energy as only particles with energy above a threshold will be able to cross the barrier! This threshold can be varied by varying the strength of the magnetic field. 

Further, the concept of magnetic confinement is the first step towards building nuclear fusion reactors. Specifically, the Tokamak Fusion Test Reactor, uses the theory behind magnetic confinement to be able to hold plasma - plasma so hot that no material container will be able to withstand the temperature! 

References : 
(1) Motion in constant and uniform B fields, Sect. 21, Ch. 3, Landau & Lifschitz Book 2 - Classical Theory of Fields
(2) Adiabatic Invariance of flux through the orbit of a particle, Sect 5, Ch. 12, J.D. Jackson - Classical Electrodynamics.
(3) Motion in a slowly varying magnetic field, Sect. 8.4, Ch. 3, T. Padmanabhan - Theoretical Astrophysics - Vol 1.

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