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(NB: Links to sub-sections will be added in due course.) My research is about fusion energy and plasma physics. If you're already lost, then you want the 'Overview for a General Audience'. Otherwise, you want the 'overview with technical detail'.
Almost everything on earth - solid, liquid and gas - is made of atoms. Atoms are tiny. A single grain of sand is made of trillions of atoms.
Each atom has an even smaller centre, called the nucleus (plural: nuclei), which has a positive electric charge. The nucleus is surrounded by electrons, which have a negative electric charge. Different "stuff" (e.g. oxygen, iron, etc.), known as chemical elements, is made of atoms with different sized nuclei.
Energy is required for nature to make a nucleus. It turns out that iron requires the least energy relative to the size of its nucleus. For atoms bigger than iron, such as uranium, splitting them into smaller atoms (closer to the size of iron) releases energy in that the fragments blast apart from each other (i.e. with increased kinetic energy). This is nuclear fission and is the process that powers the nuclear reactors of today. Similarly, for atoms smaller than iron, such as hydrogen and helium, combining ("fusing") them together to make bigger atoms (closer to the size of iron) also releases energy, again in that the resulting atom (or atoms and possibly some debris) have more kinetic energy than the original atoms. This is nuclear fusion and is the process that powers the sun.
Fusion energy is a method of generating electricity for the national grid to power our homes, businesses, public services, etc. using the energy released by the nuclear fusion process. As with nuclear energy (fission) and coal-fired power plants, the fuel undergoes a reaction (nuclear fusion, nuclear fission or, for coal, chemical combustion). The energy released by the reaction is then used to heat water to create steam, which then drives a turbine to generate electricity.
Unfortunately, it is very difficult to get a significant amount of fusion to happen. Recall, we are trying to force atomic nuclei with a positive electric charge close enough together that they fuse. Well, when it comes to electric charges, opposites attract but like charges repel each other. So, as the nuclei approach each other, the fact they both have a positive charge means they force each other away from each other, thus making it very difficult to get them close enough to fuse. The only way to overcome this electric repulsion is to make the nuclei very fast - that is, very hot - so that they smash into each other before the electric repulsion has a chance to act.
It turns out that the temperature required is a whopping 100 million degrees C! That's hotter than the centre of the sun! Incidentally, the sun is able to sustain itself at a lower temperature because it is so big. This works for two reasons: first, because it's so big, it has billions of years worth of fuel and it has its own gravity so it's not going anywhere and it's got plenty of time to do fusion reactions at a slower rate than fusion reactors on earth; secondly, the sun may have fewer fusion reactions in a given volume of space and a given amount of time compared to a fusion reactor on earth but the sun is so big that the total number of fusion reactions in a given amount of time is still a lot.
Anyway, 100 million degrees C is obviously very hot. If you tried to keep the fuel at this temperature in a solid container (glass bottle, metal box, etc) the container's walls would be vaporised instantly. Despite this seemingly impossible challenge, the fusion part of this concept (as opposed to the electricity generation part) has been successfully demonstrated in many experiments. So, that leads onto the next question...
There are two main options: (a) magnets; or (b) you try to start the reaction so quickly that it happens before the fuel has time to blow itself apart.
OK, backup a sec. Let's remember what we're trying to do here. We want to harness the energy released by fusion reactions that happen at 100 million degrees C to drive a power station. So, to get started, we need to heat the fuel. This requires and external input of energy. This will only lead to a net gain in energy if we can get the fusion reactions going such that they keep the fuel hot enough to continue fusing and we then remove (or at least reduce) the external heating - otherwise we just end up pumping in more heat than we're getting back out. It's like lighting a fire - at first, you have to provide external heat from matches or similar but then the fuel starts burning enough that it keeps itself alight without the match. This critical point, for both fire and fusion, is called ignition. To clarify an earlier point, laboratory fusion has been achieved with external heating but we have not yet achieved ignition. We have created some smouldering embers but have yet to successfully light the fire!
Hang on! What has all this got to do with containing hot fuel? Don't worry - I'm getting to that.
So, to achieve ignition, we need three things; we need the fuel to be hot enough that it fuses, dense enough that the fusion reactions can keep the fuel hot enough to continue fusing (imagine a properly built stack of fire wood compared to a load of kindling spread out on the floor) and we need to confine the fuel at those conditions for long enough for the fusion fire to really get going (imaging patiently holding multiple matches compared to giving up too soon). The temperature requirement is set (at 100 million degrees C). But we should be able to achieve ignition at very high fuel density (about 1000 times the density of a usual solid material) for a very short confinement time (some tiny fraction of a second) or at lower fuel density (less than that of air) with a longer confinement time (minutes to hours or maybe even longer). This trade-off between density and confinement time leads to the different approaches in my short answer above. With magnets, we aim to confine the fuel for a long time at low density and the other approach aims to achieve very high density but only for a short time. These two approaches are described in more detail below.
When you heat a solid enough it turns into a liquid. And when you heat a liquid enough you get a gas. When you heat a gas enough, the electrons (negatively charged outer parts of the atoms) get stripped away from their nuclei (positively charged cores of the atoms). Now we have something similar to a gas but with positively charged nuclei and negatively charged electrons all sloshing around together (instead of just neutral atoms as in a gas). This state of matter is known as a plasma.
So the bad news is we need 100 million degrees C. But the good news us that plasma exhibits some behaviour that we can use; namely, plasma responds to electric and magnetic fields.
The plasma response to electric fields is unimportant for the present discussion. But their response to magnetic fields is very useful.
Imagine a usual horseshoe magnet. Then imagine some lines coming out of one end and curving round back onto the other end. These lines represent the path that little bits of magnetic material would follow as they are attracted towards the magnet; they are the lines along which magnetic forces act and are known as magnetic field lines.
Now imagine we have a plasma near the magnet...home