Fundamentals of Plasma Physics

268 Chapter 9. MHD equilibria

9.3 Force-free fields

Although the vacuum field is the lowest energy configuration satisfying prescribed bound-
ary conditions, non-vacuum configurations do not always ‘decay’ to this lowest energy
state. This is because there also exists a family of higher energy configurations to which
the system may decay;these are the so-called force-free states. The current is not zero in
a force-free state but the magnetic force is zero because the current density is everywhere
parallel to the magnetic field. ThusJ×Bvanishes even though bothJandBare finite.
If a plasma initially in a non-force-free state evolves toward a force-free state it becomes
‘stuck’ in the force-free state because there are no forces acting to change the equilibrium.
The magnetic energy of a force-free field is not the absolute minimum energy forthe spec-
ified boundary conditions, but it is a local minimum in configuration space as sketched in
Fig.9.2. This hierarchy of states is somewhat analogous to the states of a quantum system

• the vacuum field is the analog of the ground state and the force-free states are the analogs
  of higher energy quantum states.

magnetic field

due to first w ire

firstwire

second w ire

force

force

I

force due to magnetic field

of second w ire acting on

current offirstw ire

force due to magnetic field

of first wire acting on

current of second w ire

B

I

Figure 9.3: Parallel currents attract each other

9.4 Magnetic pressure and tension

Much useful insight can be obtained by considering the force between two parallelcurrent-
carrying wires as shown in Fig.9.3. Calculation of the fieldBobserved at one wire due to
current in the other shows that theJ×Bforce is such as to push the wires together;i.e.,