276 Chapter 9. MHD equilibriathe existence in the plasma surface of an induced azimuthal current which creates a mag-
netic field in the plasma interior that exactly cancels the coil-producedtransientBzfield.
The radial confining force results from a radially inward forceJ×B=Jθˆθ×Bzzˆwhere
Bzis the magnetic field associated with the current in the coil andJθis the plasma surface
current. An alternative but equivalent point of view is to invoke the concept that opposite
currents repel and argue that the azimuthal current in the coil repels theoppositely directed
azimuthal current in the plasma surface, thereby pushing the plasma inwards and so bal-
ancing the outwards force due to plasma pressure. Theθ-pinch configuration is necessarily
transient, because the induced surface current cannot be sustained in steady-state.plasma
coil current
plasma surface current
BzBz
Bz
Bz
Bz
BzBz
Bz
Bz Bz
Figure 9.8: Theta pinch configuration. Rapidly changing azimuthal coil current induces
equal and opposite azimuthal current in plasma surface. Pressure ofBzbetween these two
currents pushes in on plasma and provides confinement.
9.8.2 Impossibility of self-confinement of current-carrying plasma inthree
dimensions: the virial theoremThe Bennett analysis showed that axial currentsflowing in an infinitely long cylindrical
plasma generate a pinch force which confines a finite pressure plasma. The inward pinch
force balances the outward force associated with the pressure gradient. Thequestion now is
whether this two-dimensional result can be extended to three dimensions;i.e., is it possible
to have a finite-pressure three-dimensional plasma, as shown in Fig.9.9, which is confined
entirely by currents circulating within the plasma? To be more specific, is it possible to
have a finite-radius plasma sphere surrounded by vacuum where the confinement of the
finite plasma pressure is entirely provided by the magnetic force of currents circulating in
the plasma;i.e., can the plasma hold itself together by its own ‘bootstraps’? The answer,