Fundamentals of Plasma Physics

9.4 Magnetic pressure and tension 269

parallel currents attract each other. Conversely, anti-parallel currents repel each other.

A bundle of parallel wires as shown in Fig. 9.4 will therefore mutuallyattract each

other resulting in an effective net force which acts to reduce the diameter of the bundle.

The bundle could be replaced by a distributed current such as the current carried by a

finite-radius, cylindrical plasma. This contracting, inward directedforce is called the pinch

force or the pinch effect.

The pinch force may be imagined as being due to a ‘tension’ in the azimuthalmagnetic

field which wraps around the distributed current. Extending this metaphor, the azimuthal

magnetic field is visualized as acting like an ‘elastic band’ which encircles the distributed

current and squeezes or pinches the current to a smaller diameter. This concept is consistent

with the situation of two permanent magnets attracting each other when the respective north

and south poles face each other. The magnetic field lines go from the north pole ofone

magnet to the south pole of the other and so one can pretend that the attraction of the two

magnets is due to tension in the field lines spanning the gap between the two magnets.

bundle ofparallelw ires

coming outof paper

radial inw ard

‘pinch’force B

Figure 9.4: Bundle of currents attract each other, giving effective radial inward pinch force.

Now consider a current-carrying loop such as is shown in Fig.9.5. Becausethe currents

on opposite sides of the loopflow in opposing directions, there will be a repulsive force

between the current element at each point and the current element on the opposite side of

the loop. The net result is a force directed to expand the diameter of the loop. This force

is called the hoop force or hoop stress. Hoop force may also be interpreted metaphorically

by introducing the concept of magnetic pressure. As shown in Fig.9.5, the magnetic field

lines linking the current are more dense inside the loop than outside, a purelygeometrical

effect resulting from the curvature of the current path. Since magnetic field strength is

proportional to field line density, the magnetic field is stronger on the insideof the loop

than on the outside, i.e.,B^2 is stronger on the inside than on the outside. The magnetic

field in the plane of the loop is normal to the plane and so one can explain the hoop force