Fundamentals of Plasma Physics

122 Chapter 3. Motion of a single plasma particle

center atz= +L/ 2 .The radiiaandbare chosen so as to intercept the field lines that
intercept the current loop (see figure). Assume the figure is somewhat exaggerated so
thatBzis approximately uniform over each of the three circular surfaces andso one
may ignore the radial dependence ofBzand therefore expressBz=Bz(z).
(a) Note thatr= (a+b)/ 2 .What is the force (magnitude and direction) on the current
loop expressed in terms ofI, Bz(0),a,bandLonly? [Hint- use the field line slope to give
a relationship betweenBrandBzat the loop radius.]
(b) For each of the circles and the current loop, express the magneticflux enclosed in
terms ofBzat the respective entity and the radius of the entity. What is the relationship
between theBz’s at these three entities?
(c) By combining the results of parts (a) and (b) above and taking the limitL→ 0 ,
show that the force on the loop can be expressed in terms of a derivative ofBz.

circle

(edge view)

circle

zaxis

yaxis

currentloop

B

r B

a

b

L/ 2 L/ 2

magnetic

field

line

Figure 3.18: Non-uniform magnetic field acting on a current loop.