Fundamentals of Plasma Physics

82 Chapter 3. Motion of a single plasma particle

(which gives the rate at which the perpendicular electric field does workon the particle),

<qv⊥·E⊥>orbit =

ωc

2 π

∫

dtqv⊥·E⊥

= −

qωc

2 π

∮

dl·E⊥

=

qωc

2

r^2 L

∂B

∂t

.

(3.98)

The substitutionv⊥dt=−dlhas been used and the minus sign is invoked because particle
motion is diamagnetic (e.g., ions have a left-handed orbit, whereas in Stokes’ theoremdl
is assumed to be a right handed line element). Averaging of Eq. (3.95) gives

〈

d

dt

(

mvL^20

2

)〉

=

mvL^20

2 B

∂B

∂t

+

mvL^20

2 B

v‖

∂B

∂s

=

mvL^20

2 B

dB

dt

(3.99)

wheredB/dt=∂B/∂t+v‖∂B/∂sis thetotal derivativeof the average magnetic field
experienced by the particle over a Larmor orbit. Defining the Larmor orbit kinetic energy
asW⊥=mv^2 L 0 / 2 ,Eq.(3.99) can be rewritten as

1

W⊥

dW⊥

dt

=

1

B

dB

dt

(3.100)

which has the solution
W⊥
B

≡μ=const. (3.101)

for magnetic fields that can be changing inbothtime and space. In plasma physics terminol-
ogy,μis called the ‘first adiabatic’ invariant, and the invariance ofμshows that the ratio of
the kinetic energy of gyromotion to gyrofrequency is a conserved quantity. The derivation
assumed the magnetic field changed sufficiently slowly for the instantaneous field strength
B(t)during an orbit to differ only slightly from the orbit-averaged field strength〈B〉the
orbit, i.e.,|B(t)−〈B〉|<<〈B〉.

3.5.4 Relation ofμconservation to other conservation relations

μconservation is both of fundamental importance and a prime example of the adiabatic
invariance of the action integral associated with a periodic motion. Theμconservation
concept unites together several seemingly disparate points of view:

1. Conservation of magnetic moment of a particle- According to electromagnetic theory
   the magnetic momentmof a current loop ism=IAwhereI is the current carried in
   the loop andAis the area enclosed by the loop. Because a gyrating particle traces out
   a circular orbit at the frequencyωc/ 2 πand has a chargeq, it effectively constitutes
   a current loop havingI =qωc/ 2 π and cross-sectional areaA=πr^2 L.Thus, the
   magnetic moment of the gyrating particle is

m=

(qω

c

2 π

)

πrL^2 =

mvL^20

2 B

=μ (3.102)

and so the magnetic momentmis an adiabatically conserved quantity.