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:
- 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.