3.5 Drift equations 813.5.3 μconservation
We now imagine being in a frame moving with the velocityv⊥gc;in this frame the only
perpendicular velocity is the cyclotron velocity (Larmor motion).Sincev⊥gcis orthogonal
toB, the parallel equation of motion is not affected by this change of frame and using
Eqs.(3.70) and (3.84) can be written as
mdv‖
dt=qE‖−mv^2 L 0
2 B∂B
∂s(3.92)
where as before,sis the distance along the magnetic field. Multiplication byv‖gives an
energy relation
d
dt(
mv^2 ‖
2)
=qE‖v‖−mvL^20
2 Bv‖∂B
∂s. (3.93)
The perpendicular force defined in Eq.(3.73) does not exist in this moving framebecause it
has been ‘transformed away’ by the change of frames. Also, recall that it was assumed that
the characteristic scale lengths ofEandBare large compared to the gyro radius (Larmor
radius). However, if the magnetic field has anabsolute time derivative, Faraday’s law states
that there must be an inductive electric field, i.e., an electric field for which
∮
E·dl= 0.
This is distinct from the static electric field that has been previously assumed and so its
consequences must be explicitly taken into account.
To understand the effect of an inductive electric field, consider a specific particle, and
dot the Lorentz equation withvto obtain
d
dt(
mv^2 ‖
2+
mvLO^2
2)
=qv‖E‖+qv⊥·E⊥ (3.94)wherev⊥is the vector Larmor orbit velocity. Subtracting Eq.(3.93) from (3.94) gives
d
dt(
mvL^20
2)
=qv⊥·E⊥+
mvL^20
2 Bv‖∂B
∂s. (3.95)
Integration of Faraday’s law over the cross-section of the Larmor orbit gives
∫
ds·∇×E=−∫
ds·∂B
∂t(3.96)
or
∮
dl·E=−πrL^2
∂B
∂t(3.97)
where it has been assumed that the magnetic field is changing sufficientlyslowly for the
orbit radius to be approximately constant during each orbit.
Equation (3.95) involves the local electric fieldE⊥but Eq.(3.97) only gives the line
integral of the electric field. This line integral can still be used if Eq.(3.95) is averaged over
a cyclotron period. The critical term is the time average over the Larmororbit ofqv⊥·E⊥