108 Chapter 3. Motion of a single plasma particle
i.e.,
dδv
dt
=
q
m[δx·∇E+δv×B]. (3.185)The difference velocity is related to the difference in positions bydδx/dt=δv.Letybe
the direction in which the electric field is non-uniform, i.e., with this choice of coordinate
systemEdepends only on theydirection. To simplify the algebra, defineEx= qEx/m
andEy=qEy/mso the components of Eq.(3.185) transverse to the magnetic field are
δx ̈ = δy∂Ex
∂y+ωcδy ̇δy ̈ = δy∂Ey
∂y
−ωcδx. ̇ (3.186)Now take the time derivative of the lower equation to obtain
δ
y=δy ̇
∂Ey
∂y+δy∂
∂y(
dEy
dt)
−ωcδ ̈x (3.187)and then substitute forδx ̈giving
δ
y=δy ̇∂Ey
∂y+δy∂
∂y(
dEy
dt)
−ωc(
δy
∂Ex
∂y+ωcδy ̇)
. (3.188)
This can be re-arranged as
δ...y
+ω^2 c(
1 −
1
ωc∂Ey
∂y)
δy ̇=ωcδy∂Ex
∂y
−δy∂
∂y(
dEy
dt)
. (3.189)
Consider the right hand side of the equation as being a forcing term for the lefthand side.
Ifω−c^1 ∂Ey/∂y < 1 , then the left hand side is a simple harmonic oscillator equation in
the variableδy ̇. However, ifω−c^1 ∂Ey/∂yexceeds unity, then the left hand side becomes
an equation with solutions that grow exponentially in time. If two particles are initially
separated by the infinitesimal distanceδyand ifω−c^1 ∂Ey/∂y < 1 the separation distance
between the two particles will undergo harmonic oscillations, but ifω−c^1 ∂Ey/∂y > 1 the
separation distance will exponentially diverge with time. It is seen thatαcorresponds to
ω−c^1 ∂Ey/∂yfor a sinusoidal wave. Exponential growth of the separation distance between
two particles that are initially arbitrarily close together is called stochastic behavior.
3.8 Motion in small-amplitude oscillatory fields
Suppose a small-amplitude electromagnetic field exists in a plasma which in addition has a
large uniform, steady-state magnetic field and no steady-state electric field. The fields can
thus be written as
E = E 1 (x,t)
B = B 0 +B 1 (x,t) (3.190)where the subscript 1 denotes the small amplitude oscillatory quantities and the subscript 0
denotes large, uniform equilibrium quantities. A typical particle in this plasma will develop
an oscillatory motion
x(t) =〈x(t)〉+δx(t) (3.191)