When discussing the Lorentz force,
we introduced the concept of gyration:
a particle in a uniform magnetic fieldBgyrates in a circular orbit around a guiding center.
Here, we will generalize this result
to more complicated situations,
for example involving electric fields.
The particle’s equation of motion
combines the Lorentz force F
with Newton’s second law:
We now allow the fields vary slowly in time and space.
We thus add deviations δE and δB:
Meanwhile, the velocity u can be split into
the guiding center’s motion ugc
and the known Larmor gyration uL around the guiding center,
such that u=ugc+uL.
We already know that mduL/dt=quL×B,
which we subtract from the total to get:
This will be our starting point.
Before proceeding, we also define
the average of ⟨f⟩ of a function f over a single gyroperiod,
where ωc is the cyclotron frequency:
Assuming that gyration is much faster than the guiding center’s motion,
we can use this average to approximately remove the finer dynamics,
and focus only on the guiding center.
Uniform electric and magnetic field
Consider the case where E and B are both uniform,
such that δB=0 and δE=0:
Dotting this with the unit vector b^≡B/∣B∣
makes all components perpendicular to B vanish,
including the cross product,
leaving only the (scalar) parallel components
ugc∥ and E∥:
This simply describes a constant acceleration,
and is easy to integrate.
Next, the equation for ugc⊥ is found by
subtracting ugc∥’s equation from the original:
Keep in mind that ugc⊥ explicitly excludes gyration.
If we try to split ugc⊥ into a constant and a time-dependent part,
and choose the most convenient constant,
we notice that the only way to exclude gyration
is to demand that ugc⊥ does not depend on time.
To find ugc⊥, we take the cross product with B,
and use the fact that B×E⊥=B×E:
Rearranging this shows that ugc⊥ is constant.
The guiding center drifts sideways at this speed,
hence it is called a drift velocityvE.
Curiously, vE is independent of q:
Drift is not specific to an electric field:
E can be replaced by a general force F/q without issues.
In that case, the resulting drift velocity vF does depend on q:
Non-uniform magnetic field
Next, consider a more general case, where B is non-uniform,
but E is still uniform:
Assuming the gyroradius rL is small compared to the variation of B,
we set δB to the first-order term
of a Taylor expansion of B around xgc,
that is, δB=(xL⋅∇)B.
We thus have:
We approximate this by taking the average over a single gyration,
as defined earlier:
To solve this, we make a crude approximation now, and improve it later.
We thus assume that ugc⊥ is constant in time,
such that the equation reduces to:
This is analogous to the previous case of a uniform electric field,
with qE replaced by F⊥,
so it is also solved by crossing with B in front,
yielding a drift:
From the definition of F⊥,
this total vF can be split into three drifts:
the previously seen electric field drift vE,
the curvature driftvc,
and the grad-B driftv∇B:
Such that vF=vE+vc+v∇B.
We are still missing a correction,
since we neglected the time dependence of ugc⊥ earlier.
This correction is called vp,
We revisit the perpendicular equation, which now reads:
We assume that vF varies much faster than vp,
such that d/dvpt is negligible.
In addition, from the derivation of vF,
we know that F⊥+qvF×B=0,
To isolate this for vp,
we take the cross product with B in front,
We thus arrive at the following correction,
known as the polarization driftvp:
In many cases vE dominates vF,
so in some literature vp is approximated as follows: