I think we will have to make some more specific assumptions about the setup.
Let's start with charges in an ordinary wire with a finite resistance.
Typically the notion of "resistance" comes about when you accelerate
charges by an electric field (this is the voltage applied to a wire,
say). This accelerates the charge carriers, typically electrons, in
the appropriate direction of the electric field (remembering that
electrons are negative).
In free space this would lead of course an accelerating charge, i.e. a
current (which is proportional to velocity of charge carriers) which
increases with time.
But Ohm's law says that V = IR, meaning for constant voltage V, we
should get constant current I? That comes about because in a real
physical material like a wire, the electrons very quickly scatter
(interact with electromagnetically) the ions in the lattice, often
scattering them back in the direction whence they came (thereby
imparting a tiny momentum to the wire in the process). It quickly
reaches a statistical equilibrium and there is a net "drift velocity"
and current.
The resistance depends on the material geometry and its intrinsic
resistivity, which depends on the precise physics of the interaction
of the material and charge carrier.
As has been pointed out, a static magnetic field will not do any work
(change of energy) on the charge carrier, and so to this first order
there doesn't seem to be any significant change of resistivity with
this applied magnetic field.
The applied magnetic field will cause the Hall effect, giving rise to
a voltage perpendicular to both the current and the applied magnetic
field. This effect is often used in sensors which measure a magnetic
field.
To go beyond this, you have to make more specific assumptions about
the structure and physics of the material. The phenomenon is
generally known
as magnetoresitance, and for most materials and conditions it is very small.
I do not know what the mechanisms are but possibly applied magnetic fields
may result in the ions in the lattice vibrating more slowly (as their
vibration has to push against the magnetic field). It is well known
that resistivity can change with temperature as that will control the
way that electrons will scatter off the ions.
However, some materials can be engineered so that magnetoresistance is
not so small. A specific case was discovered in the 1980's and known
as "Giant Magnetoresistance", referring to its magnitude which was far
larger than the previously observed values.
This has some application now in the development of highly precise
read-heads for magnetic recording (hard drives).
In particular it is a quantum-mechanical effect, usually in engineered
materials (cobalt and copper layers) with alternating ferromagnetic
layers. Ferromagnets have large regions of coherent spins (magnetic
moments). Electrons in quantum mechanics also have property of spin
which means that they are little magnets themselves, and so can feel
the effect of other magnets (say the aligned spins in the
ferromagnetic domain).
In giant magnetoresistance, the working theory is that if you have a
high external magnetic field then the various ferromagnetic domains
will align with each other and thus make the electrons less likely to
scatter. However, at lower magnetic fields, the spins may be
different at each domain boundary, and so the electron would feel a
randomly alternating pattern of spins. This will result in more
scattering and hence more resistance. Think of a coin sliding down a
smooth sheet held at an angle versus one which is corrugated randomly.
On average the coin will be bounced back by the corrugated one more.
This is a rough analogy to how things work.
There is ongoing investigation in other related phenomena, "Colossal
Magnetoresistance", and "Ballistic Magnetoresistance", which depend
intimately on the material science.
So an answer is "it is possible, with effort, to engineer significant
changes in resistance of a material as a function of applied magnetic
field, but it is not easy."
Now, in the free space of a vacuum, the motion of a charge in combined
electric and magnetic field is fully solvable. Following
Landau+Lifshitz Classical theory of fields, section 22, the motion of
the electron in two dimensions (if magnetic field H is in Z direction
and electric field in x-y plane) results in periodic functions of
time.
The *average* velocity, drift velocity can be written as v = c E x H / |H|^2,
assuming that the velocity always remains much less than the speed of light.
So if you fix your driving 'E' electric field, then the drift velocity
'v' will decrease with increasing magnetic field strength H. One
might interpret this as an increase in resistance with increasing
magnetic field strength.
However, in free space the concept of 'resistance' is not so
frequently used, until possibly when you get to plasma physics, where
all sorts of complex phenomena can arise. |
