Environmental Engineering Reference
In-Depth Information
This really is quite remarkable: even the most basic of phenomena in the study of
electricity and magnetism requires relativity theory for a consistent interpretation.
It is all the more remarkable given that the effect is sensitive to the drift speed
of the electrons in a wire, which is no more than a few millimetres per second.
Strictly speaking this ought not to have come as too great a surprise since we
already stated that Einstein was impressed by the fact that Maxwell's equations of
electromagnetism were inconsistent with Galilean relativity and that he built his
theory so as to respect Maxwell's theory. Nevertheless, this is our first concrete
illustration of the fact.
That one can view the occurrence of magnetic phenomena as a purely relativistic
effect is further illustrated by the following example. This time let us consider a
current
I
which flows as a result of the linear motion of an ensemble of charged
particles. Using Ampere's Law we can deduce the magnetic field which arises at
a distance
r
from the wire:
µ
0
I
2
πr
.
B
=
In a real wire the charged particles are electrons and they are accompanied by
positively charged ions such that the wire as a whole is electrically neutral but now
we shall consider a current of free charges. In which case there is also an electric
field at a distance
r
from the wire that is equal to
ρ
2
πrε
0
=
I
2
πε
0
vr
,
E
=
where
ρ
is the net charge per unit length of the charged particles and we have
used
I
vρ
to rewrite this in terms of the current. Now we notice that the ratio
of electric and magnetic fields is given by
=
cB
E
v
c
=
1
/c
2
. Viewed this way, it is clear that the
magnetic field is a small relativistic correction to the electric field. Even so, it is
an effect which has huge technological and commercial relevance.
and we have used the fact that
ε
0
µ
0
=
PROBLEMS 12
12.1 A rocket of initial mass
m
i
starts from rest and propels itself forwards by
emitting photons backwards. The final mass of the rocket, after its engine
has finished firing, is
m
f
. By considering the four-momenta of the rocket
before and after it emitted the photons, and the net four-momentum of the
photons, show that the final speed of the rocket,
u
, must satisfy
γ(u)
1
c
.
m
i
m
f
=
u
+
Hence deduce the final speed of the rocket.
