Environmental Engineering Reference
In-Depth Information
The right hand side of Eq. (12.31) can thus be evaluated to
P p ) 2
P π +
P p +
P p
+
=
·
( P π
2 P π
m π c 2
m p c 2
2 m π m p c 2 .
=
+
+
(12.33)
Equating Eq. (12.32) and Eq. (12.33) gives
m p c 2
m π c 2
m p c 2
2 m π m p c 2 ,
+
2 Em p
=
+
+
which can be re-arranged in order to determine the threshold energy for the incom-
ing photon:
m π c 2
2 m π m p c 2
2 m p
+
E
=
135 2
+
2
×
135
×
938
=
MeV
×
2
938
=
145 MeV.
12.4 ELECTRIC AND MAGNETIC FIELDS
As a final example we turn our attention to the subject of electromagnetism
wherein
lies
perhaps
the
most
important
application
of
relativity
theory
in
everyday life.
We start with a puzzle. Consider a wire carrying a current along its length. At
some instant in time a positively charged particle travels parallel to the wire and
in the direction of the current. Viewed from a frame in which the wire is at rest,
the charged particle is subsequently drawn towards the wire by the Lorentz force
which arises as a result of the magnetic field around the wire. Now let us consider
the same circumstance from the viewpoint of a frame of reference in which the
charged particle is at rest. In this frame, the Lorentz force is zero since the particle's
velocity is zero 2 . It therefore seems that the charged particle will remain at rest
and we have a contradiction.
The resolution to this apparent paradox lies in Einstein's theory of relativity. In
the rest frame of the charged particle, the electrons which carry the current in the
wire are closer together as a result of Lorentz contraction and hence their charge
density is greater than if they were at rest by a factor of γ(u) where u is the
speed of the electrons in the rest frame of the charged particle. The ionic lattice
against which the electrons move is also Lorentz contracted but by a lesser amount
(since the ions are at rest relative to the wire). Consequently, there is not a perfect
cancellation of the electric field due to the ions with that due to the electrons and
the positively charged particle is compelled to accelerate. Since, from the viewpoint
of the charged particle, the electron density is greater than the ionic charge density,
the postively charged particle is drawn towards the wire. We thus see that what
is a magnetic field in one frame of reference is an electric field in another frame.
2 The Lorentz force is given by F = q v × B .
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