Environmental Engineering Reference
In-Depth Information
S
where
4
p
y
=
p
y
(7.32)
and
p
x
=
UE/c
2
).
γ (U)(p
x
−
(7.33)
Now take a closer look at these equations. They are very similar to the Lorentz
transformation equations we introduced in Eq. (6.28). In fact the correspondence is
exact if we were to start from the Lorentz transformations and make the replacement
x
E/c
. The similarity is all the more striking when we
write down the transformation equation for the energy
E
:
→
p
x
,
y
→
p
y
and
ct
→
E
=
γ (U)(E
−
Ucp
x
).
(7.34)
The fact that energy and momentum transform between inertial frames in exactly
the same way as do the time and space co-ordinates is suggestive of an underlying
symmetry. Indeed such a symmetry exists, and we shall return to study it in much
more detail in Part IV.
7.2 APPLICATIONS IN PARTICLE PHYSICS
In order to explore the consequences of our new formulae for energy (Eq. (7.25))
and momentum (Eq. (7.15)) we shall use some examples taken from particle
physics. This choice is mainly motivated by the fact that, along with nuclear
physics, this is the area of physics where the new dynamics is particularly impor-
tant.
Example 7.2.1
Find the speed of an electron that has been accelerated from rest
by an electric field through a potential difference of (a) 20.0 kV (typical of a cathode
ray tube in a television set); (b) 5.00MV (typical of an X-ray machine).
Solution 7.2.1
The total energy of the electron after being accelerated through the
potential difference V is
mc
2
E
=
+
eV
γ mc
2
,
=
10
−
19
C. Hence
where e
=
1
.
60
×
eV
mc
2
.
γ
=
1
+
4
We have focussed on motion in two dimensions but it should be pretty clear that
p
z
=
p
z
if we had
considered a third spatial dimension.
