Environmental Engineering Reference
In-Depth Information
a non-invariance of the form of the laws of physics and this is indeed the case as
we can easily see. Eq. (11.18) is now replaced by
x
=
x
)
m
R
−
1
q(
R
−
1
×
B
(11.21)
since in this universe we cannot actively rotate the background magnetic field and
so
B
=
B
. This equation can be simplified to
x
=
x
×
B
).
m
q
(
R
(11.22)
Thus we see that the equation of motion for a charged particle varies depending
upon the orientation of our apparatus. In physical terms, the effective magnetic
field which appears in the Lorentz force law varies with orientation.
11.1.2 Translational symmetry
Having dealt with pure rotations, let us now focus upon the consequences of
shifting origin. Again we shall speak of two frames of reference,
T
and
T
, but
this time
T
differs from
T
in that the origin in
T
relative to the
origin in
T
, as illustrated in Figure 11.2. Clearly all vectors are once again blind
to this change of frame. In fact, as we move from
T
to
T
not only does a general
vector
lies at position
R
V
remain unchanged its components are also unchanged:
V
i
=
V
i
.
(11.23)
There is however a subtlety we ought to be sensitive to. When we speak of position
vectors we are stating a position relative to some origin. Thus when we change
frames, we should remember that we have also changed the point of reference for
position vectors.
Let us return again to the example of two massive particles acting under gravity
and in particular let us recast Eq. (11.10) in terms of position vectors measured
y'
y
x
'
O'
x'
x
R
x
O
Figure 11.2 Two different frames of reference,
T
and
T
, which differ by a translation.
