Elements of Modern Physics
24
fields. The electric field gives rise to a force, again prependicular to the line
charge. However, the magnetic field B which is prependicular to u and v gives
rise to a force q (u–v) × B which has a component parallel to the line charge.
This contradicts the result of Galilean transformations that force is invariant.
Example 2
The time-dilation of a moving clock has a dramatic manifestation in terms of the
increased lifetime of a moving particle.
In nature, unstable particles are found, whose decay is described in quantum
mechanics as a transition from the initial state to the final state. The rate of
decay is determined by the transition probability which is defined by λ,
|dN (t)| = – λ N (t) dt
(1.80)
where N (t) is the number of particles at time t, and dN (t) is the number of
particles which decay in time dt. The number of particles remaining at time t is
obtained from Eq. (1.80) to be
N (t) = N (0) e
-λt
(1.81)
The mean lifetime of the particle is then given by
0
1/
(0)
dN
t N
τ =
= λ
∫
(1.82)
Now, if the unstable particles are moving, their dilated lifetime is given
1/ 2
2
2
1
v
c
0
τ
τ =
−
(1.83)
where v is the speed of the particles, i.e., the particles live for a longer time.
The dilation of lifetime of moving particles has important implications in the
design of experiments. Consider for example, the production of K
+
-mesons by
fast-moving protons colliding with a target. For a beam of K
+
-mesons of
momentum 3 GeV/c corresponding to v ≈ 0.98645 c, the bubble chamber where
the K
+
-particles will interact with protons, is kept at a distance of 100 m. Since
τ 0 for K
+
-mesons is 1.23 × 10
–8
s, the value of τ is 7.5 × 10
–8
s so that the
fraction of K
+
-mesons reaching the chamber at t = d/v, is
2
( ) 1.1 10
(0)
N t
N
−
= ×
(1.84)
Without the time-dilation, the fraction would have been about 1.12 × 10
-12
,
so that with a typical pulse carrying about 10
3
K
+
-mesons the experiment would
24
fields. The electric field gives rise to a force, again prependicular to the line
charge. However, the magnetic field B which is prependicular to u and v gives
rise to a force q (u–v) × B which has a component parallel to the line charge.
This contradicts the result of Galilean transformations that force is invariant.
Example 2
The time-dilation of a moving clock has a dramatic manifestation in terms of the
increased lifetime of a moving particle.
In nature, unstable particles are found, whose decay is described in quantum
mechanics as a transition from the initial state to the final state. The rate of
decay is determined by the transition probability which is defined by λ,
|dN (t)| = – λ N (t) dt
(1.80)
where N (t) is the number of particles at time t, and dN (t) is the number of
particles which decay in time dt. The number of particles remaining at time t is
obtained from Eq. (1.80) to be
N (t) = N (0) e
-λt
(1.81)
The mean lifetime of the particle is then given by
0
1/
(0)
dN
t N
τ =
= λ
∫
(1.82)
Now, if the unstable particles are moving, their dilated lifetime is given
1/ 2
2
2
1
v
c
0
τ
τ =
−
(1.83)
where v is the speed of the particles, i.e., the particles live for a longer time.
The dilation of lifetime of moving particles has important implications in the
design of experiments. Consider for example, the production of K
+
-mesons by
fast-moving protons colliding with a target. For a beam of K
+
-mesons of
momentum 3 GeV/c corresponding to v ≈ 0.98645 c, the bubble chamber where
the K
+
-particles will interact with protons, is kept at a distance of 100 m. Since
τ 0 for K
+
-mesons is 1.23 × 10
–8
s, the value of τ is 7.5 × 10
–8
s so that the
fraction of K
+
-mesons reaching the chamber at t = d/v, is
2
( ) 1.1 10
(0)
N t
N
−
= ×
(1.84)
Without the time-dilation, the fraction would have been about 1.12 × 10
-12
,
so that with a typical pulse carrying about 10
3
K
+
-mesons the experiment would