1
Monday, October 17:
Multiparticle Sys
tems
For nonrelativistic charged particles, we have derived a useful formula for the
power radiated per unit solid angle in the form of electromagnetic radiation:
dP
d
Ω
=
q
2
4
πc
3
[
a
2
q
sin
2
Θ]
τ
,
(1)
where
q
is the electric charge of the particle,
~a
q
is its acceleration, and Θ
is the angle between the acceleration vector
~a
q
and the direction in which
the radiation is emitted. The subscript
τ
is a quiet reminder that for any
observer, we must use the values of
a
q
and Θ at the appropriate retarded time
τ
rather than the time of observation
t
. By integrating over all solid angles,
we found the net power radiated by a nonrelativistic charged particle:
P
=
2
q
2
3
c
3
[
a
2
q
]
τ
.
(2)
Because the charge of the electron (or proton) is small in cgs units, and the
speed of light is large in cgs units, we expect the power radiated by a single
electron (or proton) to be small, even at the large accelerations that can be
experienced by elementary particles.
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The energy that the charged particle
is radiating away has to come from somewhere.
If the only energy source
is the particle’s kinetic energy,
E
=
mv
2
q
/
2, the characteristic time scale for
energy loss is
t
E
=
E
P
=
3
c
3
m
4
q
2
"
v
2
q
a
2
q
#
τ
.
(3)
As a concrete example, consider an electron moving in a circle of radius
r
q
with a speed
v
q
=
βc
. The acceleration of the electron will be
a
q
=
β
2
c
2
/r
q
,
the power radiated will be
P
=
2
q
2
e
β
4
c
3
r
2
q
= 4
.
6
×
10

17
erg s

1
ˆ
β
0
.
01
!
4
r
q
1 cm
¶

2
.
(4)
1
In problem set 2, the electron being bombarded by red light had a maximum accel
eration of
a
q
∼
10
19
cm s

2
; the proton in Lawrence’s cyclotron had an acceleration of
a
q
∼
4
×
10
16
cm s

2
.
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The time scale for energy loss will be
t
E
=
3
cm
e
r
2
q
4
q
2
e
β
2
= 8
.
9
×
10
5
s
ˆ
β
0
.
01
!

2
r
q
1 cm
¶
2
.
(5)
It is informative to compare the time scale for energy loss with th orbital
period of the electron:
t
P
=
2
πr
q
βc
= 2
.
1
×
10

8
s
ˆ
β
0
.
01
!

1
r
q
1 cm
¶
.
(6)
For the electron’s orbit to be stable, rather than a steep inward death spiral,
we require
t
E
t
P
, which implies an orbital radius
r
q
8
πq
2
e
3
c
2
m
e
β
=
8
πr
0
3
β ,
(7)
where
r
0
≡
q
2
e
/
(
m
e
c
2
) = 2
.
8
×
10

13
cm is the
classical electron radius
.
According to
classical
electromagnetic theory, then, an electron on an
orbit smaller than
∼
βr
0
in radius will radiate away its energy and spiral in
to the origin on a time scale comparable to its orbital time. The radiation
by electrons on circular orbits was what spelled the doom of classical atomic
theory. Around 1911, Ernest Rutherford proposed a theory in which electrons
went on circular orbits around a positively charged nucleus. The radius
r
of
the orbits was determined by the requirement that the centripetal force be
provided by the electrostatic force between the electron (charge
q
e
) and the
nucleus (charge

Zq
e
):
Zq
2
e
r
2
=
m
e
β
2
c
2
r
,
(8)
or
r
=
Zq
2
e
β
2
m
e
c
2
=
Z
β
2
r
0
.
(9)
However, it was known that the radius of the hydrogen atom (
Z
= 1) was
r
∼
0
.
5
˚
A
∼
0
.
5
×
10

8
cm
∼
2
×
10
4
r
0
. This implied
β
= (
r/r
0
)

1
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 Fall '05
 RYDEN
 Electron, Electric charge, Fundamental physics concepts

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