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Supplement
—Phys411—Spring 2010
Prof. Ted Jacobson
Room 4115, (301)4056020
www.physics.umd.edu/grt/taj/411b/
[email protected]
Runaway solutions and preacceleration
With the radiation reaction force
F
rad
= (
μ
0
q
2
/
6
πc
)˙
a
, the equation of one dimensional
motion in the presence of an external force
F
(
t
) takes the form
a

τ
˙
a
=
f
(
t
)
,
(1)
where
τ
=
μ
0
q
2
/
6
πcm
, and
f
(
t
) =
F
(
t
)
/m
, where
m
is the mass of the particle. This is
equation is third order in time derivatives of position, so a solution is determined by initial
position, velocity and acceleration. In terms of
a
(
t
) however it is ﬁrst order, and can be
solved with one undetermined constant. Given that solution, the position can be found
given an initial position and velocity. Note that the equation does
not
have time reversal
symmetry.
To solve (1), let
a
(
t
) =
e
t/τ
a
1
(
t
), so
a

τ
˙
a
=

τe
t/τ
˙
a
1
. In terms of
a
1
, (1) takes the
form
˙
a
1
=

τ

1
e

t/τ
f
(
t
)
,
(2)
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This note was uploaded on 12/29/2011 for the course PHYSICS 411 taught by Professor Agshe during the Summer '11 term at Maryland.
 Summer '11
 Agshe
 Magnetism, Acceleration, Force, Radiation

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