A nonrelativistic particle of charge ze, mass m, and initial speed v0 is incident on a fixed charge...

Question

A nonrelativistic particle of charge ze, mass m, and initial speed v0 is incident on a fixed charge Ze at an impact parameter b that is large enough to ensure that the particle's deflection in the course of the collision is very small.
(a) Using the Larmor power formula and Newton's second law, calculate the total energy radiated, assuming (after you have computed the acceleration) that the particle's trajectory is a straight line at constant speed:

(b) The expression found in part a is an approximation that fails at small enough impact parameter. For a repulsive potential the closest distance of approach at zero impact parameter, rc = 2zZe2/mv20, serves as a length against which to measure b. The approximation will be valid for b >> rc. Compare the result of replacing b by rc in part a with the answer of Problem 14.5 for a head-on collision.
(c) A radiation cross section x (with dimensions of energy times area) can be defined classically by multiplying ?^?W(b) by 2 I?b db and integrating over all impact parameters. Because of the divergence of the expression at small b, one must cut off the integration at some b = bmin. If, as in Chapter, the uncertainty principle is used to specify the minimum impact parameter, one may expect to obtain an approximation to the quantum-mechanical result. Compute such a cross section with the expression from part a. Compare your result with the Bethe-Heitler formula [N–t times (15.30)].

David · Accepted Answer

Physics 506 Winter 2008
Homework Assignment #12 — Solutions
Textbook problems: Ch. 14: 14.4, 14.5, 14.8, 14.11
14.4 Using the Liénard-Wiechert fields, discuss the time-averaged power radiated per unit
solid angle in nonrelativisic motion of a particle with charge e, moving
a) along the z axis with instantaneous position z(t) = a cosω0t.
In the non-relativisitic limit, the radiated power is given by
dP (t)
dΩ
=
e2
4πc
|n̂× ~̇β|2 (1)
In the case of harmonic motion along the z axis, we take
~r = ẑa cosω0t, ~β = −ẑ
aω0
c
sinω0t, ~̇β = −ẑ
aω20
c
cosω0t
By symmetry, we assume the observer is in the x-z plane tilted with angle θ from
the vertical. In other words, we take
n̂ = x̂ sin θ + ẑ cos θ
This provides enough information to simply substitute into the power expression
(1)
n̂× ~̇β = ŷ aω
2
0
c
sin θ cosω0t ⇒
dP (t)
dΩ
=
e2a2ω40
4πc3
sin2 θ cos2 ω0t
Taking a time average (cos2 ω0t→ 1/2) gives
dP
dΩ
=
e2a2ω40
8πc3
sin2 θ
This is a familiar dipole power distribution, which looks like
-1 -0.5 0.5 1
-1
-0.5
0.5
1
z
Integrating over angles gives the total power
P =
e2a2ω40
3c3
b) in a circle of radius R in the x-y plane with constant angular frequency ω0.
Sketch the angular distribution of the radiation and determine the total power
radiated in each case.
Here we take instead
~r = R(x̂ cosω0t+ ŷ sinω0t) → ~β =
Rω0
c
(−x̂ sinω0t+ ŷ cosω0t)
~̇β = −Rω
2
0
c
(x̂ cosω0t+ ŷ sinω0t)
Then
n̂× ~̇β = −Rω
2
0
c
[ŷ cos θ cosω0t+ (ẑ sin θ − x̂ cos θ) sinω0t]
which gives
dP (t)
dΩ
=
e2R2ω40
4πc3
(cos2 θ cos2 ω0t+ sin2 ω0t)
Taking a time average gives
dP
dΩ
=
e2R2ω40
8πc3
(1 + cos2 θ)
This distribution looks like
-1 -0.5 0.5 1
-1
-0.5
0.5
1
z
The total power is given by integration over angles. The result is
P =
2e2R2ω40
3c3
14.5 A nonrelativistic particle of charge ze, mass m, and kinetic energy E makes a head-on
collision with a fixed central force field of finite range. The interaction is repulsive
and described by a potential V (r), which becomes greater than E at close distances.
a) Show that the total energy radiated is given by
∆W =
4
3
z2e2
m2c3
√
m
2
∫ ∞
rmin
∣∣∣∣dVdr
∣∣∣∣2 dr√V (rmin)− V (r)
where rmin is the closest distance of approach in the collision.
In the non-relativistic limit, we may use Lamour’s formula written in terms of ~̇p
P (t) =
2(ze)2
3m2c3
∣∣∣∣d~pdt
∣∣∣∣2 = 2(ze)23m2c3
(
dV (r)
dr
)2
(2)
where we have used Newton’s second law to write
d~p
dt
= ~F = −r̂ dV (r)
dr
The radiated energy is given by integrating power over time
∆W =
∫ ∞
−∞
P (t) dt
However, this can be converted to an integral over the trajectory of the particle.
By symmetry,

A nonrelativistic particle of charge ze, mass m, and initial speed v0 is incident on a fixed charge Ze at an impact parameter b that is large enough to ensure that the particle's deflection in the...

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