Two equal magnitude, opposite sign charges are located at either end of a molecule of mass M and length l. The molecule rotates end over end (a nonrelativistic tumbling motion) with an initial rotational period T (cT >> l). How long will it take the molecule to lose 1/10 of its rotational energy by electromagnetic radiation?

Let **E**_{0} = E_{0} **k**. The Abraham-Lorentz
force equation for a damped, charged, oscillator driven by an electric field
**E**_{0}exp(-iωt) in the dipole approximation is

d^{2}**r**'/dt^{2}
+ Γ d**r**'/dt - τ d^{3}**r**'/dt^{3} + ω_{0}^{2}
**r**' = (q/m)**E**_{0}exp(-iωt),

where Γ, τ, and ω_{0}
are constants, q is the charge and m is the mass of the oscillator.

Using
this and the expression for the radiation electric field, **E**_{rad}(**r**,t)
= -(4πε_{0})^{-1}[(q/(c^{2}r'')]**a**_{⊥}(t -
r''/c), where **r**'' = **r** - **r**'(t - |**r
**- **r**'|/c),
show that the differential cross section for scattering of radiation of
frequency ω and polarization **n **= (**θ**/θ)**
** is

dσ/dΩ = (e^{2}/(mc^{2}))^{2
}(**k**∙**n**)^{2}[ω^{4}/((ω_{0}^{2}
- ω^{2})^{2} + ω^{2}Γ_{t}^{2})],

where e^{2} = q^{2}/(4πε_{0}) and Γ_{t} = Γ
+ ω^{2}τ.

For a dipole antenna at the origin with complex electric
dipole moment **p**(t) = p_{0}exp(-iωt) **k** the electric and
magnetic fields in the far (radiation) zone are**
E**(

(a) If the real dipole moment is

(b) If there is a second dipole at -d

What is the time average? What are the minimum and maximum values of the time average with respect to variations of the phase α?

(c) A large number, N, of dipoles, as above, is located near the origin with Nkd

A thin linear antenna of length d is excited in
such a way that the sinusoidal current makes a full wavelength of oscillation as shown in
the figure.

(a) Calculate the power radiated per unit solid angle and plot the angular
distribution of the radiation.

(b) Determine the total power radiated and find a numerical value for the
radiation resistance R by setting <P> = <I^{2}>R. You can leave your
answer in integral form.

A non-relativistic positron of charge q_{e} and velocity **v**_{1}
(v_{1 }<< c) impinges head-on on a fixed nucleus of charge Zq_{e}.
The positron which is coming from far away (∞), is decelerated until it comes to
rest and then accelerated again in the opposite direction until it reaches a
terminal velocity **v**_{2}. Taking radiation loss into account
(but assuming it is small), find v_{2} as a function of v_{1}
and Z.