ACCELERATION FIELDS: THE RADIATION

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ACCELERATION FIELDS THE RADIATION
Back to the (Schwartz) formula for the
acceleration fields
Fairly complicated, isnt it? Well, we shall do
the usual thing, i.e. keep it simple!
Which is to say, do one term at the time. We will
therefore ignore the second term (in the sum at
the numerator) and only study the first one. Note
that ignoring the second term has a precise
physical meaning We assume that a charge has, oh
yes!, an acceleration but the velocity is zero
or at least it is negligible. Note also that the
term is linear in the acceleration. That is the
way to have the second term disappear beside ,
of course, having the acceleration parallel to
the velocity that case will be treated later on,
after we have studied the special
relativity. Well the whole equation is
In Jacksons form
Where the subscript a stand for acceleration.
The formula is still fairly complicated. To start
with, let us evaluate the energy flow, i.e. the
Poyntings vector.
But Ba is perpendicular to Ea, and they are both
perpendicular to . Which amounts to say
that
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Beside the 1/r2 dependence, S and g have a radial
direction as e, from the radiating charge in its
retarded point to the observer. There is a piece
of field(s) which is emitted by the accelerating
charge and goes away carrying with itself its own
momentum and energy. Let us now make more
quantitative statements. Let us consider the case
V0 (i.e. ?0), which we assume to hold as an
approximation for those cases in which vltltc
(Non-Relativistic approximation). The only term
left is the first term in the formula, i.e.,
using this time Schwartzs formula
That is new information. This formula tells us
that for low velocities - only the component
of the charge acceleration orthogonal to the to
the line of sight generates an acceleration
field, i.e. radiation.
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In words, the generated electric field in any
point of space is parallel to , but of
opposite sign (for positive charges). The
polarization of light is then understood very
easily let - as the simple system we considered
with the approximation v0 take a charge
revolving on a circular trajectory around a fixed
point with constant (low) velocity be the plane
of the movement the x-y plane (see figure) , let
the observer be at a point on y axis. An
observer on the y axis will see the charge go
up and down periodically, with the same period
the charge has. The law of the motion is a
sin(?t) and the perpendicular acceleration will
be also of sinusoidal form, and maximum when the
charge crosses the y-z plane.
The orthogonal acceleration of the charge as seen
by the observer on the y axis will be along the x
direction, we are then in one of the cases
already discussed, the electric field will
propagate along the y axis and the electric
field will be aligned with the perceived charge
acceleration along the x axis (but in opposite
direction, for a positive charge). This radiation
is called to be linearly polarized along the
x axis.
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Let us now observe the radiating charge from a
point on the positive z axis!. The charge will be
having all the time a centripetal acceleration
orthogonal to the direction of view, directed
along a direction opposite to the charge
position. Along the z axis the radiation (we
may as well assume a wavelength between 400 and
600 nm and call it light) will be also
polarized, but this time with a circular
polarization which, for the direction of motion
indicated in the figure will be Left-Handed The
electric field, for an observer staying at the
same place on the z axis all the time, will
turn counterclockwise. Another term to indicate
the sign of circular polarization is helicity.
It can be positive or negative, positive helicity
stands for left-handed, polarization,
counterclockwise rotation. A charge as indicated
in the example emits radiation not only along the
positive axis, but also along the negative x
axis. That light will then also be circularly
polarized, but with opposite helicity. In this
case, handedness is Right-handed. Another way to
look at polarization is to imagine to have many
observation posts along the z axis, they will
record the direction of the electric field all at
the same time. It is obvious then that given the
electric field measured at a certain point, all
the points spaced out by a fixed dx will see a
field generated a time dx/c earlier that is,
moving at fixed time towards the positive x we
would see the field turning clockwise in the
example.
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So far we have put small restrictions to the
motion of the charge only that the motion be
very slow. Well now, let us still assume small
(negligible) velocity of the charge At any
instant, the charge will suffer an acceleration
along some well defined direction. Given a
direction of observation (i.e., a unitary Vector
) which makes with a an angle ?, the
component of a perpendicular to the direction of
observation is and the
flow of irradiated energy in that direction is
This formula gives the amount of energy moving
through a unitary spherical surface at distance
r from the radiating charge, as a function of
the relative directions of observation and of
acceleration. It can be easily integrated over a
spherical surface, to obtain the total energy
radiated instantly by the charge W dU/dt.
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The formula so obtained is the Larmor formula,
It is the quantity of energy irradiated per unit
time by a charge moving random in space with a
very non-relativistic velocity. The power emitted
per unit time is high, it would let the charge to
stop in a very short time. This is the reason why
the atomic model of matter met with little
success when it was proposed. It had to wait for
the quantum mechanics to be discovered.
N.B. We have been working today with Radiation or
Acceleration Fields. We have always said that in
Radiation the fields are orthogonal between
themselves and with the direction of propagation.
Then came the detailed study of the acceleration
fields, the approximation of the still charge,
the Larmor formula and the concept that only
can radiate. Well, yes!!! But.but only for
V0!! We will study later on the effect of the
second term, and its 2 main instances,
bremsstrahlung and synchrotron radiation
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