Advanced EM -Master in Physics 2011-2012

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ACCELERATION FIELDS: THE RADIATION

Back to the (Schwartz) formula for the acceleration fields:

32 )1(|'|

)'(])'('[),(

ε'β'rr

β'aε'ε'ε'aarE

c

qqt

Fairly complicated, isn’t 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

aa

a

Eε'B

εβ

'ββ'εε'E

3)''1(

})'{(

'cr

qIn Jackson’s 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 Poynting’s vector.

aaa BES 4

c

But Baa is perpendicular to is perpendicular to EEaa, , and they are both

perpendicular

to . Which amounts to say that 'ε '

4'

422 εεSa aa B

cE

c

Advanced EM -Master in Physics 2011-2012

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The vectors S and g depend on r as r-2 (we anticipate here: g is the “momentum distribution” of the fields, exactly as U is their energy distribution” . The following equation holds: ) .

2cSg Beside the 1/r2 dependence, S and g have a radial direction as ε’, 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 V=0 (i.e. Β=0), which we assume to hold as an approximation for those cases in which v<<c (Non-Relativistic approximation). The only term left is the first term in the formula, i.e., using this time Schwartz’s formula:

a’

ε’

)( ε'aε'

)(' ε'aε'a )(' ε'aε'a

')1(|'|

])'('[),(

232 rc

q

c

qt

'a

ε'β'rr

ε'ε'aarE

Where is the projection vector

of a’ normal to ε’

'a

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.

Advanced EM -Master in Physics 2011-2012

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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 v≈0 – 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(ωt); and the perpendicular acceleration will be also of sinusoidal form, and maximum when the charge crosses the y-z plane.

'a

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.)0,0,(EE

Advanced EM -Master in Physics 2011-2012

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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 δx will see a field generated a time δx/c earlier; that is, moving at fixed time towards the positive x we would see the field turning clockwise in the example.

Advanced EM -Master in Physics 2011-2012

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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 θ, the component of a perpendicular to the direction of observation is ; and the flow of irradiated energy in that direction is

'ε

''23

222

24

22

'4

)(sin'

'4εε

'aS

rc

aq

rc

qc

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.

3

22

0

33

22

2

0 23

22

'

3

2sin

2

'

)'sin'2(sin'4

'

c

aqd

c

aq

drrrc

aq

dt

dUW

sina

Advanced EM -Master in Physics 2011-2012

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The formula so obtained is the Larmor formula,

3

22

3

'2

c

aqW

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 V≈0!!

We will study later on the effect of the second term, and its 2 main instances, bremsstrahlung and synchrotron radiation

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a