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### Electromagnetic Potentials

Bees can see polarized light

polarization of blue sky enables them to navigate

Humans:

Haidinger’s Brush

Vikings:

Iceland Spar?

Many invertebrates can see polarization, e.g. the Octopus

Not to navigate (they don’t go far)

Perhaps they can see transparent jellyfish better?

unpolarized polarized

In a transparent object, each wavelength of light is polarized by

a different angle. Passing unpolarized light through a polarizer,

then the object, then another polarizer results in a colorful pattern

which changes as one of the polarizers is turned.

Polarization is also used in the entertainment industry to produce and show 3-D movies. Three-dimensional movies are actually two movies being shown at the same time through two projectors. The two movies are filmed from two slightly different camera locations. Each individual movie is then projected from different sides of the audience onto a metal screen. The movies are projected through a polarizing filter. The polarizing filter used for the projector on the left may have its polarization axis aligned horizontally while the polarizing filter used for the projector on the right would have its polarization axis aligned vertically. Consequently, there are two slightly different movies being projected onto a screen. Each movie is cast by light which is polarized with an orientation perpendicular to the other movie. The audience then wears glasses which have two Polaroid filters. Each filter has a different polarization axis - one is horizontal and the other is vertical. The result of this arrangement of projectors and filters is that the left eye sees the movie which is projected from the right projector while the right eye sees the movie which is projected from the left projector. This gives the viewer a perception of depth.

(- 45 degrees):

Circular:

Circular:

Rybicki & Lightman, Section 2.3

Consider

Transverse E-field

What is the spectrum?

Energy / time as a function of frequency

Note:

Radians / sec

Define the Fourier Transform of

and

Parseval’s Theorem for Fourier Transforms:

Proof:

Rybicki & Lightman, Chapter 3

Electromagnetic Potentials

Instead of worrying about E and B, we can use the scalar and vector

potentials

Simpler:

1 scalar and 1 vector quantity instead of 2 vector quantities.

Relativistic treatment is simpler.

Equations (1) & (2) already satisfy 2 of Maxwell’s Equations –

what about the others?

becomes

For reasons which will become clear in a minute, we re-write this

last equation as

(3)

GUAGE TRANSFORMATIONS Equations –

(1)-(4) do not determine A and φ uniquely:

one can add the gradient of an arbitrary scalar ψ to A

and leave B unchanged

Likewise E will be unchanged if you add

These are called GUAGE TRANSFORMATIONS

RETARDED POTENTIALS Equations –

It turns out that the solutions to (5) and (6) can be expressed

as integrals over sources of charge, provided you properly

take into account the fact that changes in the E and B fields

can move no faster than the speed of light.

RETARDED POTENTIALS Equations –

At point r =(x,y,z), integrate over charges at positions r’

(7)

(8)

where [ρ] evaluate ρ at retarded time:

Similar for [j]

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