Cossey L. Introduction to Mathematical Physics

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In continuous material media, energetic hypotheses should be done chapter parenergint) .

Remark:

In harmonical regime\footnote{ That means that fields satisfy following relations:

} and when there are no sources and when constitutive relations are:

• for D field

D(r,t) = ε(r,t) * E(r,t)

where * represents temporal convolution{convolution} (value of D(r,t) field at time t

depends on values of E at preceding times) and:

• for B field:

Maxwell equations imply Helmholtz equation:

Proof of this is the subject of exercise exoeqhelmoltz.

Remark:

Equations of optics are a limit case of Maxwell equations. Ikonal equation:

grad L = n

2 2

where L is the optical path and n the optical index is obtained from the Helmholtz

equation using WKB method section secWKB). Fermat principle can be deduced from

ikonal equation {\it via} equation of light ray section secFermat). Diffraction's Huyghens

principle can be deduced from Helmholtz equation by using integral methods section

secHuyghens).

Conservation of charge

Local equation traducing conservation of electrical charge is:

Modelization of charge

Charge density in Maxwell-Gauss equation in vacuum

has to be taken in the sense of distributions, that is to say that E and ρ are distributions. In

particular ρ can be Dirac distribution, and E can be discontinuous the appendix chapdistr

about distributions). By definition:

• a point charge q located at r = 0 is modelized by the distribution qδ(r) where δ(r)

is the Dirac distribution.

• a dipole{dipole} of dipolar momentum P

is modelized by distribution div

(P δ(r))

• a quadripole of quadripolar tensor{tensor} Q

i,j

is modelized by distribution

• in the same way, momenta of higher order can be defined.

Current density j is also modelized by distributions:

• the monopole doesn't exist! There is no equivalent of the point charge.

• the magnetic dipole is rot Aδ(r)

Electrostatic potential

Electrostatic potential is solution of Maxwell-Gauss equation:

This equations can be solved by integral methods exposed at section chapmethint: once

the Green solution of the problem is found (or the elementary solution for a translation

invariant problem), solution for any other source can be written as a simple integral (or as

a simple convolution for translation invariant problem). Electrical potential V (r)

created

by a unity point charge in infinite space is the elementary solution of Maxwell-Gauss

equation:

Let us give an example of application of integral method of section chapmethint:

Example:

{\tt Potential created by an electric dipole}, in infinite space:

As potential is zero at infinity, using Green's formula:

From properties of δ distribution, it yields:

Covariant form of Maxwell equations

At previous chapter, we have seen that light speed c invariance is the basis of special

relativity. Maxwell equations should have a obviously invariant form. Let us introduce

this form.

Current density four-vector

Charge conservation equation (continuity equation) is:

Let us introduce the current density four-vector:

J = (j,icρ)

Continuity equation can now be written as:

which is covariant.

Potential four-vector

Lorentz gauge condition:{Lorentz gauge}

suggests that potential four-vector is:

Maxwell potential equations can thus written in the following covariant form:

Electromagnetic field tensor

Special relativity provides the most elegant formalism to present electromagnetism:

Maxwell potential equations can be written in a compact covariant form, but also, this is

the object of this section, it gives new insights about nature of electromagnetic field. Let

us show that E field and B field are only two aspects of a same physical being, the

electromagnetic field tensor. For that, consider the equations expressing the potentials

form the fields:

and

Let us introduce the anti-symetrical tensor {tensor (electromagnetic field)} of second

order F defined by:

Thus:

Maxwell equations can be written as:

This equation is obviously covariant. E and B field are just components of a same

physical being\footnote{Electromagnetic interaction is an example of unification of

interactions: before Maxwell equations, electric and magnetic interactions were

distinguished. Now, only one interaction, the electromagnetic interaction is considered. A

unified theory unifies weak and electromagnetic interaction: the electroweak interaction .

The strong interaction (and the quantum chromodynamics) can be joined to the

electroweak interaction {\it via} the standard model. One expects to describe one day all

the interactions (the gravitational interaction included) in the frame of the great

unification {unification}. }: the electromagnetic tensor. Expressing fields in various

frames is now obvious using Lorentz transformation. For instance, it is clear why a point

charge that has a uniform translation movement in a reference frame R

produces in this

same reference frame a B field.

Optics, particular case of

electromagnetism

Ikonal equation, transport equation

WKB (Wentzel-Kramers-Brillouin) method{WKB method} is used to show how

electromagnetism (Helmholtz equation) implies geometric and physical optical. Let us

consider Helmholtz equation:

∆E + k (x)E = 0

If k(x) is a constant k

then solution of eqhelmwkb is:

General solution of equation eqhelmwkb as:

This is variation of constants method. Let us write Helmholtz equation{Helmholtz

equation} using n(x) the optical index.{optical index}

with n = v / v

. Let us develop E using the following expansion )

where is the small variable of the expansion (it corresponds to small wave lengths).

Equalling terms in yields to {\it ikonal equation

}{ikonal equation}

that can also be written:

grad S = n .

It is said that we have used the "geometrical approximation"\footnote{ Fermat principle

can be shown from ikonal equation. Fermat principle is in fact just the variational form of

ikonal equation. } . If expansion is limited at this first order, it is not an asymptotic

development ) ofE. Precision is not enough high in the exponential: If S

is neglected,

phase of the wave is neglected. For terms in k

This equation is called transport equation.{transport equation} We have done the

physical "optics approximation". We have now an asymptotic expansion of E.

Geometrical optics, Fermat principle

Geometrical optics laws can be expressed in a variational form {Fermat principle} {\it

via} Fermat principle ):

Principle: Fermat principle: trajectory followed by an optical ray minimizes the path

integral:

where n(r) is the optical index{optical index} of the considered media. Functional L is

called optical path.{optical path

}

Fermat principle allows to derive the light ray equation {light ray equation} as a

consequence of Maxwell equations:

Theorem: Light ray trajectory equation is:

Proof: Let us parametrize optical path by some t variable:

Setting:

yields:

Optical path L can thus be written:

Let us calculate variations of L:

Integrating by parts the second term:

Now we have:\footnote{ Indeed

and

}

and

so:

This is the light ray equation.

Remark:

Snell-Descartes laws{Snell--Descartes law} can be deduced from Fermat principle.

Consider the space shared into two parts by a surface S; part above S has index n

and

part under S has index n

. Let I be a point of S. Consider A

a point of medium 1 and A

point of medium 2. Let us introduce optical path\footnote{Inside each medium 1 and 2,

Fermat principle application shows that light propagates as a line}.

where and are unit vectors figure figfermat).

File:Fermat

Snell-Descartes laws can be deduced from fermat principle.

From Fermat principle, dL = 0. As u

is unitary , and it yields:

This last equality is verified by each belonging to the surface:

where is tangent vector of surface. This is Snell-Descartes equation.

Another equation of geometrical optics is ikonal equation.{ikonal equation}

Theorem: Ikonal equation

is equivalent to light ray equation:

Proof:

Let us differentiate ikonal equation with respect to s ([#References

Fermat principle is so a consequence of Maxwell equations.

Physical optics, Diffraction

Problem position

Consider a screen S

with a hole{diffraction} Σ inside it. Complementar of Σ in S

noted Σ

figure figecran).

File:Ecran

Names of the various surfaces for the considered diffraction problem.

The Electromagnetic signal that falls on Σ is assumed not to be perturbed by the screen

: value of each component U of the electromagnetic field is the value U

free

of U without

any screen. The value of U on the right hand side of S

is assumed to be zero. Let us state

the diffraction problem (Rayleigh Sommerfeld diffraction problem):

Problem:

Given a function U

free

, find a function U such that:

(∆ + k )U = 0 in Ω

U = U on Σ

free

U = 0 on Σ

Elementary solution of Helmholtz operator ∆ + k

in R

where r = | MM' | . Green solution for our screen problem is obtained using images

method{images method} section secimage). It is solution of following problem:

Problem:

Find u such that:

(∆ + k )U = δ in Ω

This solution is:

with r = | M M'

s s

| where M

is the symmetrical of M with respect to the screen. Thus:

U(M) =

∫

u(M')δ

(M')dM' =

∫

u(M')(∆ + k

(M')dM'

Ω Ω

Now using the fact that in Ω, ∆U = − k

∫

U(M')(∆ + k

(M')dM' =

∫

(U(M')∆G

(M') − G

(M')∆U(M'))dM'.

Ω Ω

Applying Green's theorem, volume integral can be transformed to a surface integral: