Div,Grad&Curl

Div-"Eminating field lines."
Divergence

del.D=(i.Ð/Ðx+j.Ð/Ðy+k.Ð/Ðz).D
D=ai+bj+ck
del.D=(Ða/Ðx+Ðb/Ðy+Ðc/Ðz)
eq 1

Divergence is a scalars quantity

del.Q=(i.Ð/Ðx+j.Ð/Ðy+k.Ð/Ðz).Q
eq2

For electric fields

Therefore,

del^H

Example, field lines around a straight current line are circular, they curl!

del^H=I/A amps

Gives the magnitude of the e.m.f.. It states that,

Induced e.m.f. is proportional to change of flux
e.m.f. proportional Ð(�)/Ðt

This gives the direction of the induced e.m.f.. The direction is such to oppose the change in flux

e.m.f.=-Ð(�)/Ðt
eq 1

The electric displacement, D, is equal to,

D=E.E
eq 2
Gauss's Law: div D=ç=del.D

AmpŠres law breaks down when

A way around this is to have a hypothetical charge density, ç, between the plates.
Therefore , del.J is no longer zero (according to AmpŠres law in it's present form).

del.J=-Ðç/Ðt
del.D=ç
Therefore, del.(J+ÐD/Ðt)=0
del.J=-Ð(del.D)/Ðt

Where D is the "Displacement current".

del.D=ç
J=àE
(Ohm's law)

=-æ.Ð/Ðt{àE + E.(ÐE/Ðt)}
=-æà.(ÐE/Ðt)-æE.(ÐýE/Ðtý) eq(1)
=del.(del.E)-delýE

i.e. no field generated within the medium to which the equation is applied.

Therefore,

Therefore,

delýE-æà.(ÐE/Ðt)-æE.(ÐýE/Ðtý)=0
or
delýH-æà.(ÐH/Ðt)-æE.(ÐýH/Ðtý)=0

In general where u is H or B,

Consider the possible solution (in a single dimension),

The above equation is in a single dimension. It is a plane wave propagating in the x direction. á is the wave number and a is the attenuation coefficient. It can be rewritten as,

We now need to find the range of differentials of u.

y=a+já
yý=aý-áý+2jaá
aý-áý+2jaá=•.æ.j.w-æE.wý

Real: aý-áý=-æE.wý
Imag: 2aá=•.æ.w

We now need to solve for a and á. Taking a simple case in a non-conducting medium, i.e. •=0. If á is zero, we do not get a progressing wave, just something that oscillates in time, it doesn't go anywhere. So a must have to be equal to zero. So, in a non-conducting medium, there is no attenuation.

w=2pi/f, á=2pi/lambda

K is the dielectric constant (the relative permativitty). Hence refractive indices from measurement of the dielectric property.

because E is not a function of x or y.

and there are no free charges generating a field.

Having chosen the electric vector to lie along the x axis.

In such cases the attenuation of the wave is non zero. •<>0, hence we write,

yý=æ•jw-æEwý
aý-áý=-æEwý

:Real 2aá=æ•w
:Imag 4aýáý=(æ•w)ý
áý=(æ•w)ý/4aý

Substitute and hence,

We have to take the positive root, or a would be negative and energy conservation would be violated.

For á we have (from a similar derivation),

For a good conductor the limiting factor is if • is large. The determining factor for a and á is,

• /Ew

Then if •¯Ew, then the respective surd term in a and á approximates to,

• /Ew

Therefore in the limit /Ew being large, we get the relationship,

Now let,

(note, this is not magnetic flux!)
So, for a good conductor � is small. If �<=1/50 then the approximation above is valid to between ñ1%. Then the material is classed as a good conductor.

e.g.

The ratio of � does have a physical meaning, it is shown as follows,

�=
=displacmenet current density/conduction current density
=(ÐD/Ðt)/•E
(J=•E)
=EwE/•E
=Ew/•

In a conducting media ÐD/Ðt is small, which leads � to be small as stated above.

Phase velocity of wave,
a)for a non-conducting medium,
b)for a conducting medium

We can see here that in a non-conducting medium the velocity is frequency indenpendant. It is then frequency dependant in a conducting medium, this phenonema is called dispersion, the different frequency components travel at different speeds in the medium.

Consider,

Now take integrals of both sides,

Therefore,

We now state,

The Poynting vector is,

For a conducting medium ...

Is defined as,

n=jwæ/(a+já)
=(jwæ/aý+áý).(a-já)
=(wæ/(aý+áý)).j(a-já)
=(wæ/(aý+áý)).(á+ja)

sqr((4*pi*10**-7)*(2*pi*10**10)/5.8e7)=0.0368961345533

sqr((4*pi*10**-7)/((1/(36*pi))*10**-9))=376.991118431

a is the rate of attenuation. For a good conductor,

We can rewrite our general form of H or E as,

Where small_delta is called the Skin Depth. Within a distance small_delta, the wave will decay by 1/e.
The magnitude of small_delta is,

We write about conductors only conducting within the skin depth and imperfect dielectrics from now on.

When a T.E.M. interacts with an ionised gas, it is mainly the E field responsible for the observed behaviour. The force acting on the electrons by a T.E.M. can be expressed as,

We make several assumptions to make our model;

In this model, the E field acting on the electrons is written as,

Integrating both sides of the above equation,

Now consider the current density,

J=•E
=N.e.x'
=Ne(eE/jwm)

N=number of charge carriers, per unit volume.

Therefore,

• =Neý/jwm
  =-jNeý/wm
• , the conductivity, is an imaginary number. Considering ohm's law, J=•E. If • is complex, then J and E are out of phase by pi/2 (looking on an Argan diagram, with J as real and E as imaginary!). It actually behaves like a pure inductance (by mathematical analogy).

Now consider,

the jwE term is a description equivalent to dielectric behaviour.

The refractive index is then written as,

Therefore, the refractive index is,

We alternatively write,

Where,

This is refered to as the Plasma frequency (or less comonly used, "Langmuire Frequency").

The refractive index can become imaginary now though. Consider the following cases

the two currents are equal in magnitude and out of phase by pi. In analogy to an LCR circuit we have resonance. A consideration of resonant behaviour is beyond the scope of this model.

Boundry conditions
The tangential component of electric field is continuous across a boundry. The Normal component of D field is countinuous across boundry.

Consider an upolarised beam incident on a mirror with reflected and refracted beams. The unpolarised incident beam is represented by two perpendicular polarisations one component is perpendicular to the plane of incidence the other is parallel.

(a) E perpendicular to plane of incidence
Plane of incidence (refracted and reflected in y=0). The boundary is in the plane z=0. We now equate the tangential componenets of E on each side of the boundary, at the boundary.

For the H vector of the TEM wave is,

For the refracted wave,

For the reflected wave,

Now equate the tangential components of the magnetic fields at the boundary,

We now need to eliminate the refracted (transmitted) component. Take equation (2),

Consider Snell's law,

and recall the following,

Then we get from equation (2),

The reflection coefficient (using intensities) is,

See Fig.1

From fig. 1,

Equate tangential components of the E and H fields, on each side of the boundry.

For tangential components of the H field

Therefore,

Therefore,

Therefore,

Therefore,

Therefore,

E is in the plane of incidence. The latter equation is known as the 2nd Fresnel equation.

Near normal incidence (i=0 deg), E''/E gives nearly 4%. Near grazing incidence (i=90 deg) E''/E is approxiamtley 1, i.e. nearly 100%.

If i+r=pi/2, then tan(i+r)=infinity, so our reflection coefficient goes to zero. The Brewster angle is the angle at which no relfection occurs.

If i+r=pi/2, then sin(r)=cos(i). Then by using Snell's law,

Upon inspection of the second Fresnel equation, if the angle of incidence is geater than the Brewster angle, the numerator is negative and there is a subsequent phase change of pi. If the angle of incidence is less than the Brewster angle, then there is no phase change (numerator stays positive).

If we now consider an TEM wave traveling from a dense to a less dense medium.

Let consider several cases now,

This is positive and hence no phase change. Now, for E parallel to plane of incidence,

This becomes negative, so we get a phase change of pi. Once i is greater than the Brewster angle tan(i+r) becomes negative too, so the ratio E''/E is positive. Therefore, there is no phase change on reflection.

From eq(1),

The problem now, which root to choose, positive or negative. If we take the negative sign, otherwise the wave in the upper (rare) medium grows in amplitude (which violates conservation laws). Use this expression i the two Fresnel equations: (viz:)
Case 1, E perpendicular to the plane of incidence,

So the 2• is the phase difference between the incident and reflected wave. So we can also write,

(ii) Where E is parallel to the plane of incidence,

Consider a beam passing from a dense medium to a rare medium. In our analysis so far we have assumed a 100% reflection of the E-field. However this is not physically sensible, there must be some finite (albeit tiny) distance over which the E-field attenuates to zero in the medium in which it emerges in. Recall the equation from much earlier,

We now write the refracted wave in the same form of the above equation,

v' is the velocity of the wave in the new (rare) medium. We can also expand the cosine function,

Therefore we write the refracted wave as,

The decay distance in systems like this is usually over several atomic distances. Hall Experiment - beam splitter !!! write up !

Recall the expression for a TEM,

We now rewrite this as,

Where,

We now write the reflection coefficient (at normal incidence) as,

Hence we can write the reflection coefficient,

So far we have only attributed dispersion to conducting media. However, we know from experiance that dispersion also occurs in dielectric materials. For example, when white light is shone through a triangular glass prism, the wavelength components are split up into a spectrum. For visible light, refractive index decreases with wavelength.

If we now turn our attention to the atoms at the boundary atoms of a material, the bound electrons on atoms. The refractive index curve is imitative of a resonance curve in an oscillatory system. The peaks of the graph

All these represent the bulk properties of the material, we now wish to make a molecular analysis (which applies to non-polar dielectrics, i.e. there is no polarization until an external field is applied, furthermore, there is no permanent dipole moment.) of such a system. We can analgously write,

Where N is the number of molecules.
The local field was analyised by Lorentz and the result obtained is shown below,

Now, assume the wavelength of the field (wave) is much greater than the inter molecular separation. Consider the equation of motion,

We already know that,

So substituting into the equation of motion,

P has the same frequency as incident wave but not neccessarily in phase. It is clear now that all of this is analogous to Forced, Damped, SHM.

Consider the incident field,

Therefore, the bulk polarization can be expressed as follows,

Now substitute into the equaiton of motion,

Therefore,

The final term in the denominator of the right hand side of the above equation is the Lorentz local field correction.

Albeit a misnomer, this is also the dielectric constant of the material. An additional relation that should be noted is,

Now let us now write,

We now write (by adding 3 to each side)

Now make the step,

So that we can now write,

We now hence write,

Also recal that the complex index is written as,

and using the binomial theorem we can write the approximate relation,

For more dense media however, we obtain the result,

If we look at two of the terms in the denominator, we make the combination

and make the quantity of an effective frequency,

Which shows how much and in what way the Lorentz local field effect affects the medium.

We are concerned for the moment with the efficiency of the scattering mechanism about the scattering centre. We assume for now that the total power scattered (or radiated) by a dipole is,

We also know that the dipole moment is,

So the oscillatory dipole moment is,

Therefore,

Hence we can rewrite the power scattered as,

For free space,

We now look at N, the mean incident power,

It can also be shown that,

Where a is the susceptibility. Therefore,

If we now look at the ratio W/N, we write this as,

We can see that • is fequency dependant, however we must see how a behaves with varying frequency. Recall the expression,

From the above, assume small damping (y=0) and that our medium is a weak non-polar gas, so there is no Lorentz effect, so we can write,

Note that we write P=N.p and P=XE for bulk quantities and p=aE for atomic quantities. Which leads us to stating a=X/N. So we can write,

So do we have • as a function of frequency (or wavelength) ?

Also recall that

We also have the relation,

Where u can be replaced by H,E,V or A.

That equation of field has no term for a source of field. If we assuime •=0 (dielectric), we can write the following,

These gives the sources of E and H. The general form is,

If Psi represents A, G is the current density. If Psi represents V, G is the charge density.

Solution of equation 1 to find V (or A) from a given distrobution of ç (or J).
If Psi is sinusoidal (and assuming G is the same),

v is the velocity of the wave from source to observer at P. This also assumes that there is a spherical wave front. Hence we can write particular solutions,

These are called the retarted potentials.

If we now look at a real example, e.g. short current element.

From fig. One,

If we want the field strength,

If we want to find the power of the field, using A, we know that,

The further evaluation of this all now depends on which coordinate system we use. Here we shall use a kind of cylindrical coordinate.

For A || to l and || to z-axis.
A is a function of r only (theta and phi do not appear).

From Fig 1,

We further write,

(The underlined terms represent unit vectors.

Thankfully, most of this reduces to zero!

Leaving us finally only with one non-zero component,
i.e.,

We expand this to,
(recall also, rý=çý+zý and ç=r.sin(theta)

Likewise,

If we look close to the source, where r is very small, we can rerwrite the above,

This is a form of the Biot Savart Law. For large values of r we have,

We can see that at near distances we simply have an oscillating field (called the induction field). At large distances however, we can readily see that the H field behaves like a progressive wave (it is called the Radiation field).

The corresponding electric field comes from the theta component. We know that,

Hence it (does!) follow that,

We can now also evaluate the radiated power (in reference to the poynting vector),

The total power is obtai.ned from integration over the surface of a sphere around the (anisotropic) radiation pattern. The total power can be expressed as,

Where the denominator gives the distrobution of total power. An aerial gain of 1.5 is characteristic of dipole radiaton.

Jenkins and White, fundamentals of geometery and optics