So, if you ever want to undertake a physics major at Monash University, you will probably want to learn some theoretical physics. Although, the title will lure you, you will enviably have to face the one and only Csaba Balazs, figure 1 below.

Figure 1: Csaba

Now, the man is a bit of a legend and somewhat of a god, but be warned, passing his class will be one of the hardest things you will ever have to do. So I am here to elucidate to you, all the key concepts and ideas you will need to get through the electrodynamics part of the course. I, am your new hope.

Most of the course deals with Maxwells Equations and then applying them to matter, so that’s where we begin, after which we will then move onto electromagnetic waves (a future post).

Now as we go through all this, it is important that you have a picture or mind map of all these things that we are going to talk about, and slowly add to it as we go on. But, before we can do any of that, we have to get through some preliminary equations.

Preliminary

The current density is given by

    \[\vec{j}=\sigma \vec{E}\]

and the Lorentz force is

    \[\vec{F}=q(\vec{v}\times\vec{B}+\vec{E}).\]

This current density vec{j} is nothing to be afraid off, it simply the amount of current that is moving per unit area. So all you need to do to find the actual current is perform an integral over the area where there is current density, j, and wallah, you have current ready to go.

Maxwell’s Equations in a Vacuum

Gauss’ Law: Electric charges produce an electric field

    \[\nabla.\vec{E}=\frac{\rho}{\epsilon_0}\]

Another Law: Magnetic charges do not exist

    \[\nabla.\vec{B}=0\]

Faraday’s Law: changing magnetic field produces an electric field

    \[\nabla \times \vec{E}=-\partial_t\vec{B}\]

Ampere-Maxwell’s Law: A changing electric field induces a magnetic field 

    \[\nabla \time \vec{B}=\mu_0\vec{j}+\epsilon_0\mu_0\partial_t\vec{E}\]

One final thing to note about the meaning of these equations is that the electric field and the magnetic field are all due to some sort of charge or current, i.e. the source of these fields are electric currents and charges. This is the picture which you should have in your mind, electric charges and currents are the sources of electric and magnetic fields. Similarly. the Lorentz force law can be re-written as

    \[\vec{f}=\rho\vec{E}+\vec{j}\times\vec{B}\]

and hence, these fields exert forces on their sources.

Maxwells Equations in Matter

Here we need to add some detail to our picture, specifically, what happens to our fields when we hit matter and vice versa. We will first deal with electric fields.

If a material is placed in an electric field it becomes polarised which leads to a net charge on the surface, and in some cases where the polarisation is not uniform, there will be accumulations of charge inside the material itself. We call these accumulations of charge due to polarisation bound charge, the one inside the material is known as the volume bound charge and the one on the surface of the material is known as the surface bound charge. Take a moment to properly grasp this effect of polarisation, and add it to the picture you’re building up in your mind, because once you do, the rest is just book keeping.

The book keeping.

Polarisation actually occurs to each individual atom in the material, separating the negative and positive parts creating a dipole, each one has it’s own dipole moment (p=qd, this is just details). Since this happens over the entire volume of the material, we define

    \[\vec{P} = \text{dipole moment per unit volume}\]

and call \vec{P} the polarisation. Now, since the bound charge arises from this polarisation, you’d assume we would be able to relate the two, and you are correct.

    \[\sigma_b=\vec{P}.\vec{n}\]

where \vec{n} is a unit vector perpendicular to the surface. Similarly for the volume

    \[\rho_b=-\nabla.\vec{P}.\]

How will this effect Gauss’ Law? We have to remember that these only came about due to some external electric field, the electric field must’ve come from some other charge configuration placed near the material. This charge configuration we call the free charge. It is the free charge that polarises the material, it displaces the electric charges in the material. For this reason we call the field due to the free charge the electric displacement, D,

    \[\nabla.\vec{D}=\rho_f.\]

Finally, if you actually want the electric field, it’s pretty simple, all it is, is Gauss’ law but taking into account all the charge present, free and bound,

    \[\nabla.\vec{E}=\frac{1}{\epsilon_0}(\rho_f+\rho_b).\]

So, the picture in your head should now contain what happens to electric fields in matter. Now we need to add in a little bit on linear dielectrics, but this too is really only a footnote in the mental picture you’re creating. The footnote is this:

In some materials, the polarisation \vec{P} is linearly proportional to the electric field \vec{E},

    \[\vec{P}=\epsilon_0\chi_e\vec{E}\]

where \chi_e is the electric susceptibility of the material. Note that we define the permittivity of the di-electric as

    \[\epsilon=\epsilon_0(1+\chi_e).\]

That is pretty much it. You can also derive, if you wish, the following relationships that hold in linear di-electrics,

    \[\vec{D}=\epsilon_0(1+\chi_e)\vec{E}\]

and

    \[\rho_b=-\frac{\epsilon_0}{\epsilon}\chi_e\rho_f.\]

Now to add magnetic fields in matter to this picture. The good news is that it is pretty much exactly the same as the above. In the presence of a magnetic field, a material becomes magnetised, which we quantify by the magnetisation, \vec{M},

    \[\vec{M} =\text{magnetic dipole per unit volume}.\]

This magnetisation is manifested through bound currents, as each magnetic dipole is really a small current loop, there will be current loops joining on the surface of the material to form the bound surface current,

    \[\vec{K}_b=\vec{M}\times\vec{n}\]

and in non-uniform magnetisation there is also bound volume current within the material

    \[\vec{J}_b=\nabla\times\vec{M}.\]

There is also a free current present, which is there because someone has set up this current, by applying a potential difference. This is opposed to the idea of the bound current which is there due to magnetisation. Now, Maxwells equations still hold applied to the entire current, specifically amperes law is

    \[\frac{1}{\mu_0}\nabla\times\vec{B}=\vec{J}_f+\vec{J}_b=\vec{J}_f+\nabla\times\vec{M}\]

and from this we see that

    \[\nabla\times(\frac{1}{\mu_0}\vec{B}-\vec{M})=\vec{J}_f.\]

The book keeping component of this section is that we define the auxiliary magnetic field,

    \[\vec{H}=\frac{1}{\mu_0}\vec{B}-\vec{M}\]

and so in a medium, amperes law becomes

    \[\nabla\times\vec{H}=\vec{J}_f.\]

Note that \vec{H} is analogous the \vec{D} field. Now to add in the linear medium, \vec{M} is linearly proportional to \vec{H}

    \[\vec{M}=\chi_m\vec{H}\]

where \chi_m is the magnetic susceptibility of the material. You can also easily derive that

    \[\vec{B}=\mu\vec{H}\]

where \mu=\mu_0(1+\chi_m) is the permeability of the material.

We are one step away from writing out Maxwells Equations in a medium, we just need to figure out what we are going to do with the non-static case of the Maxwell-ampere law.

If there is any change in the electric polarisation there will be an extra current term, the polarisation current, which will create a magnetic field.

    \[\vec{J}_p=\patial_t\vec{P}.\]

So all up, the current can be written as

    \[\vec{J}=\vec{J}_f+\vec{J}_b+\vec{J}_p=\vec{J}_f+\nabla\times\vec{M}+\partial_t\vec{P}.\]

Substituting this into Maxwell’s ampere law we get

    \[\nabla \time \vec{B}=\mu_0(\vec{J}_f+\nabla\times\vec{M}+\partial_t\vec{P})+\epsilon_0\mu_0\partial_t\vec{E}\]

which can be re-arranged to give

    \[\nabla\times\vec{H}=\vec{J}_f+\partial_t\vec{D}.\]

So now that we know this, we can write out our Maxwell Equations in a  medium:

\nabla.\vec{D}=\rho_f

\nabla.\vec{B}=0

\nabla\times\vec{E}=-\partial_t\vec{B}

\nabla\times\vec{H}=\vec{J}_f+\partial_t\vec{D}.

So, I’m going to stop here, stay tuned though, next time we will talk about energy and capacitors, and probably do some nice derivations, which, admittedly, this post lacked.

Free WordPress Themes
%d bloggers like this: