Magnetic vector potential of a rotating uniformly charged shell.

Today we will solve the problem of finding magnetic vector potential of a rotating, uniformly charged spherical shell. We won’t discuss the general idea behind the vector potential (how it follows from Helmholtz theorem, and gauge freedom etc) and how its defined. That will be part of a conceptual lecture and will be available when the same would be created. The offline version is available, but the web version will call for a special priority to be assigned.

The problem is quite well defined. We just need to follow the straightforward method of implementing the basic definitions and carrying out the required steps. But we need to be mindful of the framework in which we need to accomplish these steps.

The framework I am talking about here is the coordinate system we need to set in order to solve the problem. Notice that the problem has been stated in the spherical coordinate system (which has been discussed couple of times in this website). But we need not worry about all the aspects of this coordinate system, we will only pick on those which are immediately applicable to our problem.

While this choice of the system where the polar axis (z-axis, wrt which the polar angle θ is measured in a r, θ, φ spherical coordinate system) coincides with the angular velocity vector ω is very natural, it isn’t the most convenient for carrying out the ensuing integral for the vector potential A.

Problem 5.13 Application of Ampere’s Law.

Yesterday we saw an interesting application of the Ampere’s Law (– in magnetostatics and sometimes called Ampere’s circuital law also) for the infinite uniform surface current. Today we will see yet another display of the elegance and efficacy of this law in the following problem. This problem is inherited from Griffith’s text on Electrodynamics (3rd edition)

I have tried to be a bit more explanatory than the basic solution available (in instruction manual, if you have a copy). Thats the whole idea of this labor I have taken up. I also strongly suggest anyone who want to sharpen his saber to try the problem on his/her own effort before looking into the solution. That way one can prepare oneself for the pitfalls of one’s own understanding before taking up help and damaging the opportunity of developing of a better sense of solving such problems.

A steady current I flows down a cylindrical wire of radius a. What would be the magnetic field outside the wire and inside of it? We need to find the same in two different scenarios given.

Here are the two different scenarios.

A. Its a surface current density on the outside surface and its uniform across the surface.

B. Its a volume current density and its distributed in the volume of the  wire, but this time its not uniform. In-fact the volume current density J is directly proportional to s; the distance from the axis of the wire where we are referring the value of J.

Example 5.7; Application of Ampere’s law.

The following problem is an interesting application of Ampere’s law apart from usual applications found in honors syllabus (eg infinite straight conductor, Solenoid and Torroid). This is to be found the excellent book by Griffith on Electrodynamics. 

Find the magnetic field of an infinite uniform surface current K (vect) = K i-cap, flowing over the XY-plane.

Lets first visualize the problem. This will help us solve the problem. We chose a Cartesian coordinate system as shown. Our infinite surface current is a sheet that is concurrent with the XY-plane. We also show the Ampere loop which is a rectangle of length l parallel to the y-axis. This loop is half above the XY-plane and half below. 

Maxwell Boltzmann distribution for a classical ideal gas

i. We assume a dilute gas which is enclosed by a thermally insulated container on all sides.

Dilute gas in a thermally insulated container: Dilute means concentration of gas molecules is low. Insulated implies there is no reasonable flow of heat energy across the walls of the container.

ii. Each molecule is assumed to be a hard sphere which moves randomly in all directions such that its velocity vary from – infinity to + infinity.

Hard sphere: Remember a hard sphere is a classical analogy of a rigid sphere whose surfaces do not deform when an external object comes into contact. This essentially means the incoming object is scattered elastically that is without loss of kinetic energy, only momenta magnitude and directions are changed in accordance with the conservation of linear momentum.

iii. When molecules collide they do not lose energy or time. They bounce off each other so that ‘energy’ and ‘momenta’ are conserved.

We would like to obtain an expression for velocity distribution function. That is we would like to know the fraction of molecules having velocity between v to v+dv for all possible values of velocity.

For this we assume an ensemble of molecules in equilibrium. The ratio of number of molecules in a velocity range to the total number of molecules N gives the probability of finding a molecule in that velocity range.

v. The “phase space” of the ensemble of molecules is defined by a 6 N dimensional space, which constitutes of 3 N spatial components and 3 N velocity components of the N molecules in equilibrium. For a more advanced concept of phase space check the following statistical mechanics lecture.

Helmholtz theorem in electrodynamics, Gauge transformation.

Electromagnetic theory, lecture — IV

Topics covered in this lecture

a. Helmholtz theorem — in electrodynamics

b. Gauge transformation — of scalar and vector potential in electrodynamics

c. Coulomb and Lorentz gauge

All electromagnetic theory lectures of this series, will be found here (https://mdashf.org/category/electromagnetic-theory/)

In our previous lecture — lecture — III, we discussed in quite detail, the problem of electrostatics and magneto-statics.

We understood how deeply the Helmholtz theorems formulate the entire question of these two branches of electromagnetic phenomena.

But static problems are not sufficient for any rigorous treatment of the electromagnetic theory.

We promised in that lecture to study how Helmholtz theorems lend their magical power to understand the most general nature of electromagnetic phenomena.

In this lecture we will study precisely the applicability of Helmholtz theorems to the problem of electrodynamics and we will see how it leads to a great deal of success in advancing the ability to solve electromagnetic problems of a great variety.