Ampere’s law

To find the magnetic vector potential of a rotating, uniformly charged spherical shell. Vector potential, magnetic field intensity (mag induction) and current density are the most basic parameters of magnetostatics.

Magnetic vector potential of a rotating uniformly charged shell.

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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.

A cylindrical wire carrying current I. Two situations arise. Case A: uniform surface current on the outside surface. Case B: a volume current density that depends upon radial distance from axis of wire. Whats the magnetic field of such a configuration?

Problem 5.13 Application of Ampere’s Law.

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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.

Ampere’s law application: infinite surface current, diagram. An infinite sheet of current in xy plane with Amperian loop. We like to determine its magnetic field by Ampere's Law.

Example 5.7; Application of Ampere’s law.

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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. 

Aurora Borealis or Northern Lights at Ireland. Photo Credit; www.slate.com

The Maxwell’s equations, from nature to instruments.

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The beauty of Maxwell’s equations can be seen in how it helps us understand nature as well as instruments, at the same time. Medical devices are simply an advanced understanding that began with understanding electromagnetic waves through Maxwell’s equations.

Each of the following 4 equations has a different name, by which we call’em, but together they are called as the Maxwell’s equations. Together they constitute what I am inspired to say; the golden equations of Physics. If we do some easy tricks they will be converted into whats called as the Wave Equations (of motion) ! Yes, they describe the wave behavior “fully”. — By waves I don’t mean sound waves, but any sort of waves that move at the speed of light. Sound waves are ordinary pressure oscillations, that travel much slower than even rockets.

The 4 equations therefore describe how electromagnetic waves are created and broadcast. Hence TV radio and satellite communication were understood because these 4 equations were understood.

First two are time-independent or static equations.

Gauss law of electrostatics 
The first equation is known as Gauss law of electrostatics, it says “Electric fields (E) are a result of sources of electrostatic charges”.
Gauss law of magnetostatics
The second equation is known as Gauss law of static magnetic field ( or magnetostatic field ) it says “apparently there are no sources of magneto-static charge or single magnetic pole from which the magnetic field B is created”.
Then how are magnetic fields created? We needed to know further to find the answer. Lets look at the 3rd and 4th equations.

The last two equations are time-dependent, time varying or dynamic equations. Which is why sometimes we see electro and magneto statics and some times we see electrodynamics and magneto-dynamics or simply electrodynamics, in nomenclature of these fields of studies.

Ampere’s circuital law
The 3rd equation is called as Ampere’s circuital law. Its also whats known as Faraday’s law of electromagnetic induction and Lenz’s law. It says changing magnetic fields can produce electric fields. Not only electrostatic charge but changing magnetic fields as well produce electric fields. Although we don’t know yet how the magnetic fields were produced.
Modified Ampere Circuital law
The last equation is called as Modified Ampere Circuital law  or Ampere-Maxwell law.This provides the remaining links in the understanding of Electromagnetic field’s creation and motion. It says magnetic fields are produced by actual electric currents (I) that is; change of electrostatic charges over time produces magnetic fields. Magnetic fields are also produced by changing electrostatic fields, this type of pseudo current is called as Displacement current.