An Electromagnetic Field Tensor

 From a mathematical perspective electromagnetic theory consists of laws of nature expressed in terms of vector analysis.

But what would we see if we looked at EM theory from a more physical perspective focusing on the forces and changes in momentum instead? We can start with the Lorentz force which involves electric and magnetic fields. The electric force produces changes in the motion of a charged particle in the direction of the electric field. The magnetic force involves changes in position and gives the deflection of the path of motion. So for differential changes in time and position we can associate changes in momentum or impulses acting on the particle.

It turns out that the fields are the coefficients of the differential changes in position and time that give the changes in momentum. We can add the change in energy or work done on the particle to the momentum changes to get a 4-dimension picture of what's happening. And we end up with a field tensor, the result of applying the chain rule to each component of the momentum. So the field tensor can be expressed in terms of 4-dimensional gradients of the components of a momentum flow field describing the paths of identical test particles in neighboring positions. This appears to have been the approach that Maxwell adopted for EM theory.

A Short Timeline for Early Electromagnetic Theory

  The introduction to electromagnetic theory usually involves mathematical statements of the physical laws governing the relations between forces, charges and their motion and the more abstract fields. Here are some Wikipedia articles dealing with some of the more important ones.

1785 Coulomb's law

1813 Gauss' Law

1820 Biot-Savart law

1820 Oersted's law

1823 Ampere's force law

1831 Faraday's law of induction

1855 Ampere's circuital law (Maxwell)

In the twenty years between 1855 and 1873 Maxwell wrote a number of works on electricity and magnetism attempting to develop mathematically Faraday's impressionistic lines of force approach to the subject.

1855 Maxwell, On Faraday's Lines of Force

1861 Maxwell, On Physical Lines of Force

1865 Maxwell, A Dynamical Theory of the Electromagnetic Field

1873 Maxwell, A Treatise on Electricity & Magnetism Vol 1, Vol 2

Part III of A Dynamical Theory of the Electromagnetic Field deals with the General Equations of the Electromagnetic Field in which Maxwell introduces such concepts as "electromotive force," "electromagnetic momentum," "magnetic force" and "electromotive force in a circuit." These deal with force fields rather than electric and magnetic fields and his approach appears to be less abstract than our modern theory of the subject. One can compare Maxwell's formulas with those in modern notation.

Why the Assumption of an Impulse Acting on Light May Have Worked

  If one looks at the change in phase in the plane of refraction along the boundary between the two media one sees that the two components of the wavevector are the same for both wave functions thus enabling us to match them along this line.

When we tried to explain refraction by the assumption of a vertical impulse acting on photons at the boundary we inadvertently assumed that the component of the wavevector on the boundary in the plane of refraction remained unchanged since the momentum of the photon is proportional to its wavevector.

A Physical Explanation of Refraction

  One can trace Snell's law of refraction to the boundary conditions acting on the electromagnetic wave equations for the electromagnetic fields. To show this we can assume that the incident, reflected and refracted rays are simple plane waves.

In what follows let ψ represent an arbitrary electric or magnetic field component. In the first medium the field is a sum of the incident and reflected fields and there is only one field in the second medium. The relation between the angular frequency, wavelength and phase velocity, ω=kv_ph=kc/n allows us to express the wave vectors for each medium in terms of the magnitude, k=nω/c, of the corresponding wave vector in a vacuum. We assume the plane of incidence is the x,y-plane with horizontal axis x and vertical axis y. Along the boundary between the media y=0. For reflection we know that the reflected angle is equal to the angle of incidence so we can set β equal to α. This allows us to simplify the field equation for the first medium slightly. If a field is continuous along the boundary then its derivative will also be the same for both media.

Equating these functions we get two equations for the wave amplitudes A_k and the second equation yields Snell's law after eliminating the common factors.

So the continuity of the fields appears to be the physical cause for the "broken" or refracted light path. The wave vectors do not depend on the values for the incident fields but we can expect the amplitudes of the reflected and refracted rays to depend on them.

Interpreting Snell's Law of Refraction

  Back to Hamilton's Theory of Systems of Rays. How does one explain Snell's law of refraction where the index of refraction n=sini/sinr? Mathematically it's just a description of the relation between the paths of the incident ray and the refracted ray. Is there anything significant about the length of the lines corresponding to the sines? They're the altitudes of the triangles with vertices i, n̂₁, and the origin and that for r, n̂₂ and the origin but the ratio of the areas of these triangles is also equal to the index of refraction.

These triangles are isosceles so we could draw the altitude as perpendicular to the rays instead.

What we want is some sort of physical explanation for the law of refraction. We can find this in Feynman's Lectures on Physics, Vol I where the index of refraction is attributed to a phase change due to secondary waves caused by forced oscillations of the electrons in a plate of the transparent body and not a change in the speed of light in the refracting medium. The derivation assumed that the index of refraction differed by a small amount from that of a vacuum which is 1. The relation for a denser medium is found in Vol II. This explanation is essentially that of the classical dispersion theory introduced by Paul Drude a little over a century ago. A formula for normal dispersion can be found in his Lehrbuch der Optik of 1900.

Equivalent Quaternion Simplifies Calculations

Using an equivalent rotation quaternion requires less calculation to obtain the same results. In this slightly modified version of the previous rotation one just needs to keep track of the poles, their rotation quaternions and changes to the required data.

After computing the equivalent rotation quaternion we can rotate the data points for the vertices of the tetrahedron.

In the plot below the color code red, green, blue indicates the vertices a, b and c and the axes î, ĵ, k̂ respectively. The fourth vertex was originally the origin before it was translated to the center of the tetrahedron.

In Excel the worksheet is automatically recalculated when the contents of a cell is changed so when the index is changed by pressing either the shift right or shift left command button the plot is also recalculated and we can observe the resulting rotation.

Note one needs to be careful not to confuse the axes used to determine a rotation with axes that are rotated which are treated as data. Here the original î and k̂ axes were used to determine the pole p̂' while the rotated k̂'' axis was used for p̂''.

Doing Stepped Rotations in Excel

  I've been trying to get Excel to do some simple rotation videos using command buttons to step a plot through the rotations and Power Point's record tool to capture a video. This is the best system that I have been able to come up with so far.

The image in the Blogger player is a little initially but clicking on it clears things up considerably. It doesn't appear that QGraphics is ready quite yet.