Sunday, December 25, 2022

Maxwell Had 22 Quaternions - Heaviside Mangled And Crippled These Into 4 Classical Equations...,

In this video, we're looking at how there are two sides to every Maxwell, equation, and therefore there are two ways of understanding each of Maxwell's equations.

Maxwell's equations of electromagnetism fall under the umbrella of classical physics, [NO THEY DO NOT!!!!] and describe how electric and magnetic fields are allowed to behave within our universe (assuming the equations are correct of course). Electric and magnetic fields show how electrically charged, and magnetic objects respectively, exert forces on each other.

Each of Maxwell's equations is a differential equation that can be written in one of two forms - the differential form, and the integral form. In this video, we look at two of these equations, and how each of them has two variations. We begin by studying the first Maxwell equation, which says (in the differential form) that the divergence of any magnetic field is always equal to zero.

The physical interpretation of the above statement is that if we consider any closed volume of space, the net magnetic field passing either in or out of the region must always be zero. We can never have a scenario where more magnetic field enters or leaves any closed region of space. The divergence of the magnetic field simply measures how much field is entering or leaving the volume overall. And this must be equal to zero.

Conversely, this same equation can be written in integral from (i.e. from a slightly different perspective). The integral equation says that the integral of B.dS is equal to zero. B is once again the magnetic field, and dS is a small element of the surface surrounding the volume discussed above. This method breaks up the outer surface covering the volume into very small pieces, counts the amount of magnetic field passing the surface element, and then adds up the contributions from all the elements making up the surface. This addition of contributions is given by the surface integral over the closed surface. In other words, the integral form of this Maxwell equation states the same thing as the differential form but looks at it from a slightly different perspective. Note: the integral must be a closed integral i.e. there should be no holes or breaks in the surface.

We also see a similar sort of thing with the second Maxwell equation, which looks at the behavior of electric fields. The differential form states that the divergence of the electric field is equal to a charge density divided by epsilon nought, the permittivity of free space. This therefore says that for any closed volume, the net amount of field entering or leaving the volume is directly related to the density of charge enclosed within the volume. Therefore if the net charge in the volume is zero, then the net field entering or leaving it is also zero. If the net charge is positive, the divergence is greater than zero, and if the net charge is negative, the divergence is less than zero.

The integral equation states that the sum of the electric field contributions to each of the small elements making up the area surrounding the volume is equal to the total charge enclosed within the surface, divided by epsilon nought. So once again this is looking at the same scenario from a slightly different perspective.

Each Maxwell equation has these two ways of writing it, and one can easily convert from the differential form to the integral form if one knows differential calculus. It is generally simple to move between these forms, and we can use whichever one is mathematically most convenient to us at any given time.


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