While I did succeed in creating a model that allowed the user to view the field at different speeds, it only managed to show the direction and relative magnitude compared to the rest of the current fields. If I were to continue with this project, I would attempt to improve it in several ways.

-First, I would try to find a way to make Mathematica account for the absolute magnitude of the fields so that the uniform fields would appear to change as the speed was varied. This might be accomplished by using equipotential surfaces rather than vector fields.

-Second, I would write code such that the user could simply input (or maybe copy and paste) a particular E-field equation for a rest frame, and the program would interpret the equation and produce the vector plots with sliders as shown in my model.

-Third, I would model more discrete situations rather than infinite distributions of charge or current so that the changes in field would be more visible.

-Fourth, I might consider a different type of plot, since Mathematica seems to deal with this kind of plot poorly. For instance, a static plot of E versus v and B versus v at a single point in space might better show the changing relative magnitudes of the fields.

That being said, this project was successful in several regards. It emphasizes the interplay of electric and magnetic behaviors when considered in moving reference frames, and at least in the case of the point charge, it allows for an examination of the changing shape of the fields at different speeds. It also sets up the framework for being able to model more complex systems by providing a proof-of-concept using simpler cases. Therefore, this project successfully modeled the fundamentals of relativistic electromagnetism, and provides a good foundation for a more in-depth exploration.

The next step of my project involved applying the modeling and manipulation code to different scenarios. In this case, I chose two scenarios that started as purely electric and developed magnetic components when considered in moving reference frames. The first was a continuous case- an infinite parallel-plate capacitor, and the second was a discrete case- a single point charge.

As has been well established, the field between two capacitor plates of charge density +/-σ is E=σ/2ε0. It is unsurprising then, that the E-field of a capacitor looks like this:

It is equally unsurprising that since there is no moving charge, there is no B-field. Then, when the system is considered in a reference frame moving at speed v relative to the capacitor and the transformation equations are applied, the following fields emerge for E…

and B…

These look pretty dull, but are a few things to note about these figures. The first is that Mathematica is not able to show absolute magnitude of the vector fields: just the relative magnitude. That is, the arrows are the correct size relative to the other arrows in the box, but they are not necessarily to scale when compared to other fields. This is why the plots appear unchanged when v changes: the values of the vectors are all changing, but all at the same rate, so the pictures do not change.

The second notable feature is that (without accounting for the actual magnitudes of the fields), it is possible to see that even though the E- and B-fields both change, the direction of the net force does not. For a positive test charge at rest in the reference frame moving in the x-direction, it would experience only an electrostatic force in the z-direction in the rest case. For the moving case though, it would experience an E-field still in the z-direction and a B-field in the y-direction, both of which would result in a net force that is still in the z-direction.

Despite the simplicity of a point charge, this third case is actually the most interesting to look at because it is not infinite, but discrete. Thanks to every intro E&M course we’ve ever taken, we know that the E-field created by a test charge is E=q/r²4πε_0. The following vector plot or something similar probably appears in every intro textbook currently on the market.

Pretty boring, right? Completely spherically symmetric and falls off with r² as expected. Now for the interesting ones: the fields in a moving reference frame. The E-field actually starts losing gaining so much of a y- and z-component that it appears to flatten out in the x-direction. The following figure is for v=0.75c

The B-field does something equally strange: it becomes somewhat cylindrical. However, in my code, this only appears at very high v (greater than 0.96c).

I get the feeling that if I could make mathematica show more detail, the field would not be in a perfect cylinder though. Since two rings of vectors appear on either side of the origin, it may be that the actual field has two lobes of some sort that are symmetric about the x-axis.

The last thing to note about this model is that when v=c, the simulation returns an error because it has to divide by zero when calculating gamma. Furthermore, the expression for gamma returns imaginary numbers for v>c, so this model is not useful for considering hypothetically how a system might behave if anything were able to move faster than the speed of light.

The mathematica worksheet associated with this can be found here.

It is fairly easy to see that the electric and magnetic fields of various systems change drastically when considered in different reference frames. For example, consider a wire with a line of positive charges moving to the right at speed v, and an equal line of charges moving to the left at speed v.

This system has a net charge of zero, so there should be no electric field. However, the sum of the charges does cause a total current I = 2λv to the right, which gives a magnetic field of B = μ₀λv/πs in the ϕ direction (as dictated by the right-hand rule), for any distance s from the center of the wire.

Now, consider the same situation, but in the reference frame where q is at rest. Suddenly, the positive linear charge is much smaller than the negative one, leaving a net negative charge on the wire, which will produce an electric field. Simply by changing the reference frame, the situation switched from a purely magnetic phenomenon to a combination of electric and magnetic.

However, rather than trying to analyze the magnitude of the new charge and current values to find E and B, it is possible to simply use the following transformation equations to find out what the new E and B fields are.

$E’_x$ = $E_x$
$E’_y$ = $\gamma$ ($E_y$ – $vB_z$)
$E’_z$ = $\gamma$ ($E_z$ – $vB_y$)
$B’_x$ = $B_x$
$B’_y$ = $\gamma$ ($B_y$ + $\frac{v}{c^2}$ $E_z$)
$B’_z$ = $\gamma$ ($B_z$ – $\frac{v}{c^2}$ $E_y$)

Where $\gamma$ = $\frac{1}{\sqrt{1-{\frac{v^2}{c^2}}}}}$ (From Griffiths Introduction to Electrodynamics, 3rd Ed. p531). The only hiccup is that the original setup is in cylindrical coordinates, while the transformation equations are in Cartesian. However, Mathematica can do this automatically, so I’m going to leave it in these terms.

B-field: E-field:

The Mathematica file contains 3D vector plots of the E and B fields for the rest frame and a frame moving at v=0.25c, but so far, I haven’t figured out how to make it so that the user can vary the speed of the reference frame. Hopefully, I will figure that out soon. The key thing to note is that the model shows the interplay of E and B fields at relativistic speeds: the scenario starts out completely magnetic and ends up a mix of the two.