## Electromagnetism in 2d and 4d

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

### Electromagnetism in 2d and 4d

In 3d when there is an electric current there is a magnetic field around it. The magnetic field around an electric current has an angular direction. If an electric current goes around in circles then there will be two magnetic poles perpendicular to the direction of the electric current. Electromagnetic Radiation has an electric field pointing in one direction, a magnetic field pointing in a direction perpendicular to the electric field, and the direction of movement is perpendicular to both the electric and magnetic fields.

In 2d if there was an electric current there would be no angular direction perpendicular to it so there would be no direction for a magnetic field. I was thinking that in 2d the equivalent of Electromagnetic Radiation would have only an electric field with the direction of movement perpendicular to the electric field.

In 4d if there was an electric current then it would be possible for there to be a sphere perpendicular to it and around it so I was thinking that in 4d the magnetic field around a current would cancel it self out so that there would be no magnetic field. If there was a charge with a double rotation in 4d then there would be no direction perpendicular to the double rotation besides outward so a rotating charge in 4d with a double rotation would have no magnetic field. I was thinking that in 4d there would be two equivalents of Electromagnetic Radiation. One kind would have an electric field that would point in all directions perpendicular to the direction of motion while the other would have an electric field that would point in one direction with the direction of motion perpendicular to the electric field and with a magnetic field pointing in all directions perpendicular to both the electric field and the direction of motion. Charged particles with a double rotation would produce the first type of radiation, while charged particles with a single rotation would produce the second type of radiation.
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anderscolingustafson
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### Re: Electromagnetism in 2d and 4d

2D beings would be unfamiliar with magnetic field, from what you said about electromagnetism in 2D. Similarly, there might be some other type of field which we 3D beings are unfamiliar with which exists in the radiation in tetraspace. If the wave propagates in the W direction, we might have electric field along X direction, magnetic field along Y direction and the third unknown type of field along Z direction.
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Prashantkrishnan
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### Re: Electromagnetism in 2d and 4d

I have been spending a good deal of time of late, trying to grapple with electromagnetism and gravity in 3D.

Mostly, i have been following H A Lorentz's "theory of electrons" as an inspiration. An electron here is what we might called a charged particle.

You start off with equations like F = qE, D = qS, and D=eE for a scalar charge, and F = p×B, B=µH, H= p×E for a vector force. Lorentz' force is then F=mA = qE+p×B. Since we can for a particle, write p=qv, we get q/m = A/(E+v×B). One can apparently derive maxwell's equations from jeffimenko's equations, from the equation of continuity, by using retarded potentials. S is a radiant vector corresponding to 1/4pr² for point charge.

The underlying geometry is E3J, which is the geometry of the Minkowski space of special relativity, and one uses things like Heaviside's proof of the co-existance of a scalar and vector field if the field travels at a finite speed, say, c. This comes from the derived equation eµc²=1

You can then write the constitute equations as cD = eE, and cB = µH, whence eµ=1, ie e=1/µ.

Using what is known of gravitation and cogravitation, there is an analogue that corresponds to all of the above, so we might suppose that c is a conversion factorr of space-time, and that e is a property of electromagnetism, and there is a corresponding g that exists for graviity.

One then supposes that S and c arise from the nature of geometry, and that e and g are properties of electricity and gravity respectively. Further, that any radiant field will produce D and H like fields, but it is the conversion into E and B which makes them into charge-like fields.

We note that something like × is a feature of 3d, and that it does not translate easily to 4D (although it does not mean that it is meaningless there), and that the equation $$\beta = c/\sqrt(c^2-v^2)$$ is a feature of the J linkage of space and time. It then depends on whether Jefimenko's equations have any real meaning in 4D. These include retarded potential terms, and can be written, eg as D = q(1+d)S - dS.p/c, and H = p(1+d)×S. When c is infinite, these become the definitions above.

The coulomb law is then F=c Qq / 4pe R², and newton's gravity is F = (c/4pg) Mm/r², where p=pi. or F = cQq s / e, and F = p×(P×S) / ec.

One could suppose magnetism in 4D is F = V(p, P, S)/ec, where V() is a triple-vector cross product, and the field is not simply a vector, but a tensor. It would mean though, that ampere's law would not work.
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wendy
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### Re: Electromagnetism in 2d and 4d

Would this help as a launching point? Bucky Fuller's 3D electromagnetic field geometry. http://marvinsolit.site.aplus.net/pgs/g ... ometry.htm
Grayham
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### Re: Electromagnetism in 2d and 4d

I'm currently following some developments i have made on Lorentz's theory of electrons, and Heaviside's gravitoelectromagnetism. The idea from these is that the bulk of the field equations come from geometry (ie E3T = newtonian space-time, E3J includes space-time conversions like special relativity.). Heras has given calculations that retarded potentials + continuity -> jefimenko equations, which solve maxwell's equations. This means that the bulk of em theory is not held in the nature of electricity but of geometry, and that the equations of consist (eg cD= kE, and H = ckB ) suffice to produce EM fields from a single k.

At the moment, the most interesting implementation of the cross product is V(p,q,s), where p and q are the intensities of the vector charges, and s is directed from p to q. This would then mean a cyclic permutation of these vectors give the same value, ie V(p,q,s)= V(q,s,p), and because the radial vector is measured from p to q, V(p,q,s) = V(q,p,-s). This means, for example, H is a dyadic or order 2 tensor, and D is an order one tensor (or vector).

When p and q are parallel or anti-parallel, or in the directions of the vectors p,q, the vector product is zero. This means, for example, that ampere's law would not work in 4D, since this requires the p and p'' tp be parallel. Where P is a coil around p, then one would get a force perpendicular to both of these.

It means that we do have some sort of basis for saying that curl works in 4D, but it needs three vectors to produce an output vector. I'm still uncertain how one might do \nabla×\nabla×, though. There are still more questions than answers.
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wendy
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### Re: Electromagnetism in 2d and 4d

I was thinking about electromagnetism in 5d and thought about how in 5d if there was an electric current in 5d then there would be a glome around it and perpendicular to it. A glome can have two circles on its surface that are perpendicular to each other that share the same center and in which every point on each circle is of equal distance from every point on the other circle so in 5d the magnetic field around a moving current could have two independent directions of orientation. If there was an object with an electrical charge that had a double rotation in 5d then there would be one direction perpendicular to the direction of rotation so meaning the magnetic field could have a direction and an object with an electrical charge and double rotation could have a north and south magnetic pole.

I was thinking that in 5d as in 4d there would be two equivalents of electromagnetic radiation. One would have an electric field pointing in one direction a direction of motion and a magnetic field pointing outward in all directions perpendicular to the direction of motion and the electric field. The other would have a magnetic field pointing in one direction a direction of motion and an electric field pointing outward in every direction perpendicular to the direction of motion and the electric field.
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anderscolingustafson
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### Re: Electromagnetism in 2d and 4d

The idea of 5D occurred to me too, but you can't create a force based on that sphere as far as i know. Something might be possible, but i have not wrapped my mind around it.

In four dimensions it is relatively easy. Suppose P is a moving charge. As in three dimensions, the magnetic field is H = P×S, where S is a radiant vector r/k(r^n). But there is no cross product in 4D, so instead of H being a vector, one supposes that S can be represented by a normal vector to a glome around P. Because there is no × product, we slice the glome into sphere-slices perpendicular to the direction of velocity. You then get v×S as a vector that is a circulation around the radial to the slice of S in that slice, and the intensity, is as in 3D, qv/r^3, But it points in every direction around this sphere.

When a second moving charge p strikes B=µH, it creates a force F = p×B. where because B is acting around the perpendiculars to the sphere-slice derived in the previous paragraph, the force by B on p, is to turn the vector p around in the hedrix (tangent to the sphere), by 90 degrees, and reduce its intensity in the manner of a sine (i think that's what cross products do in 3d).

This means that we can write F = p×P×S / µ, which gives the determinate of a matrix, formed by rows F = (P, p, S, x), where S is the inverse radial vector, and x represents a vector whose elements are units (w, x, y, z), as one writes in the cross product in 3D.

When one reverses P and p, then S is reversed too, so one gets F = (p, P, -S, x), but this means that magnetism by this model must give the same force on each element, including direction.

Will need to play around a tad more.
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wendy
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### Re: Electromagnetism in 2d and 4d

In 4d the magnetic field isnt a vector field. Its a bivector field.
granpa
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### Re: Electromagnetism in 2d and 4d

Great thread on electromagnetism. It is very useful knowledge.
neevamerk
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