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Maxwell Equations in N-D

- ICEAA 2013,
- September 9-13, Turin, Italy

Author: Carlo Andrea Gonano

Co-author: Prof. Riccardo Enrico Zich

Politecnico di Milano, Italy

“Classic” Maxwell equations

Magnetic monopoles and currents

The beautiful Dirac’s Symmetrization

Cross product’s difficulties and N-D extension

Curl in 3-D & N-D extension

Revising magnetic field and monopoles

Symmetrized Maxwell eq.s in N-D

Remarks and conclusions

Questions and Extras

C.A. Gonano, R.E. Zich

- In differential form Maxwell Equations can be written as:

for electric field E

for magnetic field B

- Divergence and curl equations for both E and B

IT LOOKS AS A QUITE SYMMETRIC SET, BUT…

No “magnetic charge”and no “magnetic current”!

- Introducing magnetic monopoles and currents, “symmetric” Maxwell Eq.s would look:

C.A. Gonano, R.E. Zich

BUT WHAT IS A MAGNETIC MONOPOLE?

- Let consider field E: its “monopòles” are the electric charges, isolable and observable

- A magnet generates a field B and its poles are called North and South: can they be isolated?

- Problem: breaking a magnet you will not obtain two magnetic monopòles, but two magnets!

- This difference between fields E and B has be known for a long time…

HOWEVER, WHY THE DIVERGENCE OF B

SHOULD BE ALWAYS ZERO ?

C.A. Gonano, R.E. Zich

- In 1931 Paul A. Dirac, starting from a quantistic approach,symmetrizes Maxwell eq.s adding magneticmonopòles and currents

Symmetrized Maxwell Equations

SI units,

[rm]= [C/(m/s) ] convention

The generalized EM force per unit of volume is:

In absence of magneticmonopòles and currents we get back the “classic” Maxwell eq.s and EM force

C.A. Gonano, R.E. Zich

- The set of eq.s can be written in a more compact form, defining arrays:

anti-symm. matrix

field array

charge array

flux array

nabla array

- Symmetrized Maxwell Equations will look so:

divergence eq.s

curl eq.s

- The generalized EM force per unit of volume will be:

ACTUALLY AN ELEGANT SYMMETRIC FORMULATION…

C.A. Gonano, R.E. Zich

- The set of eq.s by Dirac isinvariantunder the duality transformation

field array

charge array

flux array

- Symmetrized Maxwell Equations are left unchanged in form and the generalized EM force is exactly the same:

C.A. Gonano, R.E. Zich

In Dirac’s formulation the E and B fields are treated in the same way, so

some consequences arise…

- Conservation for electricand magnetic charge

figure from Wikipedia

- As a moving electric charge induces a rotational Bfield, so a moving magnetic charge would induce a rotational Efield

C.A. Gonano, R.E. Zich

SO A BEAUTIFUL FORMULATION, BUT…

Nowaday, isolated “magnetic charges” have never been observed!

- In the last decades, many experiments have been brought on to detect them, without results

- On december 2009 at CERN started the Monopole and Exotics Detector At the LHC (MoEDAL)

Let’s stop for a moment and try to change our perspective:

Can we really treat Eand B fields in the same way?

Are we sure that E and B can be summed together?

C.A. Gonano, R.E. Zich

First of all…

WHAT ARE ELECTRIC AND MAGNETIC FIELDS?

- Electric field E(definition):

- Magnetic field B has not such an explicit definition…

- We can measure a magnetic force and express it as:

- For charge Q with speed v, holds:

Note the presence of cross-product: it often appears in Maxwell’s, together with the curl. Which is their role?

C.A. Gonano, R.E. Zich

FIRST OF ALL… WHAT’S “CROSS PRODUCT”?

- Operation with two vectors a and b
- In 3-D, it is variously interpreted…

Vector or oriented area?

- Gibbs -“cross product”:
- it is considered a vector … well, it has a magnitude and a direction!

- Grassmann -“wedge product”:
- it is the oriented areaof the parallelogram between a and b

Identical analitic definition in 3-D:

C.A. Gonano, R.E. Zich

- Sum and scalar product are operations easy to extend in N-D, while cross product is defined just in 3-D

- Cross product is used also to express rotation-vectors in 3-D…

- … but they can not be summed with tip-tail rule, unlike common vectors!

In fact, rotations don’t sum, because they don’t commute!

THEN, ARE THEY “TRUE” VECTORS OR NOT?

- Well, it depends on the “vector” definition…
- …however, rotation-vectors and those originated by cross-product, are frequently called axial vectors or pseudo-vectors

C.A. Gonano, R.E. Zich

Alice through the Looking-glass

- Let try to position a set of vectors in front of a mirror
- “True” vectors, like radii, velocities, forces etc. are simply reflected…
- …on the contrary of moments and pseudo-vectors in general!

Cross product does not respect reflection rules!!!

In fact, at the mirror the specular image of a right hand is a left hand and counterclock-wise appears clock-wise!

C.A. Gonano, R.E. Zich

- In “Flatland” (1884) E. A. Abbott describes life and customs of people in a 2-D world

- In this universe vectors can be summed together and projected, areas are calculated, rotations are clock-wise or counterclock-wise, reflection
- is possible…

- …but cross product does not exists

…otherwise, 2-D inhabitants should have great fantasy to imagine a 3rd dimension to contain a vector orthogonal to their plane…

- …by the way, why use a vector when, in 2-D, a single scalar number is sufficient to describe a force’s moment?

With such a definition, this operation respect all algebric properties of cross-product, but the result is a scalar!

C.A. Gonano, R.E. Zich

- Let try to see if it’s possible to construct cross-productp= aΛbin a 4-D space
- Each vector has 4 elements; because p is the unknown, we’ll have 4 scalar unknowns

- In 3-D we impose that p is perpendicular to vector a and b and that its magnitude is equal to the area between them

But these are just 3 equations, while there are 4 unknowns! Problem has 1 Degree of Indetermination!

In fact, in 4-D there is an infinity ∞1 of vectors p that satisfy these requirements!

SO, CROSS PRODUCT MAYBE EXISTS JUST IN

3-D, OR IT’S NOT A VECTOR…

C.A. Gonano, R.E. Zich

FROM WHERE START FOR N-D EXTENSION?

- Angular velocity: pseudo-vector w, or matrix W?

- Let observe the z-component for M = rΛF:

- Subscript “z” doesn’t appear neither in force nor in the arm
- The moment, rather than “around axis z”,looks to be “from x to y” …

- Let’s try to express moment in a “matrix” way:

Do you notice something?

C.A. Gonano, R.E. Zich

- By treating moment M like a matrix, we notice that:

- Expressing M as a function of dyadic products between F and r it yields so:

- Let’s note that vectors Fand r can have any dimension N !

So, this is the N-D extension for cross-product

Moreover, it can be verified that the new operator respects

all required algebric properties!

Just, the result is no more a vector, but a matrix!

C.A. Gonano, R.E. Zich

WHAT IS CURL? CAN IT BE EXTENDED IN N-D?

- Curl is a differential operator in same way analogous to cross product

Analitic definition in 3-D:

- In 3-D, curl suffers for the same problems of cross product: in fact the result is a pseudo-vector which does not respect reflection rules etc.

- The extension in N-D is instantaneous:

Curl’s definition in N-D

Even in this case it can be verified that the new operator respects all required differential properties!

C.A. Gonano, R.E. Zich

- Magnetic field Bis often involved with X-productand curl, so let’s verify if it is a “true” vector or not

- Look at Faraday law and Lorentz force equations in 3-D:

- We know, from definition, that E, FB and v are “true” vectors

- …and, using N-D notation, the previous equations will look:

Thus the magnetic field B is a not a vector, but a pseudo-vector, and, in a wider N-D view, it is a matrix or tensor!

- The use of B-tensor is not new, but it seems not to be always understood

MAYBE THIS IS THE REASON FOR WHICH

MAGNETIC MONOPOLES CAN’T BE FOUND!

C.A. Gonano, R.E. Zich

Divergence and curl for field B

SO, HOW TO RE-WRITE MAXWELL EQUATIONS IN N-DIMENSIONS?

- In 3-D the divergence of a pseudo-vector is:

- In a similar way, in N-D notation stands:

- So:

3-indices divergence

- It can be easily demonstrated that the 3-D curlof a pseudo-vector correspond to:

with

In general, curl of a pseudo-vector is a true vector

C.A. Gonano, R.E. Zich

- Thus, “classic” Maxwell Equations in N-D become:

- EMforce per unit of volume will look:

IS IT STILL POSSIBLE TO INTRODUCE

MAGNETIC MONOPOLES AND CURRENTS IN N-D?

Yes, but “divergence” and “curl” for B have now different meanings!

C.A. Gonano, R.E. Zich

- With N-D notation, “divergence” for B is no more a scalar but a 3-tensor, with indices i,j,k completely arbitrary

- So, no guarantee that rm assumes a single value:

The existence of magnetic monopole cannot be excluded at all, but its interpretation would be quite different from the electric one!

- Anyway, magnetic “charge” would conserve:

- …but the meaning of EMforce looks quite obscure now:

???

Thus, in N-D the set of “symmetrized” Maxwell and Lorentz equations reveals to be not so “symmetric”

C.A. Gonano, R.E. Zich

Maybe the absence of magnetic monopòles is caused not by the lack of experimental devices, but by a “linguistic” problem!

- In 3-D notation we have the impression that electric E and magneticB fields are mathematically “similar”, but only the first one is a “true” vector!

- In N-D notation instead it becomes clear that B actually is a tensor

- Thus we can not treat them “in the same way”! The initial apparent “symmetry” was misleading

Pseudo-vectors, like vorticity, momenta, rotation axis, spin etc. are spreadly used in Physics and Engineering.

That of “magnetic monopoles” is just an example to show the potentiality of a different “language” valid in every dimension

C.A. Gonano, R.E. Zich

- Magnetic monopoles and current would symmetrize Maxwell’s
- Symmetrized Maxwell’s and EM force would be so invariant under the duality transf.

- In 3-D both X-product and curl produce pseudo-vectors, while with N-D notation they generate matrices

- So magnetic field B is not a vector but a matrix
- Interpretation of magnetic monopòles changed thanks to a notation invariant in every dimension

STILL MUCH TO DO!

THIS IS JUST THE BEGINNING

C.A. Gonano, R.E. Zich

- Relativistic symmetrised Maxwell’s
- History of “magnetic particles”
- Dirac’s analysis
- No experimental evidence

- History of vector analisys
- Uses for cross-product

C.A. Gonano, R.E. Zich

History of “magnetic particles”

WHY THE DIVERGENCE OF B SHOULD BE ALWAYS ZERO ?

- In early XIX cent., Gauss and Weber already considered the question
- In his “Wirbelbewegung”(1858) H. von Helmholtz calculates the force exterted on a “magnetic particle” by an electric current

- In “A Treatise on Electricity and Magnetism” (1873) J. C. Maxwell reports that experimentally magnetic flux F(B) is always zero across a closed surface

- In 1894 Pierre Curie defends the possible existence of “magnetic charge”

- In 1931 Paul A. Dirac, starting from a quantistic approach,symmetrizes Maxwell eq.s adding magneticmonopòles and currents

C.A. Gonano, R.E. Zich

- Because of duality transf. , the ratio between magnetic and electric charge can be arbitrarily changed, thus the magnetic charge can be set to zero just if all particles in the universe have the same ratio (c Qm)/Qe

- In Quantised Singularities in the Electromagnetic Field (1931), Dirac inteprets magnetic monopole as a nodal singularity at the end of semi-infinite solenoid

- In the same paper, Dirac demonstrates that the existence of just one magnetic monopole in the universe would explain the charge quantization

Dirac string, picture from http://cds.cern.ch/record/1360999

C.A. Gonano, R.E. Zich

Nowaday, isolated “magnetic charges” have never been observed, though many experiments have been brought on to detect them

- In september 2009, Science reported that J. Morris, A. Tennant et al. from the Helmholtz-Zentrum Berlin had detected a quasi-magnetic monopole in spin ice dysprosium titanate (Dy2Ti2O7)
- On december 2009 at CERN started the Monopole and Exotics Detector At the LHC (MoEDAL)

HOW TO FIND MONOPOLES?

A moving magnetic monopole would cause an E field, so it could be detected by measuring the current induced in a conducting ring

However, this would be not sufficient to prove their existence! Magnetic current could be solenoidal!

C.A. Gonano, R.E. Zich

Relativistic symmetric Maxwell’s

How the results could be “translated”?

- Using relativistic notation, symmetrised 3-D Maxwell’s will look:

- The EM tensor F is defined together with it’s Hodge dual:

Hodge dual

- Relativistic EM force would be:

Very nice, but it works just in 3-D!

In fact, in N-D the Hodge dual cannot be uniquely defined!

C.A. Gonano, R.E. Zich

- In differential form, “symmetric” Maxwell Equations would look:

BUT WHAT IS A MAGNETIC MONOPOLE?

- Let consider field E: its “monopòles” are the electric charges, isolable and observable

- A magnet generates a field B and its poles are called North and South: can they be isolated?

- Problem: breaking a magnet you will not obtain two magnetic monopòles, but two magnets!

C.A. Gonano, R.E. Zich

In Dirac’s formulation the E and B fields are treated in the same way, so

many consequences arise…

- Conservation for electric and magnetic charge

- A moving magnetic monopole would induct an E field

- Dirac demonstrated that the existence of just one magnetic monopole in the universe would explain the charge quantization

figure from Wikipedia

C.A. Gonano, R.E. Zich

- In 1773, Lagrange finds cross product analitically in order to calculate volume of tetrahedra…
- …but “vector” haven’t been “invented” yet!
- In 1799, C. F. Gauss and C. Wessel represent complex numbers like arrows on a plane.

Vol = (aΛb) Λc =

aΛ (bΛc)

Volume’s calculus for parallelogramma

with a, b, c lata

- In 1840 H.G. Grassmann introduces “external product” and a wedge Λas its symbol
- … but for Grassmann the operation’s result is not a “vector”: though, it’s a area, a volume or a signed iper-volume.
- Operation acts on many vector at time:
- first N-D“extension”

Hermann Gunther Grassmann

C.A. Gonano, R.E. Zich

- In 1843, W. R. Hamilton introduces quaternions to describe rotations in 3-D.
- In 1846 Hamilton adopts the terms scalar and vector referring to real and imaginary parts of a quaternion.
- The vectorial part of a product between quaternions with null real part is equal to cross product

p=axb

William Rowan Hamilton

- In 1881-84, J. W. Gibbs writes Elements of Vector analysis, where modern vectorial calculus is explained. In 1901 his disciple Edwin B. Wilson publishes Vector Analysis, which had a large diffusion.
- Cross product is indicated with a x and is considered a vector
- In 1885, O. Heaviside develops a vector system analogous to Gibbs’s one and applies it to electromagnetism

Josiah Willard Gibbs

C.A. Gonano, R.E. Zich

Though it’s widely used, cross product presents some“oddities”…

- Need for “clock-wise” and “right-hand” concepts

- Cross product is not so easy to use:
- wrong signs are the most frequent mistake: + or - ?
- In practice, you have to memorize long identities like:

Right-hand rule

- …AND MANY PARADOXES ARISE!

C.A. Gonano, R.E. Zich

But, for which task cross-product is useful?

- Cross product appears frequently in Physics and Engineering
- Easy-way for volume’s calculus
- Main use: moment’s definition and calculus

- …if moments are known, we can describe rotations, torsion, stresses, spin and other quantity for different subjects

Moment Mforforce Fapplied to an“arm” r

Large applications in many fields!

C.A. Gonano, R.E. Zich

- Calculus of perpendicular component:

- Double cross product:

- Moments in RationalMechanics

2nd cardinal equation of motus:

Thus, moments become matrices: paradoxes linked to rotation, reflection, 2-D , 4-D etc. for X-product are resolved

C.A. Gonano, R.E. Zich

- Institutional e-mail: carloandrea.gonano@polimi.it
- Thesis download (in italian):
- https://www.politesi.polimi.it/
- handle/10589/34061?mode=full
- https://www.politesi.polimi.it/
- bitstream/10589/34061/1/2011_12_Gonano.pdf
- Keywords (ENG):
- cross product n-d extension;
- cross product; wedge product; curl; rotor; N-D space; magnetic monopoles
- Keywords (ITA):
- prodotto vettore n-d;
- prodotto vettore; rotore; spazio N-D; monopoli magnetici

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