Magnetic monopoles and Maxwell’s Equations in N Dimensions Carlo Andrea Gonano Riccardo Enrico Zich * Abstract — The possible existence of “magnetic charge” has been widely investigated in the last two centuries, since it would allow to symmetrise Maxwell’s equations and to explain charge quanti- zation. Though, notwithstanding many studies and experiments, nowaday (2013) magnetic monopoles have never been observed. In this paper we are going to show that E and B fields exhibit very different natures, but while this is not so explicit in 3 spatial dimensions (3-D), it becomes clear in a generic number of dimensions (N- D). In particular, the N-D extension of cross product and curl brings to N-D Maxwell’s eq.s and to re- interpret magnetic field B, explaining the probable lack of magnetic monopoles. 1 Introduction Separating monopoles for electric field  E, i.e. posi- tive and negative electric charges, is a quite simple operation: for example, plastic can be charged by rubbing. A magnet generates a magnetic field  B and its poles are called North and South, but iso- late them is not so easy: breaking the magnet, you will not obtain two magnetic monopoles, but two smaller dipoles. That’s the problem of finding sin- gle magnetic charges. 1.1 Outline In this paper we are going to summarize the state of the art, then we illustrate Maxwell’s Equa- tions (from here, “Maxwell eq.s” ) symmetrised with monopoles, highlighting their invariance un- der the Duality Transformation (“Dual Transf.”) and drawing some consequences. In the second part we analyze the concept of cross-product and curl in 3-D and expose their extension in N spatial Dimen- sion (“N-D” ), leading to a different interpretation of magnetic field B. In the end we re-write Maxwell Eq.s in N-D and conclude with some remarks. 1.2 Brief historical notes The possible existence of magnetic charge was al- ready considered in the early XIX century[1] and in 1858 Hermann von Helmholtz calculated the force exterted on a “magnetic particle” by an electric * Dipartimento di Energia, Politecnico di Mi- lano, Via La Masa 34, 20156 Milano, Italy, e-mail: carloandrea.gonano@polimi.it, riccardo.zich@polimi.it current[2], using the Biot-Savart law and a fluid- dynamic analogy. On the other side, in his syn- thesis of ElectroMagnetism[3],[4] J.C. Maxwell no- ticed that experimentally the magnetic flux Φ(B) across a closed surface was always zero, so there was no evidence of magnetic charges. However, in 1894 Pierre Curie argued that their existence could not be excluded a priori [5] and in 1931 Paul A.M. Dirac showed that magnetic monopoles were con- sistent with quantum theory[6]. In 1968 Victor G. Veselago published a famous paper[7] which gave birth to the EM metamaterials sector: in its end he observed that a gas of mag- netic monopoles would exhibit negative permeabil- ity (μ< 0), making difficult to detect them. Today magnetic monopoles are still a “hot” topic: their existence is postulated by many physical theo- ries and there are over 900 academic on-line papers concerning with them (source: Scopus). 2 Symmetrising 3-D Maxwell’s Equations In modern notation, Maxwell Eq.s for free space can be written in differential form as: ⎧ ⎨ ⎩ -→ ∇ T ·  E = ρ e ε 0 -→ ∇ T ·  B =0 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ -→ ∇∧  E = - ∂  B ∂t -→ ∇∧  B = μ 0  J e + 1 c 2 ∂  E ∂t (1) Remember that “speed of light” is c =1/ √ μ 0 ε 0 . Note that (1) are divergence and curl equations both for  E and  B: it would look like a quite sym- metric set, but. . . no “magnetic charge ” ρ m and no “magnetic current”  J m ! 2.1 Symmetric 3-D Maxwell’s Equations The introduction of magnetic charge ρ m and flux  J m densities allows to symmetrise Maxwell’s Eq.s, which would look: ⎧ ⎪ ⎨ ⎪ ⎩ -→ ∇ T ·  E = ρ e ε 0 -→ ∇ T ·  B = ρ m ε 0 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ -→ ∇∧  E = -  1 ε 0  J m + ∂  B ∂t  -→ ∇∧  B = 1 c 2  1 ε 0  J e + ∂  E ∂t  (2) 978-1-4673-5707-4/13/$31.00 ©2013 IEEE 1544