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Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂, d is the dipole moment between two point charges, the volume density of these is the polarization density P.
The source free equations can be written by the action of the exterior derivative on this 2-form. But for the equations with source terms (Gauss's law and the Ampère-Maxwell equation), the Hodge dual of this 2-form is needed. The Hodge star operator takes a p-form to a (n − p)-form, where n is the number of dimensions.
Volume 2; A Treatise on Electricity and Magnetism at Internet Archive. 1st edition 1873 Volume 1, Volume 2; 2nd edition 1881 Volume 1, Volume 2; 3rd edition 1892 (ed. J. J. Thomson) Volume 1, Volume 2; 3rd edition 1892 (Dover reprint 1954) Volume 1, Volume 2; Original Maxwell Equations – Maxwell's 20 Equations in 20 Unknowns – PDF
The magnetization field or M-field can be defined according to the following equation: =. Where is the elementary magnetic moment and is the volume element; in other words, the M-field is the distribution of magnetic moments in the region or manifold concerned.
The magnetic field in the magnet frame and the electric field in the conductor frame must generate consistent results in the conductor. At the time of Einstein in 1905, the field equations as represented by Maxwell's equations were properly consistent. Newton's law of motion, however, had to be modified to provide consistent particle trajectories.
The most elementary force between magnets is the magnetic dipole–dipole interaction. If all magnetic dipoles for each magnet are known then the net force on both magnets can be determined by summing all the interactions between the dipoles of the first magnet and the dipoles of the second magnet.
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