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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.
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.
Magnets exert forces and torques on each other through the interaction of their magnetic fields.The forces of attraction and repulsion are a result of these interactions. The magnetic field of each magnet is due to microscopic currents of electrically charged electrons orbiting nuclei and the intrinsic magnetism of fundamental particles (such as electrons) that make up the mater
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. This is better illustrated through the following relation: m = ∭ M d V {\displaystyle \mathbf {m} =\iiint \mathbf {M} \,\mathrm {d} V} where m is an ordinary magnetic ...
For zero net magnetic charge density (ρ m = 0), the original form of Gauss's magnetism law is the result. The modified formula for use with the SI is not standard and depends on the choice of defining equation for the magnetic charge and current; in one variation, magnetic charge has units of webers, in another it has units of ampere-meters.
The potential magnetic energy of a magnet or magnetic moment in a magnetic field is defined as the mechanical work of the magnetic force on the re-alignment of the vector of the magnetic dipole moment and is equal to: = The mechanical work takes the form of a torque : = = which will act to "realign" the magnetic dipole with the magnetic field. [1]
Quadrupole magnets are useful as they create a magnetic field whose magnitude grows rapidly with the radial distance from its longitudinal axis. This is used in particle beam focusing. The simplest magnetic quadrupole is two identical bar magnets parallel to each other such that the north pole of one is next to the south of the other and vice ...
Ancient people learned about magnetism from lodestones (or magnetite) which are naturally magnetized pieces of iron ore.The word magnet was adopted in Middle English from Latin magnetum "lodestone", ultimately from Greek μαγνῆτις [λίθος] (magnētis [lithos]) [1] meaning "[stone] from Magnesia", [2] a place in Anatolia where lodestones were found (today Manisa in modern-day Turkey).