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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 nĚ‚, 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.
The magnetic field (marked B, indicated by red field lines) around wire carrying an electric current (marked I) Compass and wire apparatus showing Ørsted's experiment (video [1]) In electromagnetism , Ørsted's law , also spelled Oersted's law , is the physical law stating that an electric current induces a magnetic field .
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.
[1]: 117 The formula above is known as the Langevin paramagnetic equation. Pierre Curie found an approximation to this law that applies to the relatively high temperatures and low magnetic fields used in his experiments. As temperature increases and magnetic field decreases, the argument of the hyperbolic tangent decreases.
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.
Since this equation is primarily used to solve linear problems, the contributions can be added. For a very difficult geometry, numerical integration may be used. For problems where the dominant magnetic material is a highly permeable magnetic core with relatively small air gaps, a magnetic circuit approach is useful.
Assuming the external magnetic field is uniform and shares a common axis with the paramagnet, the extensive parameter characterizing the magnetic state is , the magnetic dipole moment of the system. The fundamental thermodynamic relation describing the system will then be of the form U = U ( S , V , I , N ) {\displaystyle U=U(S,V,I,N)} .
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