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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 strength (and direction) of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field. Its direction points from the south pole to north pole of the magnet (i.e., inside the magnet). The magnetic moment also expresses the magnetic force effect of a magnet.
A is the area of each pole, in m 2, L is the length of each magnet, in m, R is the radius of each magnet, in m, and; x is the separation between the two magnets, in m = relates the flux density at the pole to the magnetization of the magnet.
A charged particle beam in a quadrupole magnetic field will experience a focusing / defocusing force in the transverse direction. This focusing effect is summed up by a focusing strength which depends on the quadrupole gradient as well as the beam's rigidity [] = /, where is the electric charge of the particle and
Maxwell's equations can directly give inhomogeneous wave equations for the electric field E and magnetic field B. [1] Substituting Gauss's law for electricity and Ampère's law into the curl of Faraday's law of induction, and using the curl of the curl identity ∇ × (∇ × X) = ∇(∇ ⋅ X) − ∇ 2 X (The last term in the right side is the vector Laplacian, not Laplacian applied on ...
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A principal G-bundle over S 2 is defined by covering S 2 by two charts, each homeomorphic to the open 2-ball such that their intersection is homeomorphic to the strip S 1 ×I. 2-balls are homotopically trivial and the strip is homotopically equivalent to the circle S 1. So a topological classification of the possible connections is reduced to ...
The electromagnetic tensor, conventionally labelled F, is defined as the exterior derivative of the electromagnetic four-potential, A, a differential 1-form: [1] [2] = . Therefore, F is a differential 2-form— an antisymmetric rank-2 tensor field—on Minkowski space. In component form,