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Ferrofluid is a liquid that is attracted to the poles of a magnet. It is a colloidal liquid made of nanoscale ferromagnetic or ferrimagnetic particles suspended in a carrier fluid (usually an organic solvent or water). [1] Each magnetic particle is thoroughly coated with a surfactant to inhibit clumping. Large ferromagnetic particles can be ...
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,
The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field quantities. [1] = (/ / / / / /) and the result of raising its indices is = = (/ / / / / /), where E is the electric field, B the magnetic field, and c the speed of light.
This is often described by saying that the electric field and magnetic field are two interrelated aspects of a single object, called the electromagnetic field. Indeed, the entire electromagnetic field can be represented in a single rank-2 tensor called the electromagnetic tensor; see below.
If the matter field is taken so as to describe the interaction of electromagnetic fields with the Dirac electron given by the four-component Dirac spinor field ψ, the current and charge densities have form: [2] = † = †, where α are the first three Dirac matrices. Using this, we can re-write Maxwell's equations as:
This is violated for Minkowski space with a line removed, which can model a (flat) spacetime with a point-like monopole on the complement of the line. In the differential form formulation on arbitrary space times, F = 1 / 2 F αβ dx α ∧ dx β is the electromagnetic tensor considered as a 2-form, A = A α dx α is the potential 1 ...
The electromagnetic field is a covariant antisymmetric tensor of degree 2, which can be defined in terms of the electromagnetic potential by =.. To see that this equation is invariant, we transform the coordinates as described in the classical treatment of tensors: ¯ = ¯ ¯ ¯ ¯ = ¯ (¯) ¯ (¯) = ¯ ¯ + ¯ ¯ ¯ ¯ ¯ ¯ = ¯ ¯ ¯ ¯ = ¯ ¯ = ¯ ¯.
An observer at rest with respect to a system of static, free charges will see no magnetic field. However, a moving observer looking at the same set of charges does perceive a current, and thus a magnetic field. That is, the magnetic field is simply the electric field, as seen in a moving coordinate system.