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In the following, it is assumed that the system is 2-dimensional with as the invariant axis, i.e. produces 0 for any quantity. Then the magnetic field can be written in cartesian coordinates as = (,, (,)), or more compactly, = ^ + ^, where (,) ^ is the vector potential for the in-plane (x and y components) magnetic field.
The presence of a magnetic toroidic dipole moment T in condensed matter is due to the presence of a magnetoelectric effect: Application of a magnetic field H in the plane of a toroidal solenoid leads via the Lorentz force to an accumulation of current loops and thus to an electric polarization perpendicular to both T and H.
As a simple example from the physics of magnetically confined plasmas, consider an axisymmetric system with circular, concentric magnetic flux surfaces of radius (a crude approximation to the magnetic field geometry in an early tokamak but topologically equivalent to any toroidal magnetic confinement system with nested flux surfaces) and denote the toroidal angle by and the poloidal angle by .
Applications to the boussinesq-equations, Schmitt, B. J. and von Wahl, W; in The Navier–Stokes Equations II — Theory and Numerical Methods, pp. 291–305; Lecture Notes in Mathematics, Springer Berlin/ Heidelberg, Vol. 1530/ 1992. Anelastic Magnetohydrodynamic Equations for Modeling Solar and Stellar Convection Zones, Lantz, S. R. and Fan ...
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.
Similarly, if only the magnetic field (B) is non-zero and is constant in time, the field is said to be a magnetostatic field. However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using Maxwell's equations .
The classic applications of toroidal coordinates are in solving partial differential equations, e.g., Laplace's equation for which toroidal coordinates allow a separation of variables or the Helmholtz equation, for which toroidal coordinates do not allow a separation of variables
In thermodynamics and thermal physics, the theoretical formulation of magnetic systems entails expressing the behavior of the systems using the Laws of Thermodynamics. Common magnetic systems examined through the lens of Thermodynamics are ferromagnets and paramagnets as well as the ferromagnet to paramagnet phase transition. It is also ...