Frozen mirror image method (or method of frozen images) is an extension of the

method of images The method of images (or method of mirror images) is a mathematical tool for solving differential equations, in which the domain of the sought function is extended by the addition of its mirror image with respect to a symmetry hyperplane. As a resu ...

for

magnet A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nicke ...

- superconductor systems that has been introduced by Alexander Kordyuk in 1998 to take into account the magnetic

flux pinning Flux pinning is a phenomenon that occurs when flux vortices in a type-II superconductor are prevented from moving within the bulk of the superconductor, so that the magnetic field lines are "pinned" to those locations. The superconductor must be a ...

phenomenon. The method gives a simple representation of the magnetic field distribution generated by a magnet (a system of magnets) outside an infinitely flat surface of a perfectly hard (with infinite

pinning force Pinning force is a force acting on a pinned object from a pinning center. In solid state physics, this most often refers to the vortex pinning, the pinning of the magnetic vortices ( magnetic flux quanta, Abrikosov vortices) by different kinds of ...

)

type-II superconductor In superconductivity, a type-II superconductor is a superconductor that exhibits an intermediate phase of mixed ordinary and superconducting properties at intermediate temperature and fields above the superconducting phases. It also features the ...

in more general field cooled (FC) case, i.e. when the superconductor goes into superconducting state been already exposed to the magnetic field. The difference from the mirror image method, which deals with a perfect type-I superconductor (that completely expels the magnetic field, see the

Meissner effect The Meissner effect (or Meissner–Ochsenfeld effect) is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state when it is cooled below the critical temperature. This expulsion will repel a ne ...

), is that the perfectly hard superconductor screens the variation of the external magnetic field rather than the field itself.

Description

The name originates from the replacement of certain elements in the original layout with imaginary magnets, which replicates the boundary conditions of the problem (see

Dirichlet boundary conditions In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differentia ...

). In a simplest case of the

magnetic dipole In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric ...

over the flat superconducting surface (see Fig. 1), the magnetic field, generated by a dipole moved from its initial position (at which the superconductor is cooled to the superconducting state) to a final position and by the screening currents at the superconducting surface, is equivalent to the field of three magnetic dipoles: the real one (1), its mirror image (3), and the mirror image of it in initial (FC) position but with the magnetization

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

inversed (2).

Applications

The method is shown to work for the bulk

high temperature superconductors High-temperature superconductors (abbreviated high-c or HTS) are defined as materials that behave as superconductors at temperatures above , the boiling point of liquid nitrogen. The adjective "high temperature" is only in respect to previ ...

(HTSC), which are characterized by strong pinning and used for calculation of the interaction in magnet-HTSC systems such as superconducting

magnetic bearings A magnetic bearing is a type of bearing that supports a load using magnetic levitation. Magnetic bearings support moving parts without physical contact. For instance, they are able to levitate a rotating shaft and permit relative motion with ve ...

, superconducting

flywheels A flywheel is a mechanical device which uses the conservation of angular momentum to store rotational energy; a form of kinetic energy proportional to the product of its moment of inertia and the square of its rotational speed. In particular, assu ...

MAGLEV Maglev (derived from '' magnetic levitation''), is a system of train transportation that uses two sets of electromagnets: one set to repel and push the train up off the track, and another set to move the elevated train ahead, taking advantage ...

, for

spacecraft A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite, spacecraft are used for a variety of purposes, including communications, Earth observation, meteorology, navigation, space colonization, p ...

applications, as well as a

textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textbook ...

model for

science education Science education is the teaching and learning of science to school children, college students, or adults within the general public. The field of science education includes work in science content, science process (the scientific method), som ...

.{{cite journal , last1 = Saito , first1 = Y. , year = 2009 , title = Observation of magnetic field lines in the vicinity of a superconductor with the naked eye , journal =

European Journal of Physics The ''European Journal of Physics'' is a peer-reviewed, scientific journal dedicated to maintaining and improving the standard of physics education in higher education. The journal, published since 1980, is now published by IOP Publishing on beha ...

, volume = 31 , issue = 1 , pages = 229–238 , doi = 10.1088/0143-0807/31/1/020 , bibcode = 2010EJPh...31..229S , arxiv =0805.3990, s2cid = 56360791

References

Demos

Superconducting levitation with strong pinningMagnetic levitation (YouTube)
Electromagnetism Condensed matter physics Superconductivity Applied and interdisciplinary physics

Description

Applications

See also

References

Demos