Method of images (or mirror images) is a mathematical tool for solving
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s in which the domain of the sought
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
is extended by the addition of its mirror image with respect to a symmetry
hyperplaneA hyperplane is a concept in geometry. It is a generalization of the concept of a plane into a different number of dimensions. Analogous with a plane which defines a two-dimensional subspace in a three-dimensional space, a hyperplane defines a k-dimensional subspace within an n-dimensional space,...
, with the purpose of facilitating the solution of the original problem. It is used in
electrostaticsElectrostatics is the branch of science that deals with the phenomena arising from stationary or slow-moving electric charges.Since classical antiquity it was known that some materials such as amber attract light particles after rubbing. The Greek word for amber, ήλεκτρον , was the source of the...
to simply calculate or visualize the distribution of the electric field of a charge in the vicinity of the conducting surface.
Method of images (or mirror images) is a mathematical tool for solving
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s in which the domain of the sought
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
is extended by the addition of its mirror image with respect to a symmetry
hyperplaneA hyperplane is a concept in geometry. It is a generalization of the concept of a plane into a different number of dimensions. Analogous with a plane which defines a two-dimensional subspace in a three-dimensional space, a hyperplane defines a k-dimensional subspace within an n-dimensional space,...
, with the purpose of facilitating the solution of the original problem. It is used in
electrostaticsElectrostatics is the branch of science that deals with the phenomena arising from stationary or slow-moving electric charges.Since classical antiquity it was known that some materials such as amber attract light particles after rubbing. The Greek word for amber, ήλεκτρον , was the source of the...
to simply calculate or visualize the distribution of the electric field of a charge in the vicinity of the conducting surface. It is based on the fact that the tangential component of the electrical field to the surface of a
conductorIn science and engineering, an electrical conductor is a material which contains movable electric charges. In metallic conductors, such as copper or aluminum, the movable charged particles are electrons...
is zero, and that some field E with and in some region is uniquely defined by its normal component over the surface which confines this region (the uniqueness theorem).
The method may also be used in
magnetostaticsMagnetostatics is the study of static magnetic fields. In electrostatics, the charges are stationary, whereas here, the currents are stationary or dc...
for calculating the magnetic field of a magnet which is close to a superconducting surface. Here, the component of the magnetic field
normal to the superconductor is zero. Another use is in
fluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
: the movement of a
vortexA vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...
near a wall may be calculated using an image vortex.
A textbook example is an infinite flat conducting surface (see Fig. 1). In this case the field distribution between the surface and a charge is the same as between this charge and another charge (imaginary charge), which is the mirror image of the real charge in respect to the surface but has the opposite sign. Evidently, in case of an electrical dipole, the vector of the mirrored dipole will have the opposite sign. Therefore, the force between the electrical charge or system of charges and the conducting surface is attractive. See also
Method of image chargesThe method of image charges is a basic problem-solving tool in electrostatics...
.
In case of magnet-superconductor pair (the superconductor here is ideal, into which the magnetic field does not penetrate), the mirror image of the magnet will have a
magnetizationMagnetization, M, is defined as the quantity of magnetic moment per unit volume, V:Here, N is the number of magnetic moments in the sample...
vector which is mirrored but of the same sign (see Fig. 2). This can be thought as due to additional sign change upon mirroring of an axial vector which the magnetization is. The force between the magnet and the superconducting surface is therefore repulsive.
Further reading
- James Jeans (1908) The Mathematical Theory of Electricity and Magnetism, Chapter 8, Cambridge University Press
Cambridge University Press is a printer and publisher granted a Royal Letters Patent by Henry VIII in 1534. It is the world's oldest continually operating book publisher...
.