The method of images (or method of mirror images) is a mathematical tool for solving
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, in which the domain of the sought
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
is extended by the addition of its mirror image with respect to a symmetry
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
. As a result, certain
boundary conditions are satisfied automatically by the presence of a mirror image, greatly facilitating the solution of the original problem. The domain of the function is not extended. The function is made to satisfy given boundary conditions by placing singularities outside the domain of the function. The original singularities are inside the domain of interest. The additional (fictitious) singularities are an artifact needed to satisfy the prescribed but yet unsatisfied boundary conditions.
Method of image charges
The
method of image charges is used in
electrostatics
Electrostatics is a branch of physics that studies electric charges at rest (static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
to simply calculate or visualize the distribution of the electric field of a charge in the vicinity of a conducting surface. It is based on the fact that the tangential component of the electrical field on the surface of a
conductor is zero, and that an
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
E in some region is uniquely defined by its
normal component
In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the no ...
over the surface that confines this region (the
uniqueness theorem In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems ...
).
Magnet-superconductor systems
The method of images may also be used in
magnetostatics
Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equati ...
for calculating the magnetic field of a magnet that is close to a superconducting surface. The
superconductor in so-called
Meissner state is an ideal
diamagnet
Diamagnetic materials are repelled by a magnetic field; an applied magnetic field creates an induced magnetic field in them in the opposite direction, causing a repulsive force. In contrast, paramagnetic and ferromagnetic materials are attract ...
into which the magnetic field does not penetrate. Therefore, the normal component of the magnetic field on its surface should be zero. Then the image of the magnet should be mirrored. The force between the magnet and the superconducting surface is therefore repulsive.
Comparing to the case of the
charge dipole above a flat conducting surface, the mirrored
magnetization vector can be thought as due to an additional sign change of an
axial vector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its ...
.
In order 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 in
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 ...
s, the
frozen mirror image method
Frozen mirror image method (or method of frozen images) is an extension of the method of images for magnet- superconductor systems that has been introduced by Alexander Kordyuk in 1998 to take into account the magnetic flux pinning phenomenon. Th ...
can be used.
Mass transport in environmental flows with non-infinite domains
Environmental engineers are often interested in the reflection (and sometimes the absorption) of a contaminant plume off of an impenetrable (no-flux) boundary. A quick way to model this reflection is with the method of images.
The reflections, or ''images'', are oriented in space such that they perfectly replace any mass (from the real plume) passing through a given boundary.
A single boundary will necessitate a single image. Two or more boundaries produce infinite images. However, for the purposes of modeling mass transport—such as the spread of a contaminant spill in a lake—it may be unnecessary to include an infinite set of images when there are multiple relevant boundaries. For example, to represent the reflection within a certain threshold of physical accuracy, one might choose to include only the primary and secondary images.
The simplest case is a single boundary in 1-dimensional space. In this case, only one image is possible. If as time elapses, a mass approaches the boundary, then an image can appropriately describe the reflection of that mass back across the boundary.
Another simple example is a single boundary in 2-dimensional space. Again, since there is only a single boundary, only one image is necessary. This describes a smokestack, whose effluent "reflects" in the atmosphere off of the impenetrable ground, and is otherwise approximately unbounded.
Finally, we consider a mass release in 1-dimensional space bounded to its left and right by impenetrable boundaries. There are two primary images, each replacing the mass of the original release reflecting through each boundary. There are two secondary images, each replacing the mass of one of the primary images flowing through the opposite boundary. There are also two tertiary images (replacing the mass lost by the secondary images), two quaternary images (replacing the mass lost by the tertiary images), and so on ad infinitum.
For a given system, once all of the images are carefully oriented, the concentration field is given by summing the mass releases (the ''true'' plume in addition to all of the images) within the specified boundaries. This concentration field is only physically accurate within the boundaries; the field outside the boundaries is non-physical and irrelevant for most engineering purposes.
Mathematics
This method is a specific application of
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
s. The method of images works well when the boundary is a flat surface and the distribution has a geometric center. This allows for simple mirror-like reflection of the distribution to satisfy a variety of boundary conditions. Consider the simple 1D case illustrated in the graphic where there is a distribution of
as a function of
and a single boundary located at
with the real domain such that
and the image domain
. Consider the solution
to satisfy the
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
for any
, but not necessarily the boundary condition.
Note these distributions are typical in models that assume a
Gaussian distribution. This is particularly common in environmental engineering, especially in atmospheric flows that use
Gaussian plume models.
Perfectly reflecting boundary conditions
The mathematical statement of a perfectly reflecting boundary condition is as follows:
This states that the derivative of our scalar function
will have no derivative in the normal direction to a wall. In the 1D case, this simplifies to:
This condition is enforced with positive images so that:
where the
translates and reflects the image into place. Taking the derivative with respect to
:
Thus, the perfectly reflecting boundary condition is satisfied.
Perfectly absorbing boundary conditions
The statement of a perfectly absorbing boundary condition is as follows:
This condition is enforced using a negative mirror image:
And:
Thus this boundary condition is also satisfied.
References
{{DEFAULTSORT:Method Of Images
Electromagnetism
Electrodynamics
Electricity
Magnetism
Superconductivity