The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in
electrostatics
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...
. The name originates from the replacement of certain elements in the original layout with fictitious charges, which replicates the boundary conditions of the problem (see
Dirichlet boundary conditions or
Neumann boundary conditions
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.
When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative appli ...
).
The validity of the method of image charges rests upon a corollary of 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 ...
, which states that the electric potential in a volume ''V'' is uniquely determined if both the charge density throughout the region and the value of the
electric potential
Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
on all boundaries are specified. Alternatively, application of this corollary to the
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
of
Gauss' Law shows that in a volume ''V'' surrounded by conductors and containing a specified charge density ''ρ'', the electric field is uniquely determined if the total charge on each conductor is given. Possessing knowledge of either the electric potential or the electric field and the corresponding boundary conditions we can swap the charge distribution we are considering for one with a configuration that is easier to analyze, so long as it satisfies
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
in the region of interest and assumes the correct values at the boundaries.
Reflection in a conducting plane
Point charges
The simplest example of method of image charges is that of a point charge, with charge ''q'', located at
above an infinite
grounded (i.e.:
) conducting plate in the ''xy''-plane. To simplify this problem, we may replace the plate of equipotential with a charge −''q'', located at
. This arrangement will produce the same electric field at any point for which
(i.e., above the conducting plate), and satisfies the boundary condition that the potential along the plate must be zero. This situation is equivalent to the original setup, and so the force on the real charge can now be calculated with
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
between two point charges.
The potential at any point in space, due to these two point charges of charge +''q'' at +''a'' and −''q'' at −''a'' on the ''z''-axis, is given in
cylindrical coordinates
A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infinite ...
as
:
The
surface charge density on the grounded plane is therefore given by
:
In addition, the ''total'' charge induced on the conducting plane will be the integral of the charge density over the entire plane, so:
:
The total charge induced on the plane turns out to be simply −''q''. This can also be seen from the
Gauss's law, considering that the dipole field decreases at the cube of the distance at large distances, and the therefore total flux of the field though an infinitely large sphere vanishes.
Because electric fields satisfy the
superposition principle
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So th ...
, a conducting plane below multiple point charges can be replaced by the mirror images of each of the charges individually, with no other modifications necessary.
Electric dipole moments
The image of an electric dipole moment p at
above an infinite grounded conducting plane in the ''xy''-plane is a dipole moment at
with equal magnitude and direction rotated azimuthally by π. That is, a dipole moment with Cartesian components
will have in image dipole moment
. The dipole experiences a force in the ''z'' direction, given by
:
and a torque in the plane perpendicular to the dipole and the conducting plane,
:
Reflection in a dielectric planar interface
Similar to the conducting plane, the case of a planar interface between two different
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an Insulator (electricity), electrical insulator that can be Polarisability, polarised by an applied electric field. When a dielectric material is placed in an electric field, electric ...
media can be considered. If a point charge
is placed in the dielectric that has the dielectric constant
, then the interface (with the dielectric that has the dielectric constant
) will develop a bound polarization charge. It can be shown that the resulting electric field inside the dielectric containing the particle is modified in a way that can be described by an image charge inside the other dielectric. Inside the other dielectric, however, the image charge is not present.
Unlike the case of the metal, the image charge
is not exactly opposite to the real charge:
. It may not even have the same sign, if the charge is placed inside the stronger dielectric material (charges are repelled away from regions of lower dielectric constant). This can be seen from the formula.
Reflection in a conducting sphere
Point charges
The
method of images may be applied to a sphere as well. In fact, the case of image charges in a plane is a special case of the case of images for a sphere. Referring to the figure, we wish to find the potential inside a grounded sphere of radius ''R'', centered at the origin, due to a point charge inside the sphere at position
(For the opposite case, the potential outside a sphere due to a charge outside the sphere, the method is applied in a similar way). In the figure, this is represented by the green point. Let ''q'' be the point charge of this point. The image of this charge with respect to the grounded sphere is shown in red. It has a charge of and lies on a line connecting the center of the sphere and the inner charge at vector position
. It can be seen that the potential at a point specified by radius vector
due to both charges alone is given by the sum of the potentials:
:
Multiplying through on the rightmost expression yields:
:
and it can be seen that on the surface of the sphere (i.e. when ), the potential vanishes. The potential inside the sphere is thus given by the above expression for the potential of the two charges. This potential will ''not'' be valid outside the sphere, since the image charge does not actually exist, but is rather "standing in" for the surface charge densities induced on the sphere by the inner charge at
. The potential outside the grounded sphere will be determined only by the distribution of charge outside the sphere and will be independent of the charge distribution inside the sphere. If we assume for simplicity (
without loss of generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
) that the inner charge lies on the z-axis, then the induced charge density will be simply a function of the
polar angle ''θ'' and is given by:
:
The total charge on the sphere may be found by integrating over all angles:
:
Note that the reciprocal problem is also solved by this method. If we have a charge ''q'' at vector position
outside of a grounded sphere of radius ''R'', the potential outside of the sphere is given by the sum of the potentials of the charge and its image charge inside the sphere. Just as in the first case, the image charge will have charge −''qR''/''p'' and will be located at vector position
. The potential inside the sphere will be dependent only upon the true charge distribution inside the sphere. Unlike the first case the integral will be of value −''qR''/''p''.
Electric dipole moments
The image of an
electric point dipole is a bit more complicated. If the dipole is pictured as two large charges separated by a small distance, then the image of the dipole will not only have the charges modified by the above procedure, but the distance between them will be modified as well. Following the above procedure, it is found that a dipole with dipole moment
at vector position
lying inside the sphere of radius ''R'' will have an image located at vector position
(i.e. the same as for the simple charge) and will have a simple charge of:
:
and a dipole moment of:
:
Method of inversion
The method of images for a sphere leads directly to the method of inversion.
If we have a
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that i ...
of position
where
are the
spherical coordinates
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are
* the radial distance along the line connecting the point to a fixed point ...
of the position, then the image of this harmonic function in a sphere of radius ''R'' about the origin will be
:
If the potential
arises from a set of charges of magnitude
at positions
, then the image potential will be the result of a series of charges of magnitude
at positions
. It follows that if the potential
arises from a charge density
, then the image potential will be the result of a charge density
.
See also
*
Kelvin transform
*
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
*
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem relating the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the volume ...
*
Flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phe ...
*
Gaussian surface
*
Schwarz reflection principle
*
Uniqueness theorem for Poisson's equation
The uniqueness theorem for Poisson's equation states that, for a large class of boundary conditions, the equation may have many solutions, but the gradient of every solution is the same. In the case of electrostatics, this means that there is a u ...
*
Image antenna
In telecommunications and antenna design, an image antenna is an electrical mirror-image of an antenna element formed by the radio waves reflecting from a conductive surface called a ground plane, such as the surface of the earth. It is used as ...
*
Surface equivalence principle
*
Schottky effect
References
Notes
Sources
*
*
Further reading
*
*
* {{cite book, last1=Purcell, first1= Edward M., authorlink1=Edward Mills Purcell, title= Berkeley Physics Course, Vol-2: Electricity and Magnetism (2nd ed.), publisher=
McGraw-Hill
McGraw Hill is an American education science company that provides educational content, software, and services for students and educators across various levels—from K-12 to higher education and professional settings. They produce textbooks, ...
Electromagnetism
Electrostatics