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PPT No. 8
•Uniqueness Theorem
* Method of Images
Uniqueness Theorem
The uniqueness theorem states that
There is only one solution to Poisson's equation which
satisfies a given, well-posed set of boundary conditions.
Uniqueness Theorem
In the context of Electrostatics
the Uniqueness Theorem may be stated as followsIf the distribution of charges within a region of space and
the potentials at the boundaries to this region are given
then there is one and only one solution
for the electric potential V
Uniqueness Theorem
The fact that
the solutions to Poisson's equation are unique,
has a great value.
It implies that
a solution to this equation is
the only possible solution
Uniqueness Theorem
A consequence of the uniqueness theorem is that
if the given electrostatics problem is replaced by
an easier, analogue problem
with the same charges and boundary conditions
then the solution of new easier problem
is also the solution of initial harder problem.
Uniqueness theorem serves as
the basis for the method of images.
Uniqueness Theorem
In electrostatics problems
if a charge density is given or
a uniform density can be assumed
because of the high symmetry of the problem,
then V and E can be calculated.
Uniqueness Theorem & The Method of Images
However in problems involving conducting surfaces,
the charge distribution given by surface charge density σ
is either not known or its form cannot be assumed
because the symmetry of the problem is not sufficiently high.
The problem then involves finding σ in addition to V and E.
It is for these types of problems that
the method of images is particularly useful.
The Method of Images
The method of image charges is
a basic device for problem-solving in Electrostatics
The name has origin in the method in which
certain elements in the original configuration
are replaced by imaginary charges,
such that
the boundary conditions of the problem are satisfied.
The Method of Images
In electrostatics,
it is required to find electric potential and field
in the region of space when a charge /charge distribution (q)
and boundary conditions on the electric potential
on the conducting surfaces are specified.
Charges (q’) are induced on the surface of the conductor
The Method of Images
The resultant potential depends upon initial & inducedboth charges (q and q’).
The evaluation of potential becomes problematic as
the amount of induced charge q’ and
its distribution pattern is not known. .
The Method of Images
This situation is bypassed
by constructing an analogues device
called as the “image” of
the given charge distribution
A charge distribution is added
in the excluded region of space
to produce the correct boundary conditions
for the electric potential on the surfaces
The Method of Images
The additional charge distribution is
fictitious having no real existence.
So it is a virtual image.
The electric potential is then
simply the sum of
the potentials due to actual &
image charge distributions
The Method of Images
The method of images has analogy
to the optical images produced in mirrors
Hence it is known as the method of images &
method of mirror charges
The Method of Images
The essence of the method is
to construct an appropriate “auxiliary problem” i.e.
to determine the equivalent image charge distribution and
its strategic location so that it keeps
the surface an equipotential surface with required potential
and later the Laplace or Poisson’s equation with
the given boundary conditions can be solved more easily. .
The Method of Images
In the method of Images,
the uniqueness theorem plays a crucial role.
It states that
the solution to Laplace ‘s /Poisson’s equation
in some volume is uniquely determined if the potential
is specified on the boundary surface enclosing the volume.
The Method of Images
The method of images is applicable to a single point charge
& also to different static charge configurations
near a grounded conductor.
It can be used to compute
Electric Potential and Field,
Surface Charge Density,
Force,
Energy
for the following cases
The Method of Images
Applications of the method of Images in cases
* (i) A Point charge near an infinite conducting plane,
* (ii) A Point charge near a grounded conducting sphere,
* (iii) Conducting sphere in a uniform electric field
* (iv) Dielectric sphere in a uniform field
* (v) A spherical cavity in a dielectric medium
An Application of the Method of Images to A Point
Charge and an Infinite Plane Conducting Surface
Fig A point charge q at a distance d from an infinite plane conducting surface
is equivalent to a combination of a charge q, and its image charge -q
without the conductor in the Analogue problem
An Application of the Method of Images
Consider a point charge q held a distance d from
an infinite, grounded, conducting plate lying in the x-y plane,
Let the point charge be located at coordinates (0, 0, d).
The point charge induces surface charges on the plate.
Amount & distribution pattern of induced charges unknown =>
the scalar potential above the plane cannot be found directly.
An Application of the Method of Images
The conducting plate is an equipotential surface.
The potential of the plate is zero, as it is grounded and
the potential at ∞ is zero
(usual boundary condition for the scalar potential).
it is required to solve Poisson's equation in the region z>0,
for a single point charge q at position (0, 0, d)
subject to the boundary conditions
as
An Application of the Method of Images
Consider an analogue problem where
a charge q is located at (0, 0, d ) and
a charge –q is located at (0, 0, -d), and
conductor is not present.
The scalar potential for the analogue problem is given by
An Application of the Method of Images
and
For
satisfies Poisson's equation for
a charge at (0,0,d), in the region z > 0.
is a solution to the original problem in the region z > 0
is the correct solution in the region z > 0.
An Application of the Method of Images
According to the uniqueness theorem
there is only one solution to Poisson's equation which
satisfies a given, well-posed set of boundary conditions.
must be the correct potential in the region
though
is completely wrong in the region z< 0
Ф= 0 in this region as the grounded plate shields
the region z< 0 from the point charge,
An Application of the Method of Images
As the potential in the region z > 0 is known,
the distribution of charges induced on
the conducting plate can be worked out.
The relation between
the electric field immediately above a conducting surface
and the density of charge on the surface is given by
An Application of the Method of Images
An Application of the Method of Images
The charge induced on the plate has
the opposite sign to the given point charge.
The charge density on the plate is
symmetric about the z-axis, and is largest
where the plate is closest to the point charge.
An Application of the Method of Images
The total charge induced on the plate is
An Application of the Method of Images
The total charge induced on the plate is
An Application of the Method of Images
The total charge induced on the plate is
equal and opposite to the point charge which induces it.
The point charge induces charges of the opposite sign
on the conducting plate.
This gives rise to a force of attraction
between the charge and the plate.
Since the potential, and, hence, the electric field,
in the vicinity of the point charge is the same as
in the analogue problem,
then the force on the charge must be the same.
An Application of the Method of Images
In the analogue problem,
there are two charges +q and -q
a net distance 2d apart.
The force on the charge at position (0, 0,d)
(i.e., the real charge) is
An Application of the Method of Images
For the analogue problem the potential energy is just
The fields on opposite sides of the conducting plate are
mirror images of one another in the analogue problem.
So are the charges (apart from the change in sign).
This is why the technique of replacing conducting surfaces
by imaginary charges is called the method of images.
The potential energy of a set of charges is
equivalent to the energy stored in the electric field.
An Application of the Method of Images
In the analogue problem,
the fields on either side of the x-y plane are
mirror images of one another, so
An Application of the Method of Images
In the real problem
An Application of the Method of Images
To
A Point Charge near Grounded Spherical Conductor
A Point Charge near Grounded Spherical Conductor
Consider a grounded spherical conductor of radius a
placed at the origin.
Let a charge q be placed outside the sphere at (b,0,0) ,
where b > a.
To calculate the force of attraction between
the sphere and the charge,
consider an analogue problem in which
the sphere is replaced by an image charge -q’
placed somewhere on the x-axis at (c,0,0) .
A Point Charge near Grounded Spherical Conductor
In the analogue problem,
the electric potential throughout space is given by
The image charge is chosen so as to make
the surface Ф=0 corresponds to the surface of the sphere.
A Point Charge near Grounded Spherical Conductor
Setting the above expression to zero, and
performing algebraic calculations,
it is found that the Ф=0 surface satisfies
The surface of the sphere satisfies
A Point Charge near Grounded Spherical Conductor
The surface of the sphere satisfies
The above two equations can be made identical by setting
and
and
or
A Point Charge near Grounded Spherical Conductor
According to the uniqueness theorem,
the potential in the analogue problem
is identical with that in the real problem, outside the sphere.
(In the real problem, the potential inside the sphere is zero.)
Hence, the force of attraction between the sphere and
the original charge in the real problem is the same as
the force of attraction between
the two charges in the analogue problem. Hence
Applications of the Method of Images
There are many other image problems,
each of which involves replacing a conductor with
an imaginary charge (or charges)
which mimics the electric field in some region
(but not everywhere).