Work in the electric field. (Работа в электрическом поле)
The field performs work on charges.
Conversely, to move the charge against the field requires one to do work with opposite sign.
T
he
force
The work:
or
For infinitely remote point A:
.
In infinitely remote point U=0 from Coulomb's law.
The work against field forces
;
this work increases the field energy.
Dielectric polarization. (Диэлектрическая поляризация)
Dielectric materials are polarized when an electric field is applied to them. As a result, electric displacement becomes greater than under the free space condition:
The polarization is defined as the dipole moment
per unit volume.
In isotropic and linear materials, the polarization is directly proportional to the electric field and the proportionality constant does not depend on the field
,
where
is the
susceptibility of
dielectric material
(диэлектрическая восприимчивость)
(читается как "хи" на русском, "хай"
на английском).
Dielectric material characteristics. (Характеристики диэлектриков)
In the linear dielectric material, the field intensity is proportional to the displacement
is the dielectric permittivity (dielectric
constant)
is the relative dielectric permittivity
(dielectric constant)
In the case of extremely high electric field the electrons will accelerate and collide with molecular lattice structure, which causes permanent damage to the material. This phenomenon is known as dielectric breakdown (пробой диэлектрика).
Properties of dielectric materials. (Свойства диэлектрических материалов)
Poisson’s and Laplace’ s equations. (Уравнения Пуассона и Лапласа)
and in the absence of free charges
(Gauss theorem) because
.
For the linear dielectric
Laplace’s equation
In general case
and for the linear dielectric
Poisson’s equation
is Laplace operator.
We are talking about only static fields, which don't depend on time. Otherwise, these equations don't work moreover, the potential simply doesn't exist.
If dielectric permittivity is not constant, we can split our system on several (linear – permittivity is constant and doesn't depend on the field intensity or displacement) systems and consider separately.
Лекция 2. Boundary conditions for the Laplace or Poisson equations (Граничные уравнения для уравнений Лапласа и Пуассона)
Laplace's (Poisson's) equation has unique solution:
These colors mean:
1-st type boundary conditions (Dirichlet boundary
conditions):
.
It means, that we simply define a value of the potential on the border. It may be constant, but also that may be a function of the point. So, in any point of this line we should define different potential values.
2-nd type boundary conditions (Neumann boundary
conditions):
.
A normal derivative of potential at the border is equal to something. This "something" may be zero and really that is most often used case. But in general theory may be equal any kind of function, but this function should be predefined before we start solving.
3-rd type boundary conditions:
.
If we are now (it's not often) the property of potential in such a form then we can apply them to the part of the boundary, or the whole boundary.
Normal derivative of potential plus potential is itself is equal to something.
If the whole boundary is defined by these conditions, then mathematic tells us: the solution of the problem exists, and this solution is unique.
It not property of electromagnetic fields. It is only property of some mathematical operator. The same conclusion for Poisson's equation.