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.

3. Electrostatic Energy (Электростатическая энергия) Electrostatic Energy (Электростатическаяэнергия)

The force in electric field:

The work (from mechanics):

Definition of the potential difference:

For infinitely remote point "a":

The work is done by the electric field without any external force. This is insolated system.

Energy conservation law (total energy can't be changed):

(W – potential energy, A – work).

But what exactly work do? This work in insulated system (where only charges and electric field exist) transformed into kinetic energy if there no other participles of all this event. (честно, я не понял, почему он тут вдруг начал говорить про кинетическую энергию, хотя до этого всё время говорил про потенциальную)

So, we have got:

But if there external force which moves a charge and moves so that velocity of charge is constant (or velocity at the start = velocity at the finish point), so there is no change of kinetic energy. In this case, the external force applied to electric charge look like that (equal but with the opposite sign): . And after all we get the same relation: .

Virtual experiment. (Эксперимент по нахождению энергии системы)

Consider three points P₁, P₂, and P₃ in a charge free space. The point charges Q₁ ,Q₂ , and Q₃ are brought from infinity to those points, respectively and in turn (соответственно и по очереди).

No work is done in bringing point charge Q₁ from infinity to point P₁ (because there wasn't electric field in the beginning). Then we have

U₂₁ is the potential induced at point P₂ due to Q₁,

U₃₁ is the potential induced at point P₃ due to Q₁ ,

U₃₂ is the potential induced at point P₃ due to Q₂.

If we want to find a potential in any point, we need to summarize the potentials, which induced by all charges.

Now we reverse this experiment and we move at the beginning charge Q₃, then Q₂, then Q₁.

U₂₃ is the potential induced at point P₂ due to Q₃,

U₁₂ is the potential induced at point P₁ due to Q₂ ,

U₁₃ is the potential induced at point P₁ due to Q₃.

And if we add two equation we get:

or

U₁ is the potential induced at point P₁ due to Q₂ and Q₃,

U₂ is the potential induced at point P₂ due to Q₁ and Q₃ ,

U₃ is the potential induced at point P₃ due to Q₁ and Q₂.

Consequences (Следствия)

In general case:

If the charges are distributed over the space:

for the space charges (U – function because it distributed somehow over the space, where we do integrate). These charges really may exist.

for the surface charges (for example – conducting electrode)

for the line charges (it doesn't exist in our world, because we can't compress the charges into infinitely thin line, but in practice we can have ALMOST linear charges (energy transfer lines – wires) and we can approximate it as a thin line? but bad consequences may happen because of approximation – it may give us an infinite W_E and we can't to calculate it)

Substituting in the expression for the energy of charges distributed in the volume (now it's not only electrostatic field, but in electric field this consequence work too):

(for any field)

Applying the Gauss theorem (not Gauss law for field! по-русскиэтоформулаОстроградского - Гаусса) to the first term:

We have static system, which is limited (not infinitely). So, we have a point, which is very far from system. So far, that system may seemed like a point (may be charge point, may be not). And if we are going to infinity our potential and displacement induced by this point are inversed proportional to the radius.

Area of a spherical surface

Energy:

Finally, we get:

Also, we can conclude something (but it is a kind of a sumption [предположение]). Нельзя говорить, что мы выделим в области объём, и там будет какая-то энергия. Энергия W_E=Q*U говорит нам, что на самом деле энергия концентрируется в заряде, а через объёмную плотность энергии формула говорит, что энергия концентрируется вокруг заряда. Что из этого правильно и нет – может быть потом он расскажет, а может и нет.

Energy per unit volume (volume density of electric energy):

For isotropic medium: