4. Power and Joule’s Law (Энергия и закон Джоуля-Ленца)

The Joule’s Law is a physical law that gives a quantitative assessment of the thermal effect of an electric current.

Consider a charge Q moves at a velocity v by an electric field E to a distance l. In this case, the expression of the work done is:

Power is the energy, which dissipated or consumed or generated over time unit:

, where Q – total charge:

So – Joule's law in differential form

For a constant cross-section conductor, the expression of the volume is:

– Joule's law in integral form

5. Continuity Equation (Уравнения непрерывности) ContinuityEquation (Уравнение непрерывности)

Consider a closed surface, where a wire carries a current I_in in the surface and the current I_out goes out from the surface.

General definitions:

Gausstheorem (которая Остроградского-Гаусса):

And we can conclude continuity equation:

for steady (постоянные) currents (charge is constant)

It is a differential form. It is one of the important laws in circuit theory. The current, which crosses some closed area, is equal to zero. It is the 1-st Kirchhoff's law.

But we may have , it is true only in time depending fields. But the 1-st Kirchhoff's law is valid both in steady currents, and in time depending currents. So, we have some contradiction or not?

Let's look at this with more attention.

J is the current of conductivity.

If displacement current is taken into account

δ – total current density.

,

The Gauss Law for the field displacement

Time derivative:

continuity equation: or

(differential and integral forms and it is the really the 1-st Kirchhoff's law)

Болееподробно:

and

Divergence of delta is equal to zero always and independently.

Continuity equation take place in any kind of system, but we can say that for conductivity current it is now always case, because in principle if our volume (object) accumulates (накапливает) current, then this divergence of J will not be equal to zero.

6. Electric field induced by the charged wire placed above the flat boundary between two different dielectrics (Электрическое поле, индуцируемое заряженным проводом, расположенным над плоской границей между двумя различными диэлектриками)

There are 2 half-spaces with different permittivity ε₁ and ε₂ and the point that corresponds to the infinitely long charged line with the linear charge density τ. This charged line suspended above this surface at the high h. We need to find the field intensity distribution everywhere in the space in both sub-spaces (above the interface and below the partition boundary.

The electrostatic problem is described by the Poisson and Laplace equations. This equation has a unique solution in the case when we have defined proper boundary conditions. This may be definition of the potential at the border of the considered space; or normal derivative of the electric potential or intensity of the electric field because the normal derivative of the potential is really the normal component of the electric field intensity or if the medium has the constant dielectric permittivity, then it is identical to the case when we shall define a normal component of the field displacement.

– the field induced by charged line source. This expression works only in the case when there are no surfaces. There is only one space with one dielectric permittivity is everywhere the same.

Let's suppose that the electric field in the upper half-space above the interface may be calculated as the superposition of two electric field. The first of them induced by initial wire itself and second of them induced by another wire, which is placed under the surface and which has a charge density ₁.

Dielectric constant is the same in both half-spaces ε₁. Let us place the image into the point of . Charge density of the image is ₁. So we can find field intensity in the boundaries:

Let’s suppose that the dielectric permittivity of the whole space is ε₂. Now there should not be any charge in the lower space. But above the surface there will be a charge with unknown density ₂. This charge, which is placed above the surface at the distance h. Field intensity at the boundaries:

Boundary conditions are:

Boundary conditions for horizontal component of the field intensity:

In the upper half-space:

In the lower half-space:

First relation: or

Boundary conditions for vertical component of the field intensity:

In the upper half-space

In the lower half-space

Second relation:

Combining with the first relation:

If the boundary conditions are satisfied then the solution is unique one. There are no other solutions. That is why such mirror reflection is the only one possible.

7. Magnetic field induced by the wire with a current placed above the flat boundary between two media with different magnetic permeabilities (Магнитное поле, индуцируемое проводом с током, расположенным над плоской границей между двумя средами с разной магнитной проницаемостью)