Continuity Equation (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:

Gauss theorem (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)

Более подробно:

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.

Лекция 3. Static magnetic field (Статическое магнитное поле)

Basic property of static (or stationary) magnetic field that it shall not depend on time.

Variables and units (Переменные и единицы измерения)

Variable	Symbol	Units
Flux density Индукция магнитного поля	B	Tesla	[T]
Field intensity Напряжённость поля	H	Ampere/meter	[A/m]
Magnetic permeability Магнитная проницаемость	μ	Henry/meter	[H/m]
Inductance Индуктивность	L	Henry	[H]
Flux Поток	Ф	Weber	[Wb]
Flux linkage Потокосцепление	ψ	Weber	[Wb]
Scalar magnetic potential Скалярный магнитный потенциал	Um	Ampere	[A]
Vector magnetic potential Векторный магнитный потенциал	A	T·m (Wb/m)	[T·m]
Magnetization Намагниченность	M	Ampere/meter	[A/m]

Main Relations (Основные соотношения)

(! attention to difference between electric and magnetic field. In electric field coefficient ε is called dielectric permittivity, in magnetic μ - magnetic permeability)

(μ is often it is scalar coefficient. May be constant, may depend on H or B. But sometimes μ tensor value. In such relation tensor will change value of the vector or even direction of the vector in anisotropic medium cases)