33. Boundary conditions in electric and magnetic fields
1) First type boundary conditions
L
et’s
assume we know 1-st type boundary conditions.
W
hat
does that mean? We know potential distribution over the boundary, so
we simply do not need to solve this equation in the boundary nodes,
because we already know the solution, that’s why this second term,
which includes the integral over the whole boundary is not used at
all.
It is not necessary for the nodes which are placed inside the problem domain, and we already know the solution for all external nodes. That’s a lucky situation and so we completely got view of this second term. In principle, sometimes when we work with the second type boundary conditions the situation is more complicated. The 1-st type boundary conditions keep a symmetry of the main problem matrix.
2) 2-nd type boundary conditions (Второй тип граничных условий)
2-nd type boundary conditions assume the normal derivative of the potential is equal to some function.
And we have here an integral along the boundary and we can substitute the boundary conditions of the second type inside this relation and we shall get a final result.
T he equation for the external nodes, which are placed at the boundary, are more complex. The exception corresponds to the case when this function F2 is equal to zero, in such a case the form of the coefficients will be identical because this derivative is equal to zero the first integral in the right part also will be equal to zero. The 2-nd type boundary conditions of the dU/dn = 0 is most often used boundary condition, so in such a case if we have such a situation than we can say the expression for the coefficients within the finite element method is always the same and it is described by these simple relations.
34. Main equations of electromagnetic field in integral form.
1) Gauss’s Law
Definition of the electric flux:
T otal electric flux passing any closed surface is equal to the total charge enclosed by that surface.
Gauss law for the field displacement Gauss law for the field intensity
2) Electric potential
The
integral
solely (только)
depends on the points A and B but not on the particular path taken
from A to B.
Definition of the potential:
3) Ohm’s Law
4) Joule’s law
P – power
5) Gauss’s Law for for the magnetic field:
There are no magnetic charges in the nature:
6) Amphere’s Law
I – is the current crossing the surface limited by this contour
7) Faraday’s Law
The electromotive force induced in a closed contour is equal to rate of changing magnetic flux in time with opposite signs
35. Main equations of electromagnetic field in differential form.
1) Gauss’s Law
Definition of the electric flux:
T otal electric flux passing any closed surface is equal to the total charge enclosed by that surface.
Gauss law for the field displacement Gauss law for the field intensity
2) Electric potential
– nabla
3) Laplace’s equation
(When )
– delta (Laplace operator)
4) Poisson’s equation
– nabla
5) Ohm’s Law
– current density
– conductivity
6) Joule’s law
P – power
7) continuity equation
for steady currents (charge is constant):
8)
Gauss’s law for magnetism
(i.e., magnetic field has no sources)
9) Ampere’s law
10) Scalar magnetic potential
Um – magnetic potential. The unit is – A (Ampere)
11)
12) Differential equation for the vector magnetic potential
13) Faraday’s Law of induction
The curl of electric field E is equal to the negative rate of change of the magnetic field B
