or is shorter: where superficial integral All volume of V is extended to the sum of platforms of dS1 and dS it is possible to divide into elementary cylinders of the considered look and to write the same ratios for each of them. Summarizing these ratios, we will receive:
The equation (it turns out 1 by integration of a ratio (on any surface of S with the subsequent transformation of the left part according to Stokes's theorem in integral on the contour of limiting S surface. The equation (it turns out 1 in the same way from a ratio (. The equations (1 and (turn out 1 from ratios (and (by integration on any volume of V with the subsequent transformation of the left part according to Ostrogradsky-Gauss's theorem in integral on the closed surface of S limiting volume
The first two composed in (3 and (3 represent the potential of the uniform external field created by external sources. The second is a potential of the electric field created by the electric sphere polarized by an external field. Out of the sphere is a potential of a dipole with the dipolar moment. In the sphere the polarized sphere creates uniform electric field with intensity
Knowing a vector rotor in each point of some (not necessarily flat) S surfaces, it is possible to calculate circulation of this vector on the contour limiting S (the contour can also be not flat). For this purpose a surface on very small elements. In view of their trifle these elements can be considered flat. Therefore according to (the 3rd circulation of a vector on the contour limiting can be presented in the form.
Opening of current of shift allowed Maxwell to create the uniform theory of the electric and magnetic phenomena. This theory explained all the experimental facts known at that time and predicted a number of the new phenomena which existence was confirmed subsequently. The conclusion about existence of the electromagnetic waves extending with velocity of light was the main consequence of the theory of Maxwell.
As in practice it is almost always necessary to solve Maxwell's equations (– (in piecewise and continuous environments, boundary conditions (2 it is necessary to consider as an integral part of the equations of Maxwell (– (.
Let's construct infinitely narrow cylinder, one now of forming which the piece 1 Let of dσ - the area of its cross section is (size positive). Multiplying the previous ratio by dσ. As dσdx is the elementary volume of dV shaded in drawing, as a result it will turn out:
Thus, Maxwell's equations (-(have to be added with boundary conditions (1, (1, (2 and (These conditions mean a continuity of tangential components of a vector (2 and a normal component of a vector (1 upon transition through limit of the section of two environments. The normal component of a vector upon transition through limit of the section tests jump, a tangential component of a vector if there are superficial currents (
Ratio (3 carries the name of the theorem of Stokes. Its sense consists that circulation of a vector on any contour is equal to a vector stream through any surface of S limited to this contour.
Here – the solution of the equation out of the sphere, and – in the sphere. Instead of a boundary condition of a continuity of tangential components of electric field it is possible to use a potential continuity condition equivalent to it
As an example of the solution of electrostatic tasks it is possible to calculate the electric field created by a dielectric sphere of radius of R, being in uniform electric field. The electrostatics equations in dielectric (2 at =0 have an appearance:
In a boundary condition (2 there is an area density of current, superfluous in relation to magnetization currents. If currents are absent, it is necessary to put = Considering that, and there is an area density of current of magnetization, will write down a formula (2 in a look:
In total 8 equations in which 12 functions (on three components of vectors enter turned out.) As the number of the equations are less than number of known functions, equations (-(there is not enough for finding of fields on the set distributions of charges and currents. To perfrom calculation of fields, it is necessary to add Maxwell's equations with the equations connecting and with, and also page. These equations have an appearance.