§ 3 the Law of the total current.
Vortex nature of the magnetic field
 Circulation of vector (or ) in a closed loop is the integral over a closed contour L scalar product of vectors (or )and ,where  vectors of the unit length of the contour .
;
,
where –projection of vector on vector .
;
;
.
Law of the total current:
Circulation of vector an arbitrary closed loop is the sum of current covered by the circuit
;
.
Positive
are those currents, the direction of which to the direction of passage
of obeys the right hand rule. Currents, whose direction is opposite to
bypass, taken with the minus sign.
.
 In contrast to the electric field, for which the circulation of the vector equal zero and the electrostatic field is potential, the circulation of the magnetic field is not zero , if a path
on which we consider the circulation covers currents. Field, the
circulation of which is nonzero, is called a vortex or solenoidal.
Consequently, the magnetic field is a vortex. In vortex field force
lines are closed, therefore, there is no magnetic charges.
§ 4 A magnetic field the solenoid and toroid
Solenoid is cylindrical shell, which are wound windings of wire. Consider an infinitely long solenoid, ie solenoid which ℓ >> d, where ℓ  length , d –diameter of the
coil. Inside such a solenoid magnetic field is uniform. Uniform is a
field, the field lines are parallel and their density is constant.
Apply the law
of the total current to calculate the magnetic field of the solenoid.
Represent the contour L, which is considered by the circulation of the
vector , consisting of four related areas 12; 23, 34, 41. Then the circulation of the vector the chosen us contour L is equal to
.
;
, because and therefore , ,
,
because we have chosen the area 3  4 far enough from solenoid and one can assume that the
field far from the solenoid is zero,
, because and therefore, .
Circuit L includes N currents, where N  number of turns the solenoid, then the law of the total
current
;

 The magnetic field of an infinitely long solenoid
n  winding density  the number of turns per unit length
.
The field intensity inside the solenoid is equal to the number of turns per unit length of the
solenoid, multiplied by the current.
Toroid
 torus, with coils wound on a wire. Unlike a solenoid, which has a
magnetic field, both inside and outside, fully toroidal magnetic field
is concentrated inside the coils, ie there is no dissipation of
magnetic field energy.
,
where .
–magnetic field of toroid.
if R >>R_{turn}, then R ≈r and H = nI.
§5 Ampere force
 Ampere studied the effect of magnetic field on the currentcarrying conductors and found that the force , with which the magnetic field acts on the element wire with current I,in a magnetic field , directly proportional to the current I and the vector product on the magnetic induction
–
The Ampere force (or Ampere's law)
The direction
of the Ampere force is situated by the rule of the vector product  on
lefthand rule: four elongated fingers of his left hand placed on the
direction of the current, vector included
in the palm, deflected at right angles to the thumb will show the
direction of force acting on a currentcarrying conductor. (You can
also determine the direction of with his right hand: turn the four fingers of the right hand of the first factor to second , thumb indicates the direction of .)
Module of Ampere force
,
where α – the angle between the vectors and .
If the field is uniform, and the currentcarrying conductor of finite size, the
,
.
At perpendicular
.
2. Definition of the unit of measurement of the current.
Any
currentcarrying conductor generates a magnetic field around itself. If
you put it in the field of the other currentcarrying conductor, the
conductor between the forces of interaction. In this case,
codirectional parallel currents attract each the opposite direction 
are repelled
Consider two infinitely long parallel conductors with currents I_{1} and I_{2}, in a vacuum at a distance d (for vacuum μ = 1). According to Ampere's law
.
A magnetic field of direct current is
,
then
,
the force per unit length of the conductor
.
The force
per unit length of the conductor between two infinitely long conductor
with a current directly proportional to the current in each conductor
and inversely proportional to the distance between them.
Definition of the unit of measurement of the current  Ampere:
Per unit
of current in the SI in place a DC current which is flowing in two
infinitely long parallel conductors infinitesimal cross section,
located in vacuum at a distance of 1 m from each other, is the force
exerted per unit length of the conductor is equal to 2·10^{7} N.
µ = 1; I_{1} = I_{2} = 1 A; d = 1 m; µ_{0} = 4π·10^{7} H/m –magnetic constant .
.
§6 The Lorentz force
Under the Ampere’s law, force acting on the current element , determined by the formula
.
Consider that the elementary current is none other than the directional movement of electric charges
,
where V –volume , n –the carrier density , j –current density , S –
crosssectional area of the
conductor, e – electron charge (e = 1,6·10^{19} C), dl  the element length of the conductor, –velocity of the electron motion .
;
;
.
Ampere force acting on the elementary current can be seen as
the resultant force of the all forces exerted by the magnetic field on
each charge separately. Then, the force acting on a moving charge in a
magnetic field, we find by dividing the number of Ampere charge in
this volume element of the conductor
.
This force is called the Lorentz force:
.
–module of the Lorentz force
The direction of the Lorentz force is determined by the lefthand rule: four fingers of his left hand  the speed, the vector enters
in the palm, deflected at right angles to the direction of the thumb
shows the Lorentz force to the positive charge. For a negative charge 
four fingers against the speed, then the same as for the positive
charge.
