The magnetic field in the vacuum
§1 A magnetic field and its characteristics.

Mrot, pm, the lines of force  and

Permanent magnets have been known 2000 years ago, but only in 1820, H. Oersted (Danish physicist) found that around a conductor with a current creates a magnetic field, which affects the magnetic needle. Later, it was found that the magnetic field is produced by moving bodies, or any charges. The magnetic field, like the electric, is a type of matter. The magnetic field has energy. By means of the magnetic field the interaction between electric currents moving charges. Experience has shown that the effects of the magnetic field on the current varies depending on the shape of the conductor, in which the current flows, the location of the conductor and the direction of the current. Therefore, in order to characterize the magnetic field, it is necessary to consider the effect on a certain current.

For the study of the electric field using a test point charge. Similarly, for the study of the magnetic field using a current loop, whose dimensions are small compared with the distance to the currents that form a magnetic field. The orientation of the contour (with a current loop) in space is characterized by the normal to the contour. The positive direction of the normal is determined by the right-hand rule: the four fingers of his right hand in the direction of the current position in the loop, deflected at right angles to the thumb indicates the direction of the normal. The magnetic field exerts on the loop with current orienting effect. The current loop is installed in a magnetic field so that it coincides with the normal direction of the magnetic field lines.


Magnetic moment of current loop is a vector equal to the product of the current flowing through the loop on the vector square .

Direction  coincides with the direction .Direction  determined by the right-hand rule.

Because current loop experiences orienting action of the field, then it in a magnetic field exerts a force couple. Rotating moment forces depends on the properties of the field at a given point and the properties of current loop


- magnetic induction vector is a quantitative measure of force of the magnetic field.
The unit of measurement of magnetic induction – Tesla

Tesla .

If at a given point of the magnetic field to make a variety of current loop with the magnetic moments of p1, p2, ... pn, then the torque will be different for each current loop M1, M2, ... Mn, but the ratio


Magnetic induction

for all current loops is the same and can serve to characterize the magnetic field.

Magnetic induction  at a given point of a uniform magnetic field is numerically equal to the maximum torque M max , acting on the current loop with the magnetic moment of one, when the normal to the to the current loop is perpendicular to direction of the field ( also determined by the Lorentz force or Ampere force).

The direction of vector  coincides with the direction of  in the case when the current loop is in equilibrium and M rot .

A magnetic field conveniently represented with lines of force of vector .Force line of vector called a line whose tangent at any point coincides with the direction of at this point. The direction of lines of force of vector  

determined by the right-hand rule. For linear conductor: thumb in the direction of the current, bent four fingers indicate the direction of the field line. For a circular coil with a current: four fingers - on the current direction, the thumb indicates the direction of the field line in the center of the coil.

Lines of magnetic induction , unlike force lines of vector , of the electric field is always closed and covered conductor. (The lines of force of vector    

begin on positive charges and end on negative, approach perpendicular to the surface charge density of the lines of force characterizes the field.)

In some cases, along with the vector  applied vector of intensity of magnetic field ,which is associated with the vector  by ratio


µ0magnetic constant ; mu 0 ,

µ - magnetic permeability of the medium - shows how many times the magnetic field in the medium more (or less) of the magnetic field in the vacuum .


where B - the magnetic field in the material, B0 - external magnetizing field.

From a comparison of the characteristics of the electric field vector (vector   and a vector   ) and magnetic field (vector  and ) it follows that intensity vector  of electric field is similar to the magnetic induction .Both determine the effect of the force fields and depend on the properties of the medium in which the fields are created.

Analogue of the electric displacement is the vector of intensity of magnetic field . Vector  which describes the magnetic field macrocurrents macrocurrents - currents flowing through a conductor), so do not depend on the properties of the medium

Tesla   (Tesla);



§ 2 he Biot-Savart-Laplace’s Law

  1. . Bio and P. Savard in experimental studies of the magnetic fields produced by a current-carrying conductor, allowed theorist Pierre Simon de Laplace in 1820 to formulate the law of Biot-Savart-Laplace. This law determines the value  of at any point relative conductor.

Magnetic induction field, created by the conductor element , in which the current I flows , at some point A, whose position relative to the  determined by the radius vector , determined by the Biot-Savart-Laplace law





 - the Biot-Savart-Laplace law (in vector form) .

Because in the Biot-Savart-Laplace law there is a vector product , then  vector

must be perpendicular to the plane of the vectors  and . The direction of vectordetermined on the right-hand rule.

Modulus (magnitude) of the vector is equal to

 - the Biot-Savart-Laplace law (in scalar form)

where α –the angle between the  and .

2.The principle of superposition of fields:

Magnetic induction of the resulting field, multi-currents (or moving charges), equal to the geometric (vector) sum of the magnetic induction generated by each current separately.

3. Application of the Biot-Savart-Laplace’s law to the calculation of magnetic fields.

a) A magnetic field of the direct current





Since the induction created by different elementary sections, which we have broken conductor at this point have the same direction, we can sum the geometric vectors  replace the scalar summation

 - magnetic induction linear conductor of finite length.

���� ���������� �������� �����- intensity of the magnetic field of a conductor of finite length.

In the case of an infinitely long conductor



b) The magnetic field at the center of a circular current-carrying conductor






α = 90°; sin α = 1.







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