Magnetic field and magnetic induction

Magnetic field and magnetic induction

The conversion of electrical energy in induction-type machines is carried out using electromagnetic fields. A magnetic field is called space, the state and properties of which are determined by the movement of electric charges. Two manifestations are inherent in a magnetic field: force action on moving electric charges and excitation of an electric field when its state changes. The magnetic field at each point is characterized by two quantities: the magnetic induction vector and the intensity vector H.

B Magnetic induction can be determined by the mechanical force acting on the charges moving in the field, as well as by the electromotive force induced in a moving conductor or in a stationary conductor located in an alternating magnetic field. It is known from experience that a force acts on a linear conductor with current i and length l in a uniform magnetic field, the value of which is defined as

Magnetic field and magnetic induction
Magnetic field and magnetic induction

where is the magnetic field induction vector, is the vector whose direction corresponds to the location of the conductor.

Β l Expression (1.1) allows one to define induction as a force acting on a conductor one meter long, located perpendicular to the vector along which a current of one ampere flows. The unit of induction is Tesla [T] = [V⋅s / m Β 2].

In the general case, the force acting on an arbitrary conductor with current is equal to Experience shows that in a homogeneous medium around a rectilinear conductor with current a magnetic field is formed,

Magnetic field and magnetic induction
Magnetic field and magnetic induction

the induction vector of which at each point in space is located in the plane perpendicular to the conductor and coincides in the direction tangent to the circle with the center on the axis of the conductor (Fig. 1.1). The magnitude of induction at any point A can be determined on the basis of BioSavar’s law, if we assume that each current element creates its own magnetic field and idl total induction at this point is equal to the geometric sum of elementary components

Magnetic field and magnetic induction
Magnetic field and magnetic induction

where s is a vector having a value equal to the distance from the element to point A and directed to this point.

dl Thus, elementary magnetic induction in a homogeneous medium is directed normally to the plane passing through d and s, is proportional to the magnitude of the elementary current, inversely proportional to the square of the distance s and proportional to the sine of the angle between the directions of the current id and. dB l i dl ⋅ l s The proportionality coefficient k depends on the physical properties of the medium and on the choice of units of measurement. In the international SI system of units, the coefficient k for vacuum is, where 0 / (4) k = μ π 7 0 4 10 −μ = π⋅ Gn / m is the magnetic constant or magnetic permeability of the vacuum. For any other medium, where is the absolute magnetic permeability, usually represented through the magnetic permeability of a vacuum using a dimensionless quantity μ called the relative magnetic permeability.

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In the future, we will see that the magnetic permeability of all media except ferromagnetic differs insignificantly from the magnetic permeability of vacuum, and therefore, in practical calculations, we can take 0 aμ ≈μ and μ≈ for them. 1 The directions of elementary magnetic inductions created by all elementary currents at point A coincide. Therefore, when determining the total induction, they can be summed up arithmetically

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From fig. 1.1 it follows that

From here

Magnetic field and magnetic induction
Magnetic field and magnetic induction

The value of induction at point A, lying on the axis of the ring conductor and remote from its plane by a distance l (Fig. 1.2, a), can be determined

summing up the elementary magnetic inductions, taking into account the fact that the angle between the current directions id and s is 90 °. Then

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Elementary induction can be decomposed into two components, one of which is directed along the axis of the ring conductor

and the other is perpendicular to this axis

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Elementary currents located diametrically create oppositely directed transverse components that are mutually compensated. Therefore, the total magnetic induction at point A on the axis of the ring conductor is determined by summing the longitudinal components

Using expression (1.8), we can determine the value of magnetic induction on the axis of a cylindrical coil (solenoid) of length l, consisting of w turns located closely adjacent to each other (Fig. 1.2, b). Dividing the entire length l into elements, each of which is a ring current dx widx /l, we can determine the induction by summing

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Expression (1.9) with the fact that

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can bring to mind

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In the case of an infinitely long solenoid 12 0; α1 = α2 = π and the magnetic induction on its axis will be equal to

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The calculated vector value equal to the ratio of magnetic induction to the magnetic permeability of the medium at the same point is called the magnetic field strength

In a homogeneous medium, the magnetic field can be determined through elementary components

repeating the calculations done to determine the induction. Then the magnetic field strengths of the linear conductor and the infinitely long solenoid at the points corresponding to Fig. 1.1 and 1.2 will be equal respectively

Magnetic field and magnetic induction
Magnetic field and magnetic induction

From the expressions (1.13), (1.14) it follows that the unit of the magnetic field is A / m.

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