Find the magnetic flux f. Change in magnetic flux. Magnetic flux quantization

The picture shows a uniform magnetic field. Homogeneous means the same at all points in a given volume. A surface with area S is placed in a field. The field lines intersect the surface.

Determination of magnetic flux:

Magnetic flux Ф through the surface S is the number of lines of the magnetic induction vector B passing through the surface S.

Magnetic flux formula:

here α is the angle between the direction of the magnetic induction vector B and the normal to the surface S.

From the magnetic flux formula it is clear that the maximum magnetic flux will be at cos α = 1, and this will happen when vector B is parallel to the normal to the surface S. The minimum magnetic flux will be at cos α = 0, this will happen when vector B is perpendicular to the normal to the surface S, because in this case the lines of vector B will slide along the surface S without intersecting it.

And according to the definition of magnetic flux, only those lines of the magnetic induction vector are taken into account that intersect a given surface.

Magnetic flux is measured in webers (volt-seconds): 1 wb = 1 v * s. In addition, Maxwell is used to measure magnetic flux: 1 wb = 10 8 μs. Accordingly, 1 μs = 10 -8 vb.

Magnetic flux is a scalar quantity.

ENERGY OF THE MAGNETIC FIELD OF CURRENT

Around a conductor carrying current there is a magnetic field that has energy. Where does it come from? The current source included in the electrical circuit has a reserve of energy. At the moment of closing the electrical circuit, the current source spends part of its energy to overcome the effect of the self-inductive emf that arises. This part of the energy, called the current’s own energy, goes to the formation of a magnetic field. The energy of the magnetic field is equal to the intrinsic energy of the current. The self-energy of the current is numerically equal to the work that the current source must do to overcome the self-induction emf in order to create a current in the circuit.

The energy of the magnetic field created by the current is directly proportional to the square of the current. Where does the magnetic field energy go after the current stops? - stands out (when a circuit with a sufficiently large current is opened, a spark or arc may occur)

4.1. Law of electromagnetic induction. Self-induction. Inductance

Basic formulas

· Law of electromagnetic induction (Faraday's law):

, (39)

where is the induction emf; is the total magnetic flux (flux linkage).

· Magnetic flux created by current in the circuit,

where is the inductance of the circuit; is the current strength.

· Faraday's law as applied to self-induction

· Induction emf, which occurs when the frame rotates with current in a magnetic field,

where is the magnetic field induction; is the area of the frame; is the angular velocity of rotation.

Solenoid inductance

, (43)

where is the magnetic constant; is the magnetic permeability of the substance; is the number of turns of the solenoid; is the cross-sectional area of the turn; is the length of the solenoid.

Current strength when opening the circuit

where is the current established in the circuit; is the inductance of the circuit; is the resistance of the circuit; is the opening time.

Current strength when closing the circuit

. (45)

Relaxation time

Examples of problem solving

Example 1.

The magnetic field changes according to the law , where = 15 mT,. A circular conducting coil with a radius = 20 cm is placed in a magnetic field at an angle to the direction of the field (at the initial moment of time). Find the induced emf arising in the coil at time = 5 s.

Solution

According to the law of electromagnetic induction, the inductive emf arising in a coil is , where is the magnetic flux coupled in the coil.

where is the area of the turn; is the angle between the direction of the magnetic induction vector and the normal to the contour:.

Let's substitute the numerical values: = 15 mT,, = 20 cm = = 0.2 m,.

Calculations give .

Example 2

In a uniform magnetic field with induction = 0.2 T, there is a rectangular frame, the moving side of which, length = 0.2 m, moves at a speed = 25 m/s perpendicular to the field induction lines (Fig. 42). Determine the induced emf arising in the circuit.

Solution

When conductor AB moves in a magnetic field, the area of the frame increases, therefore, the magnetic flux through the frame increases and an induced emf occurs.

According to Faraday's law, where, then, but, therefore.

The “–” sign indicates that the induced emf and induced current are directed counterclockwise.

SELF-INDUCTION

Each conductor through which electric current flows is in its own magnetic field.

When the current strength changes in the conductor, the m.field changes, i.e. the magnetic flux created by this current changes. A change in magnetic flux leads to the emergence of a vortex electric field and an induced emf appears in the circuit. This phenomenon is called self-induction. Self-induction is the phenomenon of the occurrence of induced emf in an electrical circuit as a result of a change in current strength. The resulting emf is called self-induced emf

Manifestation of the phenomenon of self-induction

Circuit closure When there is a short circuit in the electrical circuit, the current increases, which causes an increase in the magnetic flux in the coil, and a vortex electric field appears, directed against the current, i.e. A self-induction emf arises in the coil, preventing the increase in current in the circuit (the vortex field inhibits the electrons). As a result L1 lights up later, than L2.

Open circuit When the electrical circuit is opened, the current decreases, a decrease in the flux in the coil occurs, and a vortex electrical field appears, directed like a current (trying to maintain the same current strength), i.e. A self-induced emf arises in the coil, maintaining the current in the circuit. As a result, L when turned off flashes brightly. Conclusion in electrical engineering, the phenomenon of self-induction manifests itself when the circuit is closed (the electric current increases gradually) and when the circuit is opened (the electric current does not disappear immediately).

INDUCTANCE

What does self-induced emf depend on? Electric current creates its own magnetic field. The magnetic flux through the circuit is proportional to the magnetic field induction (Ф ~ B), the induction is proportional to the current strength in the conductor (B ~ I), therefore the magnetic flux is proportional to the current strength (Ф ~ I). The self-induction emf depends on the rate of change of current in the electrical circuit, on the properties of the conductor (size and shape) and on the relative magnetic permeability of the medium in which the conductor is located. A physical quantity showing the dependence of the self-induction emf on the size and shape of the conductor and on the environment in which the conductor is located is called the self-induction coefficient or inductance. Inductance - physical. a value numerically equal to the self-inductive emf that occurs in the circuit when the current changes by 1 Ampere in 1 second. Inductance can also be calculated using the formula:

where Ф is the magnetic flux through the circuit, I is the current strength in the circuit.

SI units of inductance:

The inductance of the coil depends on: the number of turns, the size and shape of the coil and the relative magnetic permeability of the medium (possibly a core).

SELF-INDUCTION EMF

The self-inductive emf prevents the current from increasing when the circuit is turned on and the current from decreasing when the circuit is opened.

To characterize the magnetization of a substance in a magnetic field, it is used magnetic moment (P m ). It is numerically equal to the mechanical torque experienced by a substance in a magnetic field with an induction of 1 Tesla.

The magnetic moment of a unit volume of a substance characterizes it magnetization - I , is determined by the formula:

I=R m /V , (2.4)

Where V - volume of the substance.

Magnetization in the SI system is measured, like intensity, in Vehicle, a vector quantity.

The magnetic properties of substances are characterized volumetric magnetic susceptibility - c O , dimensionless quantity.

If any body is placed in a magnetic field with induction IN 0 , then its magnetization occurs. As a result, the body creates its own magnetic field with induction IN " , which interacts with the magnetizing field.

In this case, the induction vector in the medium (IN) will be composed of vectors:

B = B 0 + B " (vector sign omitted), (2.5)

Where IN " - induction of the own magnetic field of a magnetized substance.

The induction of its own field is determined by the magnetic properties of the substance, which are characterized by volumetric magnetic susceptibility - c O , the following expression is true: IN " = c O IN 0 (2.6)

Divide by m 0 expression (2.6):

IN " /m O = c O IN 0 /m 0

We get: N " = c O N 0 , (2.7)

But N " determines the magnetization of a substance I , i.e. N " = I , then from (2.7):

I = c O N 0 . (2.8)

Thus, if a substance is in an external magnetic field with a strength N 0 , then the induction inside it is determined by the expression:

B=B 0 + B " = m 0 N 0 +m 0 N " = m 0 (N 0 + I)(2.9)

The last expression is strictly true when the core (substance) is completely in an external uniform magnetic field (closed torus, infinitely long solenoid, etc.).

What is magnetic flux?

The picture shows a uniform magnetic field. Homogeneous means the same at all points in a given volume. A surface with area S is placed in a field. The field lines intersect the surface.

Magnetic flux definition

Magnetic flux definition:

Magnetic flux Ф through the surface S is the number of lines of the magnetic induction vector B passing through the surface S.

Magnetic flux formula

Magnetic flux formula:

here α is the angle between the direction of the magnetic induction vector B and the normal to the surface S.

And according to the definition of magnetic flux, only those lines of the magnetic induction vector are taken into account that intersect a given surface.

Magnetic flux is a scalar quantity.

Magnetic flux is measured

Magnetic flux is measured in webers (volt-seconds): 1 wb = 1 v * s.

In addition, Maxwell is used to measure magnetic flux: 1 wb = 10 8 μs. Accordingly, 1 μs = 10 -8 vb.

Right hand or gimlet rule:

The direction of the magnetic field lines and the direction of the current creating it are interconnected by the well-known rule of the right hand or gimlet, which was introduced by D. Maxwell and is illustrated by the following drawings:

Few people know that a gimlet is a tool for drilling holes in wood. Therefore, it is more understandable to call this rule the rule of a screw, screw or corkscrew. However, grabbing the wire as in the picture is sometimes life-threatening!

Magnetic induction B:

Magnetic induction- is the main fundamental characteristic of the magnetic field, similar to the electric field strength vector E. The magnetic induction vector is always directed tangentially to the magnetic line and shows its direction and strength. The unit of magnetic induction in B = 1 T is taken to be the magnetic induction of a uniform field, in which a section of conductor with a length of l= 1 m, with a current strength in it in I= 1 A, the maximum force of the Ampere acts from the field side - F= 1 H. The direction of the Ampere force is determined by the left-hand rule. In the CGS system, magnetic field induction is measured in gauss (G), in the SI system - in tesla (T).

Magnetic field strength H:

Another characteristic of the magnetic field is tension, which is an analogue of the electric displacement vector D in electrostatics. Determined by the formula:

Magnetic field strength is a vector quantity, is a quantitative characteristic of the magnetic field and does not depend on the magnetic properties of the medium. In the CGS system, magnetic field strength is measured in oersteds (Oe), in the SI system - in amperes per meter (A/m).

Magnetic flux F:

Magnetic flux Ф is a scalar physical quantity that characterizes the number of magnetic induction lines penetrating a closed circuit. Let's consider a special case. IN uniform magnetic field, the magnitude of the induction vector of which is equal to ∣B ∣, is placed flat closed loop area S. The normal n to the contour plane makes an angle α with the direction of the magnetic induction vector B. The magnetic flux through the surface is the quantity Ф, determined by the relation:

In the general case, magnetic flux is defined as the integral of the magnetic induction vector B through a finite surface S.

It is worth noting that the magnetic flux through any closed surface is zero (Gauss's theorem for magnetic fields). This means that the magnetic field lines do not break off anywhere, i.e. the magnetic field has a vortex nature, and also that it is impossible for the existence of magnetic charges that would create a magnetic field in the same way that electric charges create an electric field. In the SI, the unit of magnetic flux is Weber (Wb), in the CGS system it is Maxwell (Mx); 1 Wb = 10 8 μs.

Definition of inductance:

Inductance is a coefficient of proportionality between the electric current flowing in any closed circuit and the magnetic flux created by this current through the surface of which this circuit is the edge.

Otherwise, inductance is a proportionality coefficient in the self-induction formula.

In SI units, inductance is measured in henry (H). A circuit has an inductance of one henry if, when the current changes by one ampere per second, a self-inductive emf of one volt appears at the circuit terminals.

The term "inductance" was proposed by Oliver Heaviside, a self-taught English scientist in 1886. Simply put, inductance is the property of a current-carrying conductor to accumulate energy in a magnetic field, equivalent to capacitance for an electric field. It does not depend on the magnitude of the current, but only on the shape and size of the conductor carrying the current. To increase the inductance, the conductor is wound in coils, the calculation of which is what the program is dedicated to

Ampere's law is used to establish the unit of current, the ampere.

Ampere - the strength of a current of constant magnitude, which, passing through two parallel straight conductors of infinite length and negligibly small cross-section, located at a distance of one meter, one from the other in a vacuum, causes a force of .

(2.4.1)

Here ; ; ;

Let us determine from here the dimension and magnitude in SI.

, hence

, or .

From the Biot-Savart-Laplace law, for a straight conductor with current , Same you can find the dimension of the magnetic field induction:

Tesla is the SI unit of induction. .

Gauss– unit of measurement in the Gaussian system of units (GHS).

1 T is equal to the magnetic induction of a uniform magnetic field, in which a flat circuit with a current having a magnetic moment,torque is applied.

Tesla Nikola(1856–1943) – Serbian scientist in the field of electrical and radio engineering. He had a huge number of inventions. He invented the electric meter, frequency meter, etc. He developed a number of designs for multiphase generators, electric motors and transformers. He designed a number of radio-controlled self-propelled mechanisms. Studied the physiological effects of high frequency currents. In 1899 he built a 200 kW radio station in Colorado and a 57.6 m high radio antenna in Long Island (Wardenclyffe Tower). Together with Einstein and Openheimer, in 1943 he participated in a secret project to achieve the invisibility of American ships (Philadelphia experiment). Contemporaries spoke of Tesla as a mystic, clairvoyant, prophet, capable of looking into the intelligent cosmos and the world of the dead. He believed that with the help of an electromagnetic field one could move in space and control time.

Other definition: 1 T equal to magnetic induction, at which the magnetic flux through the area 1 m 2, perpendicular to the field direction,equals 1 Wb .

The unit of measurement of magnetic flux Wb, got its name in honor of the German physicist Wilhelm Weber (1804–1891), a professor at universities in Halle, Göttingen, and Leipzig.

As we already said, magnetic flux Ф through the surface S is one of the characteristics of the magnetic field(Fig. 2.5):

SI unit of magnetic flux:

. , and since , then .

Here Maxwell(Mks) is a unit of measurement of magnetic flux in the CGS named after the famous English scientist James Maxwell (1831–1879), creator of the theory of the electromagnetic field.

Magnetic field strength N measured in .

, .