Definition

The force acting on a conductor with a current in a magnetic field is called by Ampere... Its designations:. Ampere force is a vector quantity. Its direction is determined by the rule of the left hand: you should position the palm of your left hand so that the lines of force of the magnetic field enter it. The extended four fingers indicated the direction of the amperage. In this case, the bent thumb will indicate the direction of the Ampere force (Fig. 1).

Ampere's law

Ampere's elementary force is determined by Ampere's law (or formula):

where I is the current strength, is a small element of the length of the conductor, is a vector equal in magnitude to the length of the conductor, directed in the same direction as the current density vector, is the induction of the magnetic field in which the conductor with current is placed.

Otherwise, this formula for the Ampere force is written as:

where is the current density vector, dV is the volume element of the conductor.

Ampere's modulus is found in accordance with the expression:

where is the angle between the vectors of magnetic induction and the direction of the current flow. From expression (3) it is obvious that the Ampere force is maximum in the case of perpendicularity of the lines of magnetic induction of the field with respect to the conductor with current.

Forces acting on conductors with current in a magnetic field

It follows from Ampere's law that a conductor with a current equal to I is acted upon by a force equal to:

where the magnetic induction considered within a small piece of the conductor dl. Integration in formula (4) is carried out along the entire length of the conductor (l). From expression (4) it follows that a closed circuit with a current I, in a uniform magnetic field, acts on an Ampere force equal to

The Ampere force, which acts on the element (dl) of a straight conductor with a current I 1, placed in a magnetic field, which creates another straight conductor parallel to the first with a current I 2, is equal in magnitude:

where d is the distance between the conductors, H / m (or N / A 2) is the magnetic constant. Conductors with currents in the same direction attract. If the directions of the currents in the conductors are different, then they are repelled. For the parallel conductors of infinite length considered above, the Ampère force per unit length can be calculated by the formula:

Formula (6) in the SI system is used to obtain a quantitative value of the magnetic constant.

Ampere force units

The main unit of measurement for force Ampere (like any other force) in the SI system is: = H

In the SGS: = din

Examples of problem solving

Example

Exercise. A straight conductor of length l with current I is in a uniform magnetic field B. Force F acts on the conductor. What is the angle between the direction of current flow and the vector of magnetic induction?

Solution. A current-carrying conductor in a magnetic field is acted upon by the Ampere force, the modulus of which for a straight conductor with current located in a uniform field can be represented as:

where is the required angle. Hence:

Answer.

Example

Exercise. Two thin, long conductors with currents lie in the same plane at a distance d from each other. The width of the right conductor is a. Currents I 1 and I 2 flow through the conductors (Fig. 1). What is the Ampere force acting on the conductors per unit length?

Solution. As a basis for solving the problem, we take the formula for the elementary force of Ampere:

We will assume that a conductor with a current I 1 creates a magnetic field, and another conductor is in it. Let's look for the Ampere force acting on a conductor with a current I 2. Let's select in the conductor (2) a small element dx (Fig. 1), which is located at a distance x from the first conductor. The magnetic field, which creates the conductor 1 (the magnetic field of an infinite rectilinear conductor with current) at the point of location of the element dx according to the circulation theorem can be found as.

Ampere's law shows with what force the magnetic field acts on the conductor placed in it. This power is also called by Ampere.

The wording of the law:the force acting on a current carrying conductor placed in a uniform magnetic field is proportional to the length of the conductor, the magnetic induction vector, the current strength and the sine of the angle between the magnetic induction vector and the conductor.

If the size of the conductor is arbitrary, and the field is non-uniform, then the formula is as follows:

The direction of the Ampere force is determined by the left hand rule.

Left hand rule: if you position your left hand so that the perpendicular component of the magnetic induction vector enters the palm, and four fingers are extended in the direction of the current in the conductor, then set aside by 90° the thumb will indicate the direction of the Ampere force.

MP of the driving charge. MF action on a moving charge. Force of Ampere, Lorentz.

Any conductor with current creates a magnetic field in the surrounding space. In this case, the electric current is the ordered movement of electric charges. This means that we can assume that any charge moving in a vacuum or medium generates a magnetic field around itself. As a result of generalization of numerous experimental data, a law was established that determines the field B of a point charge Q moving with a constant nonrelativistic speed v. This law is given by the formula

(1)

where r is the radius vector, which is drawn from the charge Q to the observation point M (Fig. 1). According to (1), vector B is directed perpendicular to the plane in which the vectors v and r are located: its direction coincides with the direction of translational motion of the right screw as it rotates from v to r.

Fig. 1

The modulus of the magnetic induction vector (1) is found by the formula

(2)

where α is the angle between the vectors v and r. Comparing the Bio-Savart-Laplace law and (1), we see that a moving charge is equivalent in its magnetic properties to a current element: Idl = Qv

MF action on a moving charge.

It is known from experience that a magnetic field has an effect not only on conductors with current, but also on individual charges that move in a magnetic field. The force that acts on an electric charge Q moving in a magnetic field with a speed v is called the Lorentz force and is given by the expression: F = Q where B is the induction of the magnetic field in which the charge moves.

To determine the direction of the Lorentz force, we use the left hand rule: if the palm of the left hand is positioned so that the vector B enters it, and four outstretched fingers are directed along the vector v (for Q> 0, the directions I and v coincide, for Q Fig. 1 shows mutual orientation of vectors v, B (the field has a direction on us, shown by dots in the figure) and F. If the charge is negative, then the force acts in the opposite direction.

E.m.s. electromagnetic induction in the circuit is proportional to the rate of change of the magnetic flux Фm through the surface bounded by this circuit:

where k is the coefficient of proportionality. This emf does not depend on what caused the change in the magnetic flux - either the movement of the circuit in a constant magnetic field, or a change in the field itself.

So, the direction of the induction current is determined by Lenz's rule: For any change in the magnetic flux through a surface bounded by a closed conducting loop, in the latter an induction current appears in such a direction that its magnetic field counteracts the change in the magnetic flux.

A generalization of Faraday's law and Lenz's rule is the Faraday-Lenz law: The electromotive force of electromagnetic induction in a closed conducting loop is numerically equal and opposite in sign to the rate of change of the magnetic flux through the surface bounded by the loop:

The quantity Ψ = ΣΦm is called flux linkage or total magnetic flux. If the flow through each of the loops is the same (i.e., Ψ = NΦm), then in this case

German physicist G. Helmholtz proved that the Faraday-Lenz law is a consequence of the law of conservation of energy. Let the closed conducting circuit be in an inhomogeneous magnetic field. If a current I flows in the circuit, then under the action of the Ampere forces, the loose circuit will begin to move. The elementary work dA performed when the contour is moved in time dt will be

dA = IdФm,

where dФm is the change in the magnetic flux through the area of ​​the circuit during the time dt. The work of the current during the time dt to overcome the electrical resistance R of the circuit is equal to I2Rdt. The total work of the current source during this time is equal to εIdt. According to the law of conservation of energy, the work of the current source is spent on the two named works, i.e.

εIdt = IdФm + I2Rdt.

Dividing both sides of the equality by Idt, we get

Consequently, when the magnetic flux coupled to the circuit changes, the electromotive force of induction arises in the latter

Electromagnetic vibrations. Oscillatory circuit.

Electromagnetic vibrations are vibrations of such quantities, inductance as resistance, EMF, charge, current.

An oscillating circuit is an electrical circuit that consists of a capacitor, a coil and a resistor connected in series.The change in the electric charge on the capacitor plate over time is described by the differential equation:

Electromagnetic waves and their properties.

In the oscillatory circuit, the process of transition of the electrical energy of the capacitor into the energy of the magnetic field of the coil takes place and vice versa. If at certain points in time to compensate for the energy loss in the circuit for resistance due to an external source, then we get continuous electrical oscillations, which can be radiated through the antenna into the surrounding space.

The process of propagation of electromagnetic waves, periodic changes in the strengths of electric and magnetic fields, in the surrounding space is called an electromagnetic wave.

Electromagnetic waves cover a wide range of wavelengths from 105 to 10 m and frequencies from 104 to 1024 Hz. By name, electromagnetic waves are divided into radio waves, infrared, visible and ultraviolet radiation, X-rays and radiation. Depending on the wavelength or frequency, the properties of electromagnetic waves change, which is a convincing proof of the dialectical-materialistic law of the transition of quantity into a new quality.

The electromagnetic field is material and has energy, momentum, mass, moves in space: in a vacuum at a speed C, and in a medium at a speed: V =, where = 8.85;

The volumetric energy density of the electromagnetic field. The practical use of electromagnetic phenomena is very wide. These are systems and means of communication, radio broadcasting, television, electronic computers, control systems for various purposes, measuring and medical devices, household electrical and radio equipment and others, i.e. something without which it is impossible to imagine modern society.

There is almost no exact scientific data on how powerful electromagnetic radiation affects human health, there are only unconfirmed hypotheses and, in general, not unfounded fears that everything unnatural is destructive. It has been proven that ultraviolet, X-ray and high-intensity radiation in many cases cause real harm to all living things.

Geometric optics. Civil defense laws.

Geometric (ray) optics uses the idealized concept of a light ray - an infinitely thin beam of light propagating rectilinearly in a homogeneous isotropic medium, as well as the concept of a point radiation source that shines uniformly in all directions. λ - light wavelength, - characteristic size

an object in the path of the wave. Geometric optics is the limiting case of wave optics and its principles are fulfilled provided the following conditions are met:

h / D<< 1 т. е. геометрическая оптика, строго говоря, применима лишь к бесконечно коротким волнам.

Geometric optics is also based on the principle of independence of light rays: the rays do not disturb each other when they move. Therefore, the movements of the rays do not prevent each of them from propagating independently of each other.

For many practical problems of optics, one can ignore the wave properties of light and consider the propagation of light to be rectilinear. In this case, the picture is reduced to considering the geometry of the path of light rays.

Basic laws of geometric optics.

Let us list the basic laws of optics following from experimental data:

1) Rectilinear propagation.

2) The law of independence of light rays, that is, two rays, crossing, do not interfere with each other in any way. This law is in better agreement with wave theory, since particles, in principle, could collide with each other.

3) The law of reflection. the incident ray, the reflected ray and the perpendicular to the interface, reconstructed at the point of incidence of the ray, lie in the same plane, called the plane of incidence; the angle of incidence is equal to the angle

Reflections.

4) The law of refraction of light.

Refraction law: the incident ray, the refracted ray and the perpendicular to the interface, reconstructed from the point of incidence of the ray, lie in the same plane - the plane of incidence. The ratio of the sine of the angle of incidence to the sine of the angle of reflection is equal to the ratio of the speeds of light in both media.

Sin i1 / sin i2 = n2 / n1 = n21

where is the relative refractive index of the second medium relative to the first medium. n21

If substance 1 is a void, a vacuum, then n12 → n2 is the absolute refractive index of substance 2. It can be easily shown that n12 = n2 / n1, in this equality on the left is the relative refractive index of two substances (for example, 1 is air, 2 is glass) , and on the right is the ratio of their absolute refractive indices.

5) The law of reversibility of light (it can be derived from Law 4). If you direct the light in the opposite direction, it will follow the same path.

From law 4) it follows that if n2> n1, then Sin i1> Sin i2. Now let us have n2< n1 , то есть свет из стекла, например, выходит в воздух, и мы постепенно увеличиваем угол i1.

Then it can be understood that when a certain value of this angle (i1) pr is reached, it turns out that the angle i2 will be equal to π / 2 (ray 5). Then Sin i2 = 1 and n1 Sin (i1) pr = n2. So Sin

Forces acting on the conductor.

In an electric field, on the surface of a conductor, namely here, electric charges are located, certain forces act from the side of the field. Since the strength of the electrostatic field on the surface of a conductor has only a normal component, the force acting on an element of the surface area of ​​the conductor is perpendicular to this element of the surface. The expression for the considered force, referred to the value of the area of ​​the element of the surface of the conductor, has the form:

(1)

where is the outer normal to the surface of the conductor, is the surface density of the electric charge on the surface of the conductor. For a charged thin spherical shell, tensile forces can cause stresses in the shell material in excess of the ultimate strength.

It is interesting that such ratios were the subject of research by such classics of science as Poisson and Laplace at the very beginning of the 19th century. In relation (1), bewilderment is caused by the factor 2 in the denominator. Indeed, why is the correct result obtained by halving the expression? Consider one particular case (Fig. 1): let a conducting ball of radius contain an electric charge on its lateral surface. It is easy to calculate the surface density of an electric charge: We introduce a spherical coordinate system (), the element of the lateral surface of the ball is defined as. The charge of a surface element can be calculated from the dependence:. The total electric charge of the ring of radius and width is determined by the expression:. The distance from the plane of the ring under consideration to the pole of the sphere (lateral surface of the ball) is ... There is a known solution to the problem of determining the component of the vector of the electrostatic field strength on the axis of the ring (the principle of superposition) at the observation point, which is at a distance from the plane of the ring:

Let us calculate the total value of the strength of the electrostatic field created by surface charges, excluding the elementary charge in the vicinity of the pole of the sphere:

Recall that near a charged conducting sphere, the strength of the external electrostatic field is

It turns out that the force acting on the charge of an element on the surface of a charged conducting ball is 2 times less than the force acting on the same charge located near the side surface of the ball, but outside it.

The total force acting on the conductor is

(5)

In addition to the force from the electrostatic field, the conductor is subjected to the action of a moment of forces

(6)

where is the radius vector of the surface element dS conductor.

In practice, it is often more convenient to calculate the force effect of the electrostatic field on the conductor by differentiating the electrical energy of the system W. The force acting on the conductor, in accordance with the definition of potential energy, is

and the magnitude of the projection of the vector of the moment of forces on some axis is equal to

where is the angle of rotation of the body as a whole around the axis under consideration. Note that the above formulas are valid if the electric energy W expressed in terms of the charges of the conductors (field sources!), and the calculation of the derivatives is carried out at constant values ​​of electric charges.

Ampere force is the force with which a magnetic field acts on a conductor, with a current placed in this field. The magnitude of this force can be determined using Ampere's law. This law defines the infinitely small force for an infinitely small section of the conductor. That makes it possible to apply this law to conductors of various shapes.

Formula 1 - Ampere's Law

B induction of the magnetic field in which the current conductor is located

I conductor current

dl infinitesimal element of the length of the current-carrying conductor

alpha the angle between the induction of an external magnetic field and the direction of the current in a conductor

The direction of the Ampere force is according to the left hand rule. The wording of this rule sounds like this. When the left hand is positioned in such a way that the lines of magnetic induction of the external field enter the palm, and four outstretched fingers indicate the direction of current flow in the conductor, while the thumb bent at a right angle will indicate the direction of the force that acts on the element of the conductor.

Figure 1 - left hand rule

Some problems arise when using the left-hand rule when the angle between field induction and current is small. It is difficult to determine where the open hand should be. Therefore, for ease of application of this rule, you can position your palm so that it does not include the magnetic induction vector itself, but its modulus.

It follows from Ampere's law that the Ampere force will be zero if the angle between the line of magnetic induction of the field and the current is zero. That is, the conductor will be located along such a line. And the Ampere force will have the maximum possible value for this system if the angle is 90 degrees. That is, the current will be perpendicular to the magnetic induction line.

Using Ampere's law, you can find the force acting in a system of two conductors. Imagine two infinitely long conductors that are spaced apart. Currents flow through these conductors. The force acting from the side of the field created by the conductor with current number one on the conductor number two can be represented as.

Formula 2 - Ampere force for two parallel conductors.

The force acting from the side of the number one conductor on the second conductor will have the same form. Moreover, if the currents in the conductors flow in one direction, then the conductor will be attracted. If in the opposite, then they will repel. There is some confusion, because currents flow in one direction, so how can they be attracted. After all, the poles and charges of the same name have always repelled. Or Ampere decided not to imitate the others and came up with something new.

In fact, Ampere did not invent anything, since if you think about it, then the fields created by parallel conductors are directed opposite each other. And why they are attracted, the question no longer arises. To determine in which direction the field created by the conductor is directed, you can use the right screw rule.

Figure 2 - Parallel conductors with current

Using parallel conductors and the expression of the ampere force for them, you can determine the unit of one ampere. If the same currents of one ampere flow through infinitely long parallel conductors located at a distance of one meter, then the forces of interaction between them will be 2 * 10-7 Newtons, for each meter of length. Using this relationship, you can express what will be equal to one Ampere.

This video describes how a permanent magnetic field created by a horseshoe magnet acts on a current-carrying conductor. In this case, the role of a conductor with current is played by an aluminum cylinder. This cylinder rests on copper rails through which an electric current is supplied to it. The force acting on a conductor with a current in a magnetic field is called the Ampere force. The direction of action of the Ampere force is determined using the left-hand rule.

The French physicist Dominique François Arago (1786-1853) at a meeting of the Paris Academy of Sciences spoke about Oersted's experiments and repeated them. Arago offered a natural, as it seemed to everyone, explanation of the magnetic action of an electric current: a conductor, as a result of an electric current flowing through it, turns into a magnet. The demonstration was attended by another academician, mathematician Andre Marie Ampere. He assumed that the essence of the newly discovered phenomenon is in the movement of the charge, and decided to carry out the necessary measurements himself. Ampere was convinced that closed currents are equivalent to magnets. On September 24, 1820, he connected two wire coils to a voltaic pole, which turned into magnets.

That. the current coil creates the same field as the strip magnet. Ampere created a prototype of an electromagnet, discovering that a steel bar placed inside a spiral with a current magnetizes, multiplying the magnetic field. Ampere suggested that the magnet is a certain system of internal closed currents and showed (both on the basis of experiments and by means of calculations) that a small circular current (loop) is equivalent to a small magnet located in the center of the loop perpendicular to its plane, i.e. any circuit with current can be replaced by a magnet of infinitely small thickness.

Ampere's hypothesis that there are closed currents inside any magnet, called. hypothesis of molecular currents and formed the basis of the theory of interaction of currents - electrodynamics.

A current-carrying conductor in a magnetic field is affected by a force that is determined only by the properties of the field in the place where the conductor is located, and does not depend on which system of currents or permanent magnets created the field. The magnetic field has an orienting effect on the frame with the current. Consequently, the torque experienced by the frame is the result of the action of forces on its individual elements.

Ampere's law can be used to determine the modulus of the magnetic induction vector. The modulus of the induction vector at a given point of a uniform magnetic field is equal to the greatest force that acts on a conductor of unit length placed in the vicinity of this point, through which a current flows per unit of current:. The value is achieved provided that the conductor is perpendicular to the induction lines.

Ampere's law is used to determine the strength of the interaction of two currents.

Between two parallel infinitely long conductors, through which direct currents flow, an interaction force arises. Conductors with equally directed currents attract, with oppositely directed currents repel.

The strength of interaction per unit length of each of the parallel conductors is proportional to the magnitudes of the currents and inversely proportional to the distance between R between them. This interaction of conductors with parallel currents is explained by the left-hand rule. The modulus of force acting on two infinite rectilinear currents and, the distance between which is equal to R.