Ampere's law. Force acting on a current-carrying conductor in a magnetic field
where I
- current strength; 
- a vector equal in absolute value to the length l conductor and coinciding in direction with the current;
- magnetic field induction.
Vector modulus 
is defined by the expression
F=BI l sin,
where α is the angle between the vectors 
and
.
The force of interaction of two straight infinitely long parallel conductors with currents I 1 and I 2 at a distance d from each other, designed for a piece of conductor with a length l is expressed by the formula
F=
Magnetic moment of the circuit with current
,
where 
- a vector equal in modulus to the area S, covered by the contour, and coinciding in direction with the normal to its plane.
Mechanical moment acting on a current-carrying circuit placed in a uniform magnetic field
.
Mechanical moment module
M= p m B sin,
where α is the angle between the vectors 
and
.
Potential (mechanical) energy of a circuit with current in a magnetic field
Force acting on a loop with current in a magnetic field (changing along the x-axis)
F=p m 
,
where 
- change in magnetic induction along the axis Oh, calculated per unit length; α - angle between vectors 
and
.
Force acting on a charge moving in a magnetic field (Lorentz force)
Force 
,
acting on charge q, moving at a speed 
in a magnetic field with induction 
Lorentz force) is expressed by the formula
,
or F= q vB sin,
where is the angle formed by the velocity vector 
moving particle and vector
magnetic field induction.
Full current law. magnetic flux. Magnetic circuits
Circulation of the magnetic induction vector
along a closed loop
,
where B i
-
projection of the vector of magnetic induction on the direction of elementary displacement 
along the contour L.
Tension vector circulation 
along a closed loop
Total current law (for magnetic field in vacuum)
,
where 0 - magnetic constant; 
- algebraic sum of the currents covered by the circuit; P - number of currents.
Total current law (for an arbitrary medium)
Magnetic flux Ф through a flat contour with an area S
a) in the case of a homogeneous field
F= BS cos; or F = B n S,
where is the angle between the normal vector 
to the contour plane and the magnetic induction vector
;
AT n
-
vector projection
to normal 
(B n =
B cos );
b) in the case of an inhomogeneous field
,
where integration is carried out over the entire surface S.
Flux linkage, i.e. total magnetic flux coupled to all turns of a solenoid or toroid
= N F,
where Ф - magnetic flux through one turn; N- the number of turns of the solenoid or toroid.
The magnetic field of a toroid whose core is composed of two parts made of substances with different magnetic permeabilities:
a) magnetic induction on the axial line of the toroid
,
where I- current strength in the toroid winding; N- the number of its turns; l 1 and l 2 - lengths of the first and second parts of the toroid core; 1 and 2 - magnetic permeability of the substances of the first and second parts of the toroid core; 0 - magnetic constant;
b) magnetic field strength on the axial line of the toroid in the first and second parts of the core
, and 
c) magnetic flux in the toroid core
or by analogy with Ohm's law (Hopkinson's formula)
,
where F m - magnetomotive force; R m - total magnetic resistance of the circuit.
d) magnetic resistance of a circuit section
Magnetic permeabilityμ , ferromagnet is related to magnetic induction AT fields in it and tension H magnetizing field ratio
The French physicist Dominique Francois Arago (1786-1853) at a meeting of the Paris Academy of Sciences spoke about Oersted's experiments and repeated them. Arago proposed a natural, as it seemed to everyone, explanation of the magnetic action of electric current: the conductor, as a result of the flow of electric current through it, turns into a magnet. Another academician, mathematician André Marie Ampère, attended the demonstration. He suggested that the essence of the newly discovered phenomenon is in the movement of the charge, and decided to take the necessary measurements himself. Ampère was sure that closed currents were equivalent to magnets. On September 24, 1820, he connected two wire spirals to a voltaic column, which turned into magnets.
That. a coil with current creates the same field as a bar magnet. Ampere created the prototype of the electromagnet, discovering that a steel bar placed inside a current-carrying spiral becomes magnetized, amplifying the magnetic field many times over. Ampere suggested that the magnet is a certain system of internal closed currents and showed (both on the basis of experiments and with the help of calculations) that a small circular current (coil) is equivalent to a small magnet located in the center of the coil perpendicular to its plane, i.e. any circuit with current can be replaced by a magnet of infinitely small thickness.
Ampere's hypothesis that inside any magnet there are closed currents, 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 subject to a force that is determined only by the properties of the field at the location 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 current. Consequently, the torque experienced by the frame is the result of the action of forces on its individual elements.
Ampère'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 a given point, through which a current flows per unit of current strength: . The value is achieved provided that the conductor is perpendicular to the lines of induction.
Ampère'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 the same direction of currents attract, those with oppositely directed currents repel.
The power of interaction per unit length of each of the parallel conductors is proportional to the magnitude of the currents and 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.
Ampère's law shows the force with which a magnetic field acts on a conductor placed in it. This force is also called by the power of Ampere.
The wording of the law:the force acting on a conductor with current 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 not uniform, then the formula is as follows:
The direction of Ampère's force is determined by the rule of the left hand.
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° thumb, will indicate the direction of Ampere's force.
MP of the driving charge. The action of the magnetic field on a moving charge. Ampere force, Lorentz.
Any conductor with current creates a magnetic field in the surrounding space. In this case, the electric current is an ordered movement of electric charges. So we can assume that any charge moving in a vacuum or medium generates a magnetic field around itself. As a result of the generalization of numerous experimental data, a law was established that determines the field B of a point charge Q moving with a constant non-relativistic velocity v. This law is given by the formula
(1)
where r is the radius vector drawn from the charge Q to the observation point M (Fig. 1). According to (1), the vector B is directed perpendicular to the plane in which the vectors v and r are located: its direction coincides with the direction of the translational movement of the right screw when it rotates from v to r.
Fig.1
The module 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 Biot-Savart-Laplace law and (1), we see that the moving charge is equivalent to the current element in terms of its magnetic properties: Idl = Qv
The action of the magnetic field on a moving charge.
It is known from experience that a magnetic field has an effect not only on current-carrying conductors, 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 rule of the left hand: if the palm of the left hand is positioned so that it includes the vector B, and four extended fingers are directed along the vector v (for Q> 0, the directions I and v coincide, for Q Fig. 1 shows the mutual orientation of the vectors v, B (the field has a direction towards us, shown by dots in the figure) and F for a positive charge.If the charge is negative, then the force acts in the opposite direction.
emf 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 by moving the circuit in a constant magnetic field, or by changing the field itself.
So, the direction of the induction current is determined by the Lenz rule: With any change in the magnetic flux through a surface bounded by a closed conducting circuit, an induction current arises in the latter 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 circuit is numerically equal and opposite in sign to the rate of change of the magnetic flux through the surface bounded by the circuit:
The value Ψ = ΣΦm is called flux linkage or total magnetic flux. If the flow penetrating each of the turns is the same (i.e., Ψ = NΦm), then in this case
 |
The German physicist G. Helmholtz proved that the Faraday-Lenz law is a consequence of the law of conservation of energy. Let a closed conducting circuit be in a non-uniform magnetic field. If a current I flows in the circuit, then under the action of Ampère's forces, the loose circuit will begin to move. The elementary work dA, performed when moving the contour during the time dt, will be
dA = IdФm,
where dФm is the change in the magnetic flux through the loop area 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
Therefore, when the magnetic flux coupled to the circuit changes, an electromotive force of induction arises in the latter
Electromagnetic vibrations. Oscillatory contour.
Electromagnetic oscillations are oscillations of such quantities as inductance, resistance, emf, charge, current strength.
An oscillatory 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 transferring the electrical energy of the capacitor into the energy of the magnetic field of the coil and vice versa takes place. If, at certain points in time, the energy losses in the circuit due to resistance due to an external source are compensated, then we will get undamped electrical oscillations that can be radiated through the antenna into the surrounding space.
The process of propagation of electromagnetic oscillations, periodic changes in the strength 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 with a speed C, and in a medium with a speed: V= , where = 8.85;
Volumetric energy density of the electromagnetic field. The practical use of electromagnetic phenomena is very wide. These are systems and means of communication, broadcasting, television, electronic computers, control systems for various purposes, measuring and medical devices, household electrical and radio equipment, and others, i.e. without which it is impossible to imagine modern society.
How powerful electromagnetic radiation affects people's health, there is almost no exact scientific data, there are only unconfirmed hypotheses and, in general, not unfounded fears that everything unnatural acts destructively. It has been proven that ultraviolet, X-ray and high-intensity radiation in many cases cause real harm to all living things.
Geometric optics. Laws of GO.
Geometric (beam) optics uses an idealized idea of a light beam - an infinitely thin beam of light propagating in a straight line in a homogeneous isotropic medium, as well as the idea of a point source of radiation that shines uniformly in all directions. λ - light wavelength, - characteristic size
object in the path of the wave. Geometric optics is the limiting case of wave optics and its principles are fulfilled under the condition:
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 moving. Therefore, the displacements of the rays do not prevent each of them from propagating independently of each other.
For many practical problems in 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 consideration of the geometry of the path of light rays.
Basic laws of geometric optics.
Let us list the basic laws of optics following from the experimental data:
1) Rectilinear propagation.
2) The law of independence of light rays, that is, two rays, intersecting, do not interfere with each other in any way. This law is in better agreement with the wave theory, since the particles could in principle collide with each other.
3) The law of reflection. the incident beam, the reflected beam and the perpendicular to the interface, restored at the point of incidence of the beam, 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.
Law of refraction: the incident beam, the refracted beam and the perpendicular to the interface, restored from the point of incidence of the beam, 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 with respect to the first medium. n21
If substance 1 is emptiness, 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, the relative refractive index of two substances (for example, 1 - air, 2 - 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 send light in the opposite direction, it will follow the same path.
It follows from law 4) that if n2 > n1 , then Sin i1 > Sin i2 . Let now we 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 will turn out that the angle i2 will be equal to π /2 (beam 5). Then Sin i2 = 1 and n1 Sin (i1)pr = n2 . So Sin
The effect of a magnetic field on a current-carrying conductor was experimentally investigated by André Marie Ampère (1820). By changing the shape of the conductors and their location in a magnetic field, Ampère was able to determine the force acting on a separate section of the current-carrying conductor (current element). In his honor, this force was named the Ampère force.
- Amp power is the force with which a magnetic field acts on a current-carrying conductor placed in it.
According to experimental data, the modulus of force F:
Proportional to conductor length l located in a magnetic field; proportional to the magnetic field induction modulus B; proportional to the current in the conductor I; depends on the orientation of the conductor in the magnetic field, i.e. on the angle α between the direction of the current and the magnetic field induction vector \(~\vec B\).
Ampere's force modulus is equal to the product of the magnetic field induction modulus B, in which the conductor with current is located, the length of this conductor l, current I in it and the sine of the angle between the directions of the current and the magnetic field induction vector
\(~F_A = I \cdot B \cdot l \cdot \sin \alpha\) ,
- This formula can be used: if the length of the conductor is such that the induction at all points of the conductor can be considered the same; if the magnetic field is uniform (then the length of the conductor can be any, but the conductor must be entirely in the field).
To determine the direction of the Ampere force, use left hand rule: if the palm of the left hand is positioned so that the magnetic field induction vector (\(~\vec B\)) enters the palm, four outstretched fingers indicate the direction of the current ( I), then the thumb bent by 90° will indicate the direction of the Ampère force (\(~\vec F_A\)) (Fig. 1, a, b).
Rice. one
Since the value B∙sin α is the modulus of the component of the induction vector perpendicular to the conductor with current, \(~\vec B_(\perp)\) (Fig. 2), then the orientation of the palm can be determined precisely by this component - the component perpendicular to the surface of the conductor must be included in open palm of the left hand.
From (1) it follows that the Ampère force is zero if the conductor with current is located along the lines of magnetic induction, and is maximum if the conductor is perpendicular to these lines.
The forces acting on a current-carrying conductor in a magnetic field are widely used in engineering. Electric motors and generators, devices for recording sound in tape recorders, telephones and microphones - all these and many other devices and devices use the interaction of currents, currents and magnets, etc.
Lorentz force
The expression for the force with which a magnetic field acts on a moving charge was first obtained by the Dutch physicist Hendrik Anton Lorenz (1895). In his honor, this force is called the Lorentz force.
- Lorentz force is the force with which a magnetic field acts on a charged particle moving in it.
The modulus of the Lorentz force is equal to the product of the modulus of the magnetic field \(~\vec B\), in which the charged particle is located, the modulus of the charge q of this particle, its velocity υ and the sine of the angle between the directions of the velocity and the magnetic field induction vector
\(~F_L = q \cdot B \cdot \upsilon \cdot \sin \alpha\).
To determine the direction of the Lorentz force, use left hand rule: if the left hand is positioned so that the magnetic field induction vector (\(~\vec B\)) enters the palm, four outstretched fingers indicate the direction of the speed of movement positively charged particle(\(~\vec \upsilon\)), then the thumb bent by 90° will indicate the direction of the Lorentz force (\(~\vec F_L\)) (Fig. 3, a). For negative particle four extended fingers are directed against the speed of the particle (Fig. 3, b).
Rice. 3 Since the value B∙sin α is the modulus of the component of the induction vector perpendicular to the velocity of a charged particle, \(~\vec B_(\perp)\), then the orientation of the palm can be determined precisely by this component - the component perpendicular to the velocity of a charged particle must enter the open palm of the left hand .
Since the Lorentz force is perpendicular to the velocity vector of the particle, it cannot change the value of the velocity, but only changes its direction and, therefore, does no work.
Movement of a charged particle in a magnetic field
1. If the speed υ charged particle with mass m directed along vector of the magnetic field, then the particle will move in a straight line at a constant speed (Lorentz force F L = 0, because α = 0°) (Fig. 4, a).
Rice. 4 2. If the speed υ charged particle with mass m perpendicular magnetic field induction vector, then the particle will move along a circle of radius R, the plane of which is perpendicular to the lines of induction (Fig. 4, b). Then Newton's 2nd law can be written in the following form:
\(~m \cdot a_c = F_L\) ,
where \(~a_c = \dfrac(\upsilon^2)(R)\) , \(~F_L = q \cdot B \cdot \upsilon \cdot \sin \alpha\) , α = 90°, because particle velocity is perpendicular to the magnetic induction vector.
\(~\dfrac(m \cdot \upsilon^2)(R) = q \cdot B \cdot \upsilon\) .
3. If the speed υ charged particle with mass m directed at an angle α (0 < α < 90°) к вектору индукции магнитного поля, то частица будет двигаться по спирали радиуса R and step h(Fig. 4c).
The action of the Lorentz force is widely used in various electrical devices:
- cathode ray tubes of TVs and monitors;
- particle accelerators;
- experimental facilities for the implementation of controlled thermonuclear;
- MHD generators
Literature
- Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Proc. allowance for institutions providing general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsia i vykhavanne, 2004. - C. 321-322, 324-327.
- Zhilko, V. V. Physics: textbook. allowance for the 11th grade. general education institutions with Russian. lang. training with a 12-year term of study (basic and advanced levels) /V. V. Zhilko, L. G. Markovich. - 2nd ed., corrected. - Minsk: Nar. asveta, 2008. - S. 157-164.
The ampere force is the force with which a magnetic field acts on a current-carrying conductor placed in this field. The magnitude of this force can be determined using Ampère's law. This law defines an infinitely small force for an infinitely small section of the conductor. This makes it possible to apply this law to conductors of various shapes.
Formula 1 - Ampère's Law
B induction of the magnetic field in which there is a conductor with current
I current in a conductor
dl an infinitesimal element of the length of a current-carrying conductor
alpha angle between the induction of an external magnetic field and the direction of current in a conductor
The direction of Ampere's force is found according to the rule of the left hand. The wording of this rule is as follows. 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 conductor element.
Figure 1 - left hand rule
Some problems arise when using the left hand rule if the angle between the field induction and the current is small. It is difficult to determine where the open palm should be. Therefore, for ease of application of this rule, the palm can be positioned so that it includes not the magnetic induction vector itself, but its module.
It follows from Ampère'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 line of magnetic induction.
Using Ampere's law, you can find the force acting in a system of two conductors. Imagine two infinitely long conductors that are at a distance from each other. Current flows 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 conductor number one 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 they are opposite, then they will repel. There is some confusion, because the currents flow in one direction, so how can they be attracted. After all, poles and charges of the same name always repel each other. Or Amper decided that it was not worth imitating the rest and came up with something new.
In fact, Ampère did not invent anything, because if you think about it, the fields created by parallel conductors are directed towards 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 with a force of one ampere flow through infinitely long parallel conductors located at a distance of one meter, then the interaction force between them will be 2 * 10-7 Newtons, for each meter long. Using this relationship, you can express what one ampere will be equal to.
This video talks about how a permanent magnetic field created by a horseshoe magnet affects a conductor with current. The role of the conductor with current in this case is performed by an aluminum cylinder. This cylinder lies on copper bars, through which an electric current is supplied to it. The force acting on a current-carrying conductor in a magnetic field is called the ampere force. The direction of the Ampère force is determined using the left hand rule.
