Forced electromagnetic oscillations. Operating principle of an alternator

Topic 3. Electrical vibrations. Alternating electric current. Main questions of the topic: 3. 1. 1. Free undamped electrical oscillations 3. 1. 2. Damped electrical oscillations 3. 1. 3. Forced electrical oscillations. Resonance 3. 1. 4. Alternating electric current.

Repetition Harmonic oscillations A – amplitude of oscillation; ω – circular frequency (ωt+φ0) – oscillation phase; φ0 – initial phase of oscillation. Differential equation of free undamped harmonic oscillations: Equation of a plane harmonic wave propagating along the X axis:

3. 1. Free undamped electrical oscillations An oscillatory circuit is a circuit consisting of a capacitor and a coil. E – electric field strength; H – magnetic field strength; q – charge; C is the capacitance of the capacitor; L – coil inductance, I – current in the circuit

- natural circular frequency of oscillations Thomson’s formula: (3) T – period of natural oscillations in the oscillatory circuit

Let's find the relationship between the amplitude values ​​of current and voltage: From Ohm's law: U=IR - wave impedance.

Electric field energy (energy of a charged capacitor) at any time: Magnetic field energy (inductor energy) at any time:

Maximum (amplitude) value of the magnetic field energy: - maximum value of the electric field energy Total energy of the oscillatory circuit at any time: The total energy of the circuit remains constant

Problem 3. 1 An oscillatory circuit consists of a capacitor and an inductor. Determine the frequency of oscillations occurring in the circuit if the maximum current in the inductor is 1.2 A, the maximum potential difference across the capacitor plates is 1200 V, the total energy of the circuit is 1.1 mJ. Given: Im = 1.2 A UCm = 1200 B W = 1.1 m J = 1.1 10 -3 J ν-?

Task In the oscillating circuit, the capacitance has increased by 8 times, and the inductance has decreased by half. How will the period of natural oscillations of the circuit change? a) will decrease by 2 times; b) will increase by 2 times; c) will decrease by 4 times; d) will increase by 4 times.

(7)

(17)

Impact on vibration the contour of the forcing E.M.S., the frequencies of which are different from ω0, will be weaker, the “sharper” the resonance curve. The “sharpness” of the resonance curve is characterized by the relative width of this curve equal to Δω/ω0, where Δω is the cycle difference. frequencies at I=Im/√ 2

Problem 3. 2 An oscillatory circuit consists of a resistor with a resistance of 100 Ohms and a capacitor with a capacity of 0.55 microns. F and coils with inductance 0.03 H. Determine the phase shift between the current through the circuit and the applied voltage if the frequency of the applied voltage is 1000 Hz. Given: R = 100 Ohm C = 0.55 microns. Ф = 5.5·10 -7 Ф L = 0.03 Hn ν = 1000 Hz φ-?

They appear in the presence of an external periodically changing force. Such oscillations appear, for example, in the presence of a periodic electromotive force in the circuit. An alternating induced emf arises in a wire frame of several turns rotating in the field of a permanent magnet.

In this case, the magnetic flux passing through the frame changes periodically. In accordance with the law of electromagnetic induction, the resulting induced emf also changes periodically. If the frame is closed to a galvanometer, its needle will begin to oscillate around the equilibrium position, indicating that alternating current is flowing in the circuit. A distinctive feature of forced oscillations is the dependence of their amplitude on the frequency of changes in the external force.

Alternating current.

Alternating current is an electric current that changes over time.

Various types of pulsed, pulsating, periodic and quasi-periodic currents are classified as alternating current. In engineering, alternating current usually means periodic or almost periodic currents of alternating direction.

Operating principle of an alternating current generator.

The most commonly used is periodic current, the strength of which varies over time according to a harmonic law (harmonic, or sinusoidal alternating current). This is the current used in factories and factories and in the lighting network of apartments. It represents forced electromagnetic oscillations. The industrial AC frequency is 50 Hz. Alternating voltage in the sockets of lighting network sockets is created by generators at power plants. The simplest model of such a generator is a wire frame rotating in a uniform magnetic field.

Magnetic induction flux F piercing a wire frame with an area S, proportional to the cosine of the angle α between the normal to the frame and the magnetic induction vector:

Ф = BS cos α.

With uniform rotation of the frame, the angle α increases in proportion to time t: α = 2πnt, Where n- rotation frequency. Therefore, the flux of magnetic induction changes harmoniously with the cyclic frequency of oscillations ω = 2πn:

Ф = BS cos ωt.

According to the law of electromagnetic induction, the induced emf in the frame is equal to:

e = -Ф" = -BS (cos ωt)" = ɛ m sin ωt,

Where ɛm= BSω is the amplitude of the induced emf.

Thus, the voltage in the AC network changes according to a sinusoidal (or cosine) law:

u = U m sin ωt(or u = U m cos ωt),

Where u— instantaneous voltage value, U m— voltage amplitude.

The current in the circuit will change at the same frequency as the voltage, but there may be a phase shift between them φ s. Therefore, in the general case, the instantaneous current value i determined by the formula:

i = I m sin(φt + φWith) ,

Where I m- current amplitude.

Current strength in an AC circuit with a resistor. If the electrical circuit consists of active resistance R and wires with negligible inductance

If an external EMF variable is included in the circuit circuit (Fig. 1), then the field strength in the coil conductor and the wires connecting the circuit elements to each other will periodically change, which means that the speed of the ordered movement of free charges in them will also periodically change, as a result the current strength in the circuit will periodically change, which will cause periodic changes in the potential difference between the plates of the capacitor and the charge on the capacitor, i.e. forced electrical oscillations will occur in the circuit.

Forced electrical oscillations- these are periodic changes in the current strength in the circuit and other electrical quantities under the influence of an alternating EMF from an external source.

The most widely used in modern technology and in everyday life is sinusoidal alternating current with a frequency of 50 Hz.

Alternating current is a current that changes periodically over time. It represents forced electrical oscillations occurring in an electrical circuit under the influence of periodically changing external emf. Period alternating current is the period of time during which the current makes one complete oscillation. Frequency AC current is the number of oscillations of alternating current per second.

In order for a sinusoidal current to exist in a circuit, the source in that circuit must create an alternating electric field that varies sinusoidally. In practice, sinusoidal EMF is created by alternating current generators operating in power plants.

Literature

Aksenovich L. A. Physics in secondary school: Theory. Tasks. Tests: Textbook. benefits for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyhavanne, 2004. - P. 396.

Mechanical vibrations.

3. Transformers.

Waves.

4. Wave diffraction.

9. Doppler effect in acoustics.

1.Magnetic phenomena

Magnetic field induction of a straight conductor carrying current.

Faraday's law

Faraday's law of electromagnetic induction is written as the following formula:

– is an electromotive force that acts along any contour;

Фв is a magnetic flux passing through a surface stretched over a contour.

For a coil placed in an alternating magnetic field, Faraday's law looks slightly different:

This is electromotive force;

N is the number of turns of the coil;

F in is the magnetic flux passing through one turn.

Lenz's rule

The induced current has such a direction that the increment of the magnetic flux it creates through the area limited by the contour and the increment of the magnetic induction flux of the external field are opposite in sign.

The induced current arising in a closed circuit with its magnetic field counteracts the change in magnetic flux that caused this current.

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

If the current in the circuit under consideration changes for some reason, then the magnetic field of this current also changes, and, consequently, the own magnetic flux penetrating the circuit. A self-inductive emf arises in the circuit, which, according to Lenz’s rule, prevents a change in the current in the circuit. This phenomenon is called self-induction, and the corresponding value is self-induced emf.

The self-induction emf is directly proportional to the inductance of the coil and the rate of change of current in it

Inductance

Inductance (from the Latin inductio - guidance, motivation) is a quantity that characterizes the relationship between a change in current in an electrical circuit and the resulting EMF (electromotive force) of self-induction. Inductance is denoted by the capital letter "L", in honor of the German physicist Lenz. The term inductance was proposed in 1886 by Oliver Heaviside.

The amount of magnetic flux passing through the circuit is related to the current strength as follows: Φ = LI. The proportionality coefficient L is called the circuit self-inductance coefficient or simply inductance. The inductance value depends on the size and shape of the circuit, as well as on the magnetic permeability of the medium. The unit of inductance is Henry (H). Additional quantities: mH, μH.

Knowing the inductance, the change in current strength and the time of this change, you can find the self-inductive emf that occurs in the circuit:

The energy of the magnetic field of the current is also expressed through inductance:

Accordingly, the greater the induction, the greater the magnetic energy accumulated in the space around the current-carrying circuit. Inductance is a kind of analogue of kinetic energy in electricity.

7. Solenoid inductance.

L - Inductance (solenoid), dimension in SI Gn

L - Length (solenoid), dimension in SI - m

N - Number (of solenoid turns

V- Volume (solenoid), dimension in SI - m3

Relative magnetic permeability

Magnetic constant Gn/m

Solenoid magnetic field energy

The energy Wm of the magnetic field of a coil with inductance L, created by current I, is equal to

Let us apply the resulting expression for the coil energy to a long solenoid with a magnetic core. Using the above formulas for the self-induction coefficient Lμ of the solenoid and for the magnetic field B created by the current I, one can obtain:

Diamagnets

Diamagnets are substances that are magnetized against the direction of an external magnetic field. In the absence of an external magnetic field, diamagnetic materials are nonmagnetic. Under the influence of an external magnetic field, each atom of a diamagnetic substance acquires a magnetic moment I (and each mole of the substance acquires a total magnetic moment), proportional to the magnetic induction H and directed towards the field.

Diamagnets include inert gases, nitrogen, hydrogen, silicon, phosphorus, bismuth, zinc, copper, gold, silver, and many other, both organic and inorganic, compounds. A person in a magnetic field behaves like a diamagnetic.

Paramagnets

Paramagnetic substances are substances that are magnetized in an external magnetic field in the direction of the external magnetic field. Paramagnetic substances are weakly magnetic substances, the magnetic permeability differs slightly from unity

Paramagnetic materials include aluminum (Al), platinum (Pt), many other metals (alkali and alkaline earth metals, as well as alloys of these metals), oxygen (O2), nitrogen oxide (NO), manganese oxide (MnO), ferric chloride (FeCl2), etc.

Ferromagnets

Ferromagnets are substances (usually in a solid crystalline or amorphous state) in which, below a certain critical temperature (Curie point), a long-range ferromagnetic order is established in the magnetic moments of atoms or ions (in non-metallic crystals) or the moments of itinerant electrons (in metallic crystals). In other words, a ferromagnet is a substance that, at a temperature below the Curie point, is capable of magnetization in the absence of an external magnetic field.

Among the chemical elements, the transition elements Fe, Co and Ni (3 d-metals) and rare earth metals Gd, Tb, Dy, Ho, Er have ferromagnetic properties.

Questions for testing in the section “Oscillations and Waves.”

Mechanical vibrations.

1. Oscillatory motion

Oscillatory motion is a motion that repeats itself exactly or approximately at regular intervals. The study of oscillatory motion in physics is especially emphasized. This is due to the commonality of the patterns of oscillatory motion of various natures and the methods of its study.

Mechanical, acoustic, electromagnetic vibrations and waves are considered from a single point of view.

Oscillatory motion is characteristic of all natural phenomena. Rhythmically repeating processes, such as the beating of the heart, continuously occur inside any living organism.

Huygens formula

4 . Physical pendulum

A physical pendulum is a rigid body fixed on a fixed horizontal axis (suspension axis) that does not pass through the center of gravity, and which oscillates about this axis under the influence of gravity. Unlike a mathematical pendulum, the mass of such a body cannot be considered pointlike.

The minus sign on the right side means that the force F is directed towards decreasing angle α. Taking into account the smallness of the angle α

To derive the law of motion of mathematical and physical pendulums, we use the basic equation of the dynamics of rotational motion

Moment of force: cannot be determined explicitly. Taking into account all the quantities included in the original differential equation of oscillations of a physical pendulum has the form:

Solution to this equation

Let us determine the length l of the mathematical pendulum at which the period of its oscillations is equal to the period of oscillations of the physical pendulum, i.e. or

From this relation we determine

Resonance

A sharp increase in the amplitude of forced oscillations as the cyclic frequency of the disturbing force approaches the natural frequency of oscillations is called resonance.

An increase in amplitude is only a consequence of resonance, and the reason is the coincidence of the external (exciting) frequency with the internal (natural) frequency of the oscillatory system.

Self-oscillations.

There are systems in which undamped oscillations arise not due to periodic external influences, but as a result of the ability of such systems to regulate the supply of energy from a constant source. Such systems are called self-oscillating, and the process of undamped oscillations in such systems is self-oscillations.

In Fig. Figure 1.10.1 shows a diagram of a self-oscillating system. In a self-oscillating system, three characteristic elements can be distinguished: oscillatory system, energy source And valve- a device that performs feedback between the oscillatory system and the energy source.

Feedback is called positive, if the energy source produces positive work, i.e. transfers energy to the oscillatory system. In this case, during the period of time while an external force acts on the oscillatory system, the direction of the force and the direction of the speed of the oscillatory system coincide, as a result of which undamped oscillations occur in the system. If the directions of force and velocity are opposite, then negative feedback, which only enhances the damping of oscillations.

An example of a mechanical self-oscillating system is a clock mechanism (Fig. 1.10.2). The running wheel with oblique teeth is rigidly attached to a toothed drum, through which a chain with a weight is thrown. At the upper end of the pendulum there is an anchor (anchor) with two plates of hard material, bent along a circular arc with the center on the axis of the pendulum. In hand watches, the weight is replaced by a spring, and the pendulum is replaced by a balancer - a handwheel connected to a spiral spring. The balancer performs torsional vibrations around its axis. The oscillatory system in a clock is a pendulum or balancer. The source of energy is a raised weight or a wound spring. The device by which feedback is provided - the valve - is an anchor that allows the running wheel to turn one tooth in one half-cycle. Feedback is provided by the interaction of the anchor with the running wheel. With each oscillation of the pendulum, a tooth of the running wheel pushes the anchor fork in the direction of movement of the pendulum, transferring to it a certain portion of energy, which compensates for energy losses due to friction. Thus, the potential energy of the weight (or twisted spring) is gradually, in separate portions, transferred to the pendulum.

Mechanical self-oscillating systems are widespread in life around us and in technology. Self-oscillations occur in steam engines, internal combustion engines, electric bells, strings of bowed musical instruments, air columns in the pipes of wind instruments, vocal cords when talking or singing, etc.

Mechanical vibrations.

1. Oscillatory motion. Conditions for the occurrence of oscillations. Parameters of oscillatory motion. Harmonic vibrations.

2. Oscillations of the load on the spring.

3. Mathematical pendulum. Huygens formula.

4. Physical pendulum. The period of free oscillations of a physical pendulum.

5. Transformation of energy in harmonic vibrations.

6. Addition of harmonic vibrations occurring along one straight line and in two mutually perpendicular directions. Lissajous figures.

7. Damped mechanical vibrations. Equation for damped oscillations and its solution.

8. Characteristics of damped oscillations: damping coefficient, relaxation time, logarithmic damping decrement, quality factor.

9. Forced mechanical vibrations. Resonance.

10. Self-oscillations. Examples of self-oscillating systems.

Electrical vibrations. Alternating current.

1. Electrical vibrations. Oscillatory circuit. Thomson's formula.

2. Alternating electric current. A frame rotating in a magnetic field. Alternator.

3. Transformers.

4. DC electric machines.

5. Resistor in the AC circuit. Effective value of emf, voltage and current.

6. Capacitor in the AC circuit.

7. Inductor in an alternating current circuit.

8. Forced oscillations in the alternating current circuit. Resonance of voltages and currents.

9. Ohm's law for an alternating current circuit.

10. Power released in the alternating current circuit.

Waves.

1. Mechanical waves. Types of waves and their characteristics.

2. Traveling wave equation. Plane and spherical waves.

3. Interference of waves. Conditions for minimum and maximum interference.

4. Wave diffraction.

5. Huygens' principle. Laws of reflection and refraction of mechanical waves.

6. Standing wave. Standing wave equation. The appearance of a standing wave. Natural frequencies of oscillations.

7. Sound waves. Sound speed.

8. Movement of bodies at a speed greater than the speed of sound.

9. Doppler effect in acoustics.

10. Electromagnetic waves. Prediction and discovery of electromagnetic waves. Physical meaning of Maxwell's equations. Hertz's experiments. Properties of electromagnetic waves. Electromagnetic wave scale.

11. Radiation of electromagnetic waves. Transfer of energy by electromagnetic wave. Umov-Poynting vector.

Questions for testing in 11th grade. Questions for the final exam.

Questions for testing in the section “Magnetism”.

1.Magnetic phenomena are any natural phenomena associated with the presence of magnetic fields (both static and waves) and no matter where, in space or in solid crystals or in technology. Magnetic phenomena do not appear in the absence of magnetic fields.

Some examples of magnetic phenomena:

Attraction of magnets to each other, generation of electric current in generators, operation of a transformer, northern lights, radio emission of atomic hydrogen at a wavelength of 21 cm, spin waves, spin glasses, etc.

An electrical circuit consisting of an inductor and a capacitor (see figure) is called an oscillatory circuit. In this circuit, peculiar electrical oscillations can occur. Let, for example, at the initial moment of time we charge the capacitor plates with positive and negative charges, and then allow the charges to move. If the coil were missing, the capacitor would begin to discharge, an electric current would appear in the circuit for a short time, and the charges would disappear. The following happens here. First, thanks to self-induction, the coil prevents the current from increasing, and then, when the current begins to decrease, it prevents it from decreasing, i.e. supports current. As a result, the self-induction EMF charges the capacitor with reverse polarity: the plate that was initially positively charged acquires a negative charge, the second - positive. If there is no loss of electrical energy (in the case of low resistance of the circuit elements), then the value of these charges will be the same as the value of the initial charges of the capacitor plates. In the future, the process of moving charges will be repeated. Thus, the movement of charges in the circuit is an oscillatory process.

To solve USE problems devoted to electromagnetic oscillations, you need to remember a number of facts and formulas regarding the oscillatory circuit. First, you need to know the formula for the period of oscillation in the circuit. Secondly, be able to apply the law of conservation of energy to an oscillatory circuit. And finally (although such tasks are rare), be able to use the dependence of the current through the coil and the voltage across the capacitor on time

The period of electromagnetic oscillations in the oscillatory circuit is determined by the relation:

where and is the charge on the capacitor and the current in the coil at this point in time, and is the capacitance of the capacitor and the inductance of the coil. If the electrical resistance of the circuit elements is small, then the electrical energy of the circuit (24.2) remains practically unchanged, despite the fact that the capacitor charge and the current in the coil change over time. From formula (24.4) it follows that during electrical oscillations in the circuit, energy transformations occur: at those moments in time when the current in the coil is zero, the entire energy of the circuit is reduced to the energy of the capacitor. At those moments in time when the capacitor charge is zero, the energy of the circuit is reduced to the energy of the magnetic field in the coil. Obviously, at these moments of time, the charge of the capacitor or the current in the coil reaches its maximum (amplitude) values.

During electromagnetic oscillations in the circuit, the charge of the capacitor changes over time according to the harmonic law:

standard for any harmonic vibrations. Since the current in the coil is the derivative of the capacitor charge with respect to time, from formula (24.4) we can find the dependence of the current in the coil on time

In the Unified State Examination in physics, problems on electromagnetic waves are often proposed. The minimum knowledge required to solve these problems includes an understanding of the basic properties of an electromagnetic wave and knowledge of the electromagnetic wave scale. Let us briefly formulate these facts and principles.

According to the laws of the electromagnetic field, an alternating magnetic field generates an electric field, and an alternating electric field generates a magnetic field. Therefore, if one of the fields (for example, electric) begins to change, a second field (magnetic) will arise, which then again generates the first (electric), then again the second (magnetic), etc. The process of mutual transformation of electric and magnetic fields into each other, which can propagate in space, is called an electromagnetic wave. Experience shows that the directions in which the electric and magnetic field strength vectors oscillate in an electromagnetic wave are perpendicular to the direction of its propagation. This means that electromagnetic waves are transverse. Maxwell's theory of electromagnetic field proves that an electromagnetic wave is created (emitted) by electric charges when they move with acceleration. In particular, the source of the electromagnetic wave is an oscillatory circuit.

Electromagnetic wave length, its frequency (or period) and propagation speed are related by a relationship that is valid for any wave (see also formula (11.6)):

Electromagnetic waves in a vacuum propagate at speed = 3 10 8 m/s, in the medium the speed of electromagnetic waves is less than in vacuum, and this speed depends on the frequency of the wave. This phenomenon is called wave dispersion. An electromagnetic wave has all the properties of waves propagating in elastic media: interference, diffraction, and Huygens’ principle is valid for it. The only thing that distinguishes an electromagnetic wave is that it does not require a medium to propagate - an electromagnetic wave can propagate in a vacuum.

In nature, electromagnetic waves are observed with frequencies that differ greatly from each other, and therefore have significantly different properties (despite the same physical nature). The classification of the properties of electromagnetic waves depending on their frequency (or wavelength) is called the electromagnetic wave scale. Let's give a brief overview of this scale.

Electromagnetic waves with a frequency less than 10 5 Hz (i.e., with a wavelength greater than several kilometers) are called low-frequency electromagnetic waves. Most household electrical appliances emit waves in this range.

Waves with a frequency between 10 5 and 10 12 Hz are called radio waves. These waves correspond to wavelengths in vacuum from several kilometers to several millimeters. These waves are used for radio communications, television, radar, and cell phones. The sources of radiation of such waves are charged particles moving in electromagnetic fields. Radio waves are also emitted by free electrons of the metal, which oscillate in an oscillatory circuit.

The region of the electromagnetic wave scale with frequencies lying in the range 10 12 - 4.3 10 14 Hz (and wavelengths from a few millimeters to 760 nm) is called infrared radiation (or infrared rays). The source of such radiation is the molecules of the heated substance. A person emits infrared waves with a wavelength of 5 - 10 microns.

Electromagnetic radiation in the frequency range 4.3 10 14 - 7.7 10 14 Hz (or wavelengths 760 - 390 nm) is perceived by the human eye as light and is called visible light. Waves of different frequencies within this range are perceived by the eye as having different colors. The wave with the smallest frequency in the visible range 4.3 10 14 is perceived as red, and the highest frequency within the visible range 7.7 10 14 Hz is perceived as violet. Visible light is emitted during the transition of electrons in atoms, molecules of solids heated to 1000 °C or more.

Waves with a frequency of 7.7 10 14 - 10 17 Hz (wavelength from 390 to 1 nm) are usually called ultraviolet radiation. Ultraviolet radiation has a pronounced biological effect: it can kill a number of microorganisms, can cause increased pigmentation of human skin (tanning), and with excessive irradiation in some cases it can contribute to the development of oncological diseases (skin cancer). Ultraviolet rays are contained in solar radiation and are created in laboratories with special gas-discharge (quartz) lamps.

Behind the region of ultraviolet radiation lies the region of x-rays (frequency 10 17 - 10 19 Hz, wavelength from 1 to 0.01 nm). These waves are emitted when charged particles accelerated by a voltage of 1000 V or more are decelerated in matter. They have the ability to pass through thick layers of matter that are opaque to visible light or ultraviolet radiation. Due to this property, X-rays are widely used in medicine to diagnose bone fractures and a number of diseases. X-rays have a detrimental effect on biological tissue. Thanks to this property, they can be used to treat cancer, although with excessive irradiation they are deadly to humans, causing a number of disorders in the body. Due to their very short wavelength, the wave properties of X-rays (interference and diffraction) can only be detected on structures comparable in size to atoms.

Gamma radiation (-radiation) is called electromagnetic waves with a frequency greater than 10 20 Hz (or a wavelength less than 0.01 nm). Such waves arise in nuclear processes. A special feature of -radiation is its pronounced corpuscular properties (i.e., this radiation behaves like a stream of particles). Therefore, -radiation is often spoken of as a flow of -particles.

IN problem 24.1.1 to establish correspondence between units of measurement, we use formula (24.1), from which it follows that the period of oscillation in a circuit with a capacitor of 1 F and an inductance of 1 H is equal to seconds (answer 1 ).

From the graph given in problem 24.1.2, we conclude that the period of electromagnetic oscillations in the circuit is 4 ms (answer 3 ).

Using formula (24.1) we find the period of oscillations in the circuit given in problem 24.1.3:
(answer 4 ). Note that, according to the electromagnetic wave scale, such a circuit emits long-wave radio waves.

The period of oscillation is the time of one complete oscillation. This means that if at the initial moment of time the capacitor is charged with the maximum charge ( problem 24.1.4), then after half the period the capacitor will also be charged with the maximum charge, but with reverse polarity (the plate that was initially positively charged will be negatively charged). And the maximum current in the circuit will be achieved between these two moments, i.e. after a quarter of the period (answer 2 ).

If you increase the inductance of the coil four times ( problem 24.1.5), then according to formula (24.1) the period of oscillations in the circuit will double, and the frequency will decrease by half (answer 2 ).

According to formula (24.1), when the capacitor capacity increases fourfold ( problem 24.1.6) the period of oscillation in the circuit doubles (answer 1 ).

When the key is closed ( problem 24.1.7) in the circuit, instead of one capacitor, two identical capacitors connected in parallel will work (see figure). And since when capacitors are connected in parallel, their capacitances add up, closing the switch leads to a doubling of the circuit capacitance. Therefore, from formula (24.1) we conclude that the period of oscillation increases by a factor of (answer 3 ).

Let the charge on the capacitor oscillate with a cyclic frequency ( problem 24.1.8). Then, according to formulas (24.3)-(24.5), the current in the coil will oscillate with the same frequency. This means that the dependence of the current on time can be represented as . From here we find the dependence of the energy of the magnetic field of the coil on time

From this formula it follows that the energy of the magnetic field in the coil oscillates with double the frequency, and, therefore, with a period half as long as the period of oscillation of charge and current (answer 1 ).

IN problem 24.1.9 We use the law of conservation of energy for the oscillatory circuit. From formula (24.2) it follows that for the amplitude values ​​of the voltage on the capacitor and the current in the coil, the following relation is true:

where and are the amplitude values ​​of the capacitor charge and the current in the coil. From this formula, using relation (24.1) for the oscillation period in the circuit, we find the amplitude value of the current

answer 3 .

Radio waves are electromagnetic waves with certain frequencies. Therefore, the speed of their propagation in a vacuum is equal to the speed of propagation of any electromagnetic waves, and in particular, X-rays. This speed is the speed of light ( problem 24.2.1- answer 1 ).

As stated earlier, charged particles emit electromagnetic waves when moving with acceleration. Therefore, the wave is not emitted only with uniform and rectilinear motion ( problem 24.2.2- answer 1 ).

An electromagnetic wave is an electric and magnetic field that varies in space and time in a special way and supports each other. Therefore the correct answer is problem 24.2.3 - 2 .

From what is given in the condition tasks 24.2.4 The graph shows that the period of this wave is - = 4 µs. Therefore, from formula (24.6) we obtain m (answer 1 ).

IN problem 24.2.5 using formula (24.6) we find

(answer 4 ).

An oscillatory circuit is connected to the antenna of the electromagnetic wave receiver. The electric field of the wave acts on the free electrons in the circuit and causes them to oscillate. If the frequency of the wave coincides with the natural frequency of electromagnetic oscillations, the amplitude of oscillations in the circuit increases (resonance) and can be recorded. Therefore, to receive an electromagnetic wave, the frequency of natural oscillations in the circuit must be close to the frequency of this wave (the circuit must be tuned to the frequency of the wave). Therefore, if the circuit needs to be reconfigured from a 100 m wave to a 25 m wave ( problem 24.2.6), the natural frequency of electromagnetic oscillations in the circuit must be increased by 4 times. To do this, according to formulas (24.1), (24.4), the capacitance of the capacitor should be reduced by 16 times (answer 4 ).

According to the scale of electromagnetic waves (see the introduction to this chapter), the maximum length listed in the condition tasks 24.2.7 radiation from a radio transmitter antenna has electromagnetic waves (answer 4 ).

Among those listed in problem 24.2.8 electromagnetic waves, X-ray radiation has the maximum frequency (answer 2 ).

An electromagnetic wave is transverse. This means that the vectors of the electric field strength and magnetic field induction in the wave at any time are directed perpendicular to the direction of propagation of the wave. Therefore, when a wave propagates in the direction of the axis ( problem 24.2.9), the electric field strength vector is directed perpendicular to this axis. Therefore, its projection onto the axis is necessarily equal to zero = 0 (answer 3 ).

The speed of propagation of an electromagnetic wave is an individual characteristic of each medium. Therefore, when an electromagnetic wave passes from one medium to another (or from a vacuum to a medium), the speed of the electromagnetic wave changes. What can we say about the other two wave parameters included in formula (24.6) - wavelength and frequency. Will they change when a wave passes from one medium to another ( problem 24.2.10)? Obviously, the frequency of the wave does not change when moving from one medium to another. Indeed, a wave is an oscillatory process in which an alternating electromagnetic field in one medium creates and maintains a field in another medium due to these very changes. Therefore, the periods of these periodic processes (and therefore the frequencies) in one and another environment must coincide (answer 3 ). And since the speed of the wave in different media is different, it follows from the above reasoning and formula (24.6) that the wavelength changes when it passes from one medium to another.