Comparison of mechanical and electromagnetic oscillations table. Analogy between mechanical and electromagnetic oscillations. Own undamped electromagnetic oscillations

Analogy between mechanical and electromagnetic oscillations

fluctuations - the process of changing the states of the system around the equilibrium point, which is repeated to some extent in time.

Fluctuations are almost always associated with the alternating transformation of the energy of one form of manifestation into another form.

Classification by physical nature :

- Mechanical (sound, vibration)
- Electromagnetic (light, radio waves, heat)

Characteristics:

Amplitude - the maximum deviation of the fluctuating value from some average value for the system, A (m)
Period - a period of time after which any indicators of the state of the system are repeated (the system makes one complete oscillation), T (sec)
Frequency - number of oscillations per unit of time, v (Hz, sec −1).

Oscillation period T and frequency v - reciprocal values;

T=1/v and v=1/T

In circular or cyclic processes, instead of the "frequency" characteristic, the concept is used circular (cyclic) frequency W (rad/sec, Hz, sec −1), showing the number of oscillations per 2P units of time:

w = 2P/T = 2PV

Electromagnetic oscillations in the circuit are similar to free mechanical oscillations (with oscillations of a body fixed on a spring).

The similarity relates to processes periodic change different sizes.
- The nature of the change in values is explained by the existing analogy in the conditions under which mechanical and electromagnetic oscillations are generated.

-Return to the equilibrium position of the body on the spring is caused by an elastic force proportional to the displacement of the body from the equilibrium position.

Proportionality factor is the stiffness of the spring k.

The discharge of the capacitor (appearance of current) is due to voltage u between the plates of a capacitor, which is proportional to the charge q.
The coefficient of proportionality is 1 / C, the inverse of the capacitance (since u = 1/C*q)

Just as, due to inertia, a body only gradually increases its speed under the action of a force, and this speed does not immediately become equal to zero after the termination of the force, electricity in the coil, due to the phenomenon of self-induction, increases gradually under the influence of voltage and does not disappear immediately when this voltage becomes zero.Loop inductance L plays the same role as body weight m in mechanics. According to the kinetic energy of the body mv(x)^2/2 responsible energy magnetic field current Li^2/2.

Charging a capacitor from a battery corresponds to a message to a body attached to a spring, potential energy when the body is displaced (for example, by hand) at a distance Xm from the equilibrium position (Fig. 75, a). Comparing this expression with the energy of the capacitor, we note that the stiffness K of the spring plays the same role in the mechanical oscillatory process as the value 1/C, the reciprocal of the capacitance during electromagnetic oscillations, and the initial coordinate Xm corresponds to the charge Qm.

Appearance in electrical circuit current i due to the potential difference corresponds to the appearance in the mechanical oscillatory system of the speed Vx under the action of the elastic force of the spring (Fig. 75,b)

The moment when the capacitor is discharged, and current strength reaches a maximum, corresponds to the passage of the body through the equilibrium position with maximum speed(Fig. 75, c)

Further, the capacitor will begin to recharge, and the body will shift to the left from the equilibrium position (Fig. 75, d). After half of the period T, the capacitor will be completely recharged and the current strength will become equal to zero. This state corresponds to the deviation of the body to the extreme left position, when its speed is zero (Fig. 75, e).

Target :

Demonstration of a new problem solving method
Development abstract thinking ability to analyze, compare, generalize
Fostering a sense of camaraderie, mutual assistance, tolerance.

The topics “Electromagnetic oscillations” and “Oscillation circuit” are psychologically difficult topics. The phenomena occurring in an oscillatory circuit cannot be described with the help of human senses. Only visualization with an oscilloscope is possible, but even in this case we will get a graphical dependence and cannot directly observe the process. Therefore, they remain intuitively and empirically obscure.

A direct analogy between mechanical and electromagnetic oscillations helps to simplify the understanding of processes and analyze changes in the parameters of electrical circuits. In addition, to simplify the solution of problems with complex mechanical oscillatory systems in viscous media. When considering this topic, the generality, simplicity and scarcity of the laws necessary to describe physical phenomena are once again emphasized.

This topic is given after studying the following topics:

Mechanical vibrations.
Oscillatory circuit.
Alternating current.

Required set of knowledge and skills:

Definitions: coordinate, velocity, acceleration, mass, stiffness, viscosity, force, charge, current, rate of change of current with time (use of this value), capacitance, inductance, voltage, resistance, emf, harmonic oscillations, free, forced and damped oscillations, static displacement, resonance, period, frequency.
Equations describing harmonic oscillations (using derivatives), energy states of an oscillatory system.
Laws: Newton, Hooke, Ohm (for AC circuits).
The ability to solve problems for determining the parameters of an oscillatory system (mathematical and spring pendulum, oscillatory circuit), its energy states, to determine the equivalent resistance, capacitance, resultant force, alternating current parameters.

Previously, as homework, students are offered tasks, the solution of which is greatly simplified when using a new method and tasks leading to an analogy. The task can be group. One group of students performs the mechanical part of the work, the other part is associated with electrical vibrations.

Homework.

1a. A load of mass m, attached to a spring with stiffness k, is removed from the equilibrium position and released. Determine the maximum displacement from the equilibrium position if the maximum speed of the load v max

1b. In an oscillatory circuit consisting of a capacitor with a capacitance C and an inductor L, the maximum value of the current I max. Determine the maximum charge value of the capacitor.

2a. A mass m is suspended from a spring of stiffness k. The spring is brought out of equilibrium by shifting the load from the equilibrium position by A. Determine the maximum x max and minimum x min displacement of the load from the point where the lower end of the unstretched spring was located and v max the maximum speed of the load.

2b. The oscillatory circuit consists of a current source with an EMF equal to E, a capacitor with a capacitance C and a coil, an inductance L and a key. Before closing the key, the capacitor had a charge q. Determine the maximum q max and q min minimum charge of the capacitor and the maximum current in the circuit I max.

An evaluation sheet is used when working in class and at home

Kind of activity	Self-esteem	Mutual evaluation
Physical dictation
comparison table
Problem solving
Homework
Problem solving
Preparation for the test

The course of lesson number 1.

Analogy between mechanical and electrical oscillations

Introduction to the topic

1. Actualization of previously acquired knowledge.

Physical dictation with mutual verification.

Dictation text

2. Check (work in dyads, or self-assessment)

3. Analysis of definitions, formulas, laws. Search for similar values.

A clear analogy can be traced between such quantities as speed and current strength. . Next, we trace the analogy between charge and coordinate, acceleration and the rate of change in current strength over time. Force and EMF characterize the external influence on the system. According to Newton's second law F=ma, according to Faraday's law E=-L. Therefore, we conclude that mass and inductance are similar quantities. It is necessary to pay attention to the fact that these quantities are similar in their physical meaning. Those. This analogy can also be obtained in the reverse order, which confirms its deep physical meaning and the correctness of our conclusions. Next, we compare Hooke's law F \u003d -kx and the definition of the capacitance of the capacitor U \u003d. We get an analogy between the rigidity (the value characterizing the elastic properties of the body) and the value of the reciprocal capacitance of the capacitor (as a result, we can say that the capacitance of the capacitor characterizes the elastic properties of the circuit). As a result, based on the formulas for the potential and kinetic energy of the spring pendulum, and , we obtain the formulas and . Since this is the electrical and magnetic energy of the oscillatory circuit, this conclusion confirms the correctness of the obtained analogy. Based on the analysis carried out, we compile a table.

Spring pendulum	Oscillatory circuit

4. Demonstration of solving problems No. 1 a and No. 1 b On the desk. analogy confirmation.

1b. In an oscillatory circuit consisting of a capacitor with a capacitance C and an inductor L, the maximum value of the current I max. Determine the maximum charge value of the capacitor.

according to the law of conservation of energy

consequently

Dimension check:

according to the law of conservation of energy

Hence

Dimension check:

Answer:

While solving problems on the board, students are divided into two groups: "Mechanics" and "Electricians" and using the table make up a text similar to the text of the tasks 1a and 1b. As a result, we notice that the text and the solution of problems confirm our conclusions.

5. Simultaneous execution on the board of solving problems No. 2 a and by analogy No. 2 b. When solving a problem 2b difficulties must have arisen at home, since similar problems were not solved in the lessons and the process described in the condition is unclear. The solution of the problem 2a there shouldn't be any problems. The parallel solution of problems on the blackboard with the active help of the class should lead to the conclusion about the existence of a new method for solving problems through analogies between electrical and mechanical vibrations.

Decision:

Let's define the static displacement of the load. Since the load is at rest

Hence

As can be seen from the figure,

x max \u003d x st + A \u003d (mg / k) + A,

x min \u003d x st -A \u003d (mg / k) -A.

Determine the maximum speed of the load. The displacement from the equilibrium position is insignificant, therefore, the oscillations can be considered harmonic. Let us assume that at the moment of the beginning of the countdown the displacement was maximum, then

x=Acos t.

For spring pendulum =.

=x"=Asin t,

with sint=1 = max.

>> Analogy between mechanical and electromagnetic oscillations

§ 29 ANALOGY BETWEEN MECHANICAL AND ELECTROMAGNETIC OSCILLATIONS

Electromagnetic oscillations in the circuit are similar to free mechanical oscillations, for example, to oscillations of a body fixed on a spring (spring pendulum). The similarity does not refer to the nature of the quantities themselves, which change periodically, but to the processes of periodic change of various quantities.

During mechanical vibrations, the coordinate of the body periodically changes X and the projection of its speed x, and with electromagnetic oscillations, the charge q of the capacitor and the current strength change i in the chain. The same nature of the change in quantities (mechanical and electrical) is explained by the fact that there is an analogy in the conditions under which mechanical and electromagnetic oscillations occur.

The return to the equilibrium position of the body on the spring is caused by the elastic force F x control, proportional to the displacement of the body from the equilibrium position. The proportionality factor is the spring constant k.

The discharge of the capacitor (appearance of current) is due to the voltage between the plates of the capacitor, which is proportional to the charge q. The coefficient of proportionality is the reciprocal of the capacitance, since u = q.

Just as, due to inertia, the body only gradually increases its speed under the action of force and this speed does not immediately become equal to zero after the termination of the force, the electric current in the coil, due to the phenomenon of self-induction, increases gradually under the action of voltage and does not disappear immediately when this voltage becomes equal to zero. The circuit inductance L plays the same role as the body mass m during mechanical vibrations. Accordingly, the kinetic energy of the body is similar to the energy of the magnetic field of the current

Charging a capacitor from a battery is similar to communicating potential energy to a body attached to a spring when the body is displaced by a distance x m from the equilibrium position (Fig. 4.5, a). Comparing this expression with the energy of the capacitor, we notice that the stiffness k of the spring plays the same role during mechanical vibrations as the reciprocal of the capacitance during electromagnetic vibrations. In this case, the initial coordinate x m corresponds to the charge q m .

The appearance of a current i in an electric circuit corresponds to the appearance of a body speed x in a mechanical oscillatory system under the action of the elastic force of a spring (Fig. 4.5, b).

The moment in time when the capacitor is discharged and the current strength reaches its maximum is similar to the moment in time when the body passes at maximum speed (Fig. 4.5, c) the equilibrium position.

Further, the capacitor in the course of electromagnetic oscillations will begin to recharge, and the body in the course of mechanical oscillations will begin to shift to the left from the equilibrium position (Fig. 4.5, d). After half the period T, the capacitor will be fully recharged and the current will become zero.

With mechanical vibrations, this corresponds to the deviation of the body to the extreme left position, when its speed is zero (Fig. 4.5, e).

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ELECTROMAGNETIC OSCILLATIONS. FREE AND FORCED ELECTRIC OSCILLATIONS IN THE OSCILLATION CIRCUIT.

Electromagnetic vibrations- interconnected fluctuations of electric and magnetic fields.

Electromagnetic oscillations appear in various electrical circuits. In this case, the magnitude of the charge, voltage, current strength, intensity fluctuate electric field, magnetic field induction and other electrodynamic quantities.

Free electromagnetic oscillationsarise in the electromagnetic system after removing it from the state of equilibrium, for example, by imparting a charge to the capacitor or by changing the current in the circuit section.

These are damped vibrations, since the energy communicated to the system is spent on heating and other processes.

Forced electromagnetic oscillations- undamped oscillations in the circuit caused by an external periodically changing sinusoidal EMF.

Electromagnetic oscillations are described by the same laws as mechanical ones, although the physical nature of these oscillations is completely different.

Electrical vibrations - special case electromagnetic, when only oscillations of electrical quantities are considered. In this case, one speaks of alternating current, voltage, power, etc.

OSCILLATORY CIRCUIT

An oscillatory circuit is an electrical circuit consisting of a series-connected capacitor with a capacitance C, an inductor with an inductance Land a resistor with resistance R. Ideal circuit - if the resistance can be neglected, that is, only the capacitor C and the ideal coil L.

State stable balance The oscillatory circuit is characterized by the minimum energy of the electric field (the capacitor is not charged) and the magnetic field (there is no current through the coil).

CHARACTERISTICS OF ELECTROMAGNETIC OSCILLATIONS

Analogy of mechanical and electromagnetic oscillations

Characteristics:	Mechanical vibrations	Electromagnetic vibrations
Quantities expressing the properties of the system itself (system parameters):	m- mass (kg) k- spring rate (N/m)	L- inductance (H) 1/C- reciprocal of capacitance (1/F)
Quantities characterizing the state of the system:	Kinetic energy (J) Potential energy (J) x - displacement (m)	Electrical energy(J) Magnetic energy (J) q - capacitor charge (C)
Quantities expressing the change in the state of the system:	v = x"(t) displacement speed (m/s)	i = q"(t) current strength - rate of change of charge (A)
Other Features: T=1/ν T=2π/ω ω=2πν	T- oscillation period time of one complete oscillation (s)
	ν- frequency - number of vibrations per unit of time (Hz)
	ω - cyclic frequency number of vibrations per 2π seconds (Hz)
	φ=ωt - oscillation phase - shows what part of the amplitude value it takes in this moment fluctuating value, i.e.the phase determines the state of the oscillating system at any time t.

where q" is the second derivative of charge with respect to time.

Value is the cyclic frequency. The same equations describe fluctuations in current, voltage, and other electrical and magnetic quantities.

One of the solutions to equation (1) is the harmonic function

This is an integral equation of harmonic oscillations.

Oscillation period in the circuit (Thomson formula):

The value φ = ώt + φ 0 , standing under the sign of sine or cosine, is the phase of the oscillation.

The current in the circuit is equal to the derivative of the charge with respect to time, it can be expressed

The voltage on the capacitor plates varies according to the law:

Where I max \u003d ωq poppy is the amplitude of the current (A),

Umax=qmax /C - voltage amplitude (V)

Exercise: for each state of the oscillatory circuit, write down the values of the charge on the capacitor, current in the coil, electric field strength, magnetic field induction, electric and magnetic energy.

§ 29. Analogy between mechanical and electromagnetic oscillations

During mechanical vibrations, the coordinate of the body periodically changes X and the projection of its speed v x, and with electromagnetic oscillations, the charge changes q capacitor and current i in the chain. The same nature of the change in quantities (mechanical and electrical) is explained by the fact that there is an analogy in the conditions under which mechanical and electromagnetic oscillations occur.

Just as, due to inertia, a body only gradually increases its speed under the action of a force, and this speed does not immediately become equal to zero after the termination of the force, the electric current in the coil, due to the phenomenon of self-induction, increases gradually under the action of voltage and does not disappear immediately when this voltage becomes equal to zero. The loop inductance L plays the same role as the mass of the body m during mechanical vibrations. Accordingly, the kinetic energy of the body is similar to the energy of the magnetic field of the current

Charging a capacitor from a battery is similar to communicating a body attached to a spring with potential energy when the body is displaced by a distance x m from the equilibrium position (Fig. 4.5, a). Comparing this expression with the energy of the capacitor, we notice that the stiffness k of the spring plays the same role during mechanical vibrations as the reciprocal of the capacitance during electromagnetic vibrations. In this case, the initial coordinate x m corresponds to the charge q m .

The appearance in the electric circuit of current i corresponds to the appearance in the mechanical oscillatory system of the speed of the body v x under the action of the elastic force of the spring (Fig. 4.5, b).

With mechanical vibrations, this corresponds to the deviation of the body to the extreme left position, when its speed is zero (Fig. 4.5, e). The correspondence between mechanical and electrical quantities during oscillatory processes can be summarized in a table.

Electromagnetic and mechanical vibrations are of different nature, but are described by the same equations.

Questions for the paragraph

1. What is the analogy between electromagnetic oscillations in a circuit and oscillations of a spring pendulum?

2. Due to what phenomenon does the electric current in the oscillatory circuit not immediately disappear when the voltage across the capacitor becomes zero?