Examples of tasks of various difficulty levels. A magnetic field

Example . A particle of mass m, carrying charge q, flies into a uniform magnetic field perpendicular to the vector lines IN(Fig. 10). Determine the radius of the circle, period and circular frequency of the charged particle.

Solution . The magnetic component of the Lorentz force bends the trajectory of the particle, but does not take it out of the plane perpendicular to the field. The absolute value of the velocity does not change, the force remains constant, so the particle moves in a circle. Equating the magnetic component of the Lorentz force to the centrifugal force

for the particle radius we obtain the equality

Particle orbital period

. (3.3.3)

Circular frequency ω revolution of the particle, that is, the number of revolutions in 2π seconds,

(3.3.3 ΄).

Answer : R = mv/(qB); ω = qB/ m; for a particular type of particle, the period and frequency depend only on the magnetic field induction.

Consider the motion of a particle moving at an angle< 90° к направлению линий вектора IN(Fig. 11). Let us determine the pitch of the spiral turn h. Speed v has two components, one of which v çç = v cosβ, is parallel IN, the other v ^ = v sin β – perpendicular to the lines of magnetic induction IN.

When a particle moves along lines IN the magnetic component of the force is zero, therefore the particle moves uniformly along the field with speed

v çç = v cosβ.

Spiral pitch

h = v çç T = v T cosβ.

Substituting the expression for T from formula (1.3.3), we obtain:

(3.3.4)

Per conductor element with current Id l The Ampere force acts in a magnetic field.

or in scalar form

dF = I dl B sinα, (3.3.5)

where α is the angle between the conductor element and the magnetic induction.

For a conductor of finite length it is necessary to take the integral:

F= I ∫ . (3.3.6)

The direction of the Ampere force, like the Lorentz force (see above), is determined by the left-hand rule. But taking into account the fact that four fingers here are directed along the current.

Example . A conductor in the form of a semiring with a radius R = 5 cm (Fig. 12) is placed in a uniform magnetic field, the lines of force of which are directed away from us (depicted by crosses). Find the force acting on the conductor if the current flowing through the conductor is I = 2 A, and the magnetic field induction B = 1 µT.

Solution . Let us use formula (3.3.6), taking into account that under the integral there is a vector product, and therefore, ultimately, a vector quantity. It is convenient to find the sum of vectors by projecting vectors - terms on the coordinate axis and adding their projections. Therefore, solving the problem in scalar form, the integral can be represented as a sum of integrals:

F = ∫ dF i, F = ∫ dF x + ∫ dF y.

Using the left-hand rule, we find the force vectors d F, acting on each element of the conductor (Fig. 12).

The first integral on the right side is equal to zero, since the sum of the projections d F is equal to zero, as follows from the figure: due to the symmetry of the picture, each positive projection corresponds to a negative one of the same magnitude. Then the required force is equal only to the second integral

F = ∫ dF y = ∫ dF cosβ,

where β is the angle between vectors d F and the OΥ axis, and the conductor length element can be represented as dl = R cos β. Since the angle is measured from the OΥ axis to the left and right, the limits of integration will be the values ​​90 0 and 90 0. Substituting dl into dF and solving the second integral, we obtain

F=

Numerical calculation gives: F = 2 2 A 10 -6 T 0.05 m = 2 10 -7 N.

Answer: F = 2 10 -7 N.

Ampere's law gives an expression for the force with which two interact infinitely long parallel to each other conductor with currents , located at a distance b from each other:

(3.3.7)

It can be shown that conductors with currents flowing in one direction are attracted and repelled in the case of antiparallel direction of currents.

On the frame ( circuit) forces act on a current in a magnetic field. Who strive to turn it this way. So that the magnetic moment R m of the frame coincided with the direction of magnetic induction. In this case, the torque M acting on a circuit of area S with current I is equal to

M = I S B sinα, (3.3.8)

where α is the angle between the magnetic induction and the normal to the frame. In vector form

M = [ P m, B].

The position at which the angle α = 0 0 . called stable equilibrium, and the position with α = 180 0 - unstable balance.

Elementary work of a magnetic field when the frame is rotated through an angle α

Option 1

A1. What explains the interaction of two parallel conductors with direct current?

1. interaction of electric charges;
2. the effect of the electric field of one conductor with current on the current in another conductor;
3. the effect of the magnetic field of one conductor on the current in another conductor.

A2. Which particle is affected by the magnetic field?

1. on a moving charged one;
2. to a moving uncharged one;
3. to a stationary charged one;
4. to an uncharged one at rest.

A4. A straight conductor 10 cm long is in a uniform magnetic field with an induction of 4 Tesla and is located at an angle of 30 0 to the magnetic induction vector. What is the force acting on the conductor from the magnetic field if the current in the conductor is 3 A?

1. 1.2 N; 2) 0.6 N; 3) 2.4 N.

A6. Electromagnetic induction is:

1. a phenomenon characterizing the effect of a magnetic field on a moving charge;
2. the phenomenon of the occurrence of electric current in a closed loop when the magnetic flux changes;
3. a phenomenon characterizing the effect of a magnetic field on a current-carrying conductor.

A7. Children swing on a swing. What type of vibration is this?

1. free 2. forced 3. Self-oscillations

A8. A body of mass m on a thread of length l oscillates with a period T. What will be the period of oscillation of a body of mass m/2 on a thread of length l/2?

1. ½ T 2. T 3. 4 T 4. ¼ T

A9. The speed of sound in water is 1470 m/s. What is the length of the sound wave with an oscillation period of 0.01 s?

1. 147 km 2. 1.47 cm 3. 14.7 m 4. 0.147 m

A10 . What is the number of oscillations in 2πs called?

1. frequency 2. Period 3. Phase 4. Cyclic frequency

A11. The boy heard an echo 10 seconds after the gun fired. The speed of sound in air is 340m/s. How far is the obstacle from the boy?

A12. Determine the period of free electromagnetic oscillations if the oscillatory circuit contains a coil with an inductance of 1 μH and a capacitor with a capacity of 36 pF.

1. 40ns 2. 3*10 -18 s 3. 3.768*10 -8 s 4. 37.68*10 -18 s

A13. The simplest oscillatory system containing a capacitor and an inductor is called...

1. self-oscillatory system 2. oscillatory system

3. Oscillatory circuit 4. Oscillatory installation

A14. How and why does the electrical resistance of semiconductors change with increasing temperature?

1. Decreases due to an increase in the speed of electron movement.

2. Increases due to an increase in the amplitude of vibrations of positive ions of the crystal lattice.

3. Decreases due to an increase in the concentration of free electric charge carriers.

4. Increases due to an increase in the concentration of free electric charge carriers.

IN 1.

AT 2. Particle of mass m , carrying charge q B circumferential radius R with speed v . What happens to the orbital radius, orbital period, and kinetic energy of the particle as its speed increases?

C1. In a coil with an inductance of 0.4 H, a self-inductive emf of 20 V arose. Calculate the change in current strength and energy of the magnetic field of the coil if this happened in 0.2 s.

Option 2

A1. The rotation of a magnetic needle near a current-carrying conductor is explained by the fact that it is affected by:

1. magnetic field created by charges moving in a conductor;
2. electric field created by conductor charges;
3. electric field created by moving charges of a conductor.

A2.

1. only electric field;
2. only magnetic field.

A4. A straight conductor 5 cm long is in a uniform magnetic field with an induction of 5 T and is located at an angle of 30 0 to the magnetic induction vector. What is the force acting on the conductor from the magnetic field if the current in the conductor is 2 A?

1. 0.25 N; 2) 0.5 N; 3) 1.5 N.

A6. The Lorentz force acts

1. on an uncharged particle in a magnetic field;
2. to a charged particle at rest in a magnetic field;
3. on a charged particle moving along the lines of magnetic induction field.

A7. For a square frame with an area of ​​2 m 2 at a current of 2 A, the maximum torque is 4 N∙m. What is the magnetic field induction in the space under study?

1. Tl; 2) 2 T; 3) 3T.

A8. What type of oscillation is observed when a pendulum swings in a clock?

1. free 2. forced

A9. The speed of sound in air is 330m/s. What is the frequency of sound vibrations if the wavelength is 33 cm?

1. 1000Hz 2. 100Hz 3. 10Hz 4. 10,000Hz 5. 0.1Hz

A10 Determine the period of free electromagnetic oscillations if the oscillatory circuit contains a capacitor with a capacity of 1 μF and an inductance coil of 36 H.

1. 4*10 -8 s 2. 4*10 -18 s 3. 3.768*10 -8 s 4. 37.68*10 -3 s

A11 . Determine the frequency of emitted waves by a system containing a coil with an inductance of 9 H and a capacitor with an electrical capacity of 4 F.

1. 72πHz 2. 12πHz 3. 36Hz 4. 6Hz 5. 1/12πHz

A12. Which characteristic of a light wave determines its color?

1. by wavelength 2. by frequency

3. By phase 4. By amplitude

A13. Undamped oscillations occurring due to an energy source located inside the system are called...

1. free 2. forced

3. Self-oscillations 4. Elastic vibrations

A14. Pure water is a dielectric. Why is an aqueous solution of NaCl a conductor?

1. Salt in water breaks down into charged Na ions+ and Cl - .

2. After the salt dissolves, the NaCl molecules transfer charge

3. In solution, electrons are removed from the NaCl molecule and transfer charge.

4. When interacting with salt, water molecules break down into hydrogen and oxygen ions

IN 1. Establish correspondence between physical

Moves in a uniform magnetic field with induction B circumferential radius R with speed v. What happens to the orbital radius, orbital period, and kinetic energy of the particle as the particle's charge increases?

For each position in the first column, select the corresponding position in the second and write down the selected numbers in the table under the corresponding letters

C1. At what angle to the magnetic field lines with an induction of 0.5 Tesla should a copper conductor with a cross section of 0.85 mm move? 2 and a resistance of 0.04 Ohm, so that at a speed of 0.5 m/s an induced emf equal to 0.35 V is excited at its ends? (copper resistivity ρ= 0.017 Ohm∙mm 2 /m)

Option 3

A1. Magnetic fields are created:

1. both stationary and moving electric charges;
2. stationary electric charges;
3. moving electric charges.

A2. The magnetic field affects:

1. only on stationary electric charges;
2. only on moving electric charges;
3. both moving and stationary electric charges.

A4. What force acts from a uniform magnetic field with an induction of 30 mT on a straight conductor 50 cm long located in the field, carrying a current of 12 A? The wire forms a right angle with the direction of the magnetic field induction vector.

1. 18 N; 2) 1.8 N; 3) 0.18 N; 4) 0.018 N.

A6. What do the four outstretched fingers of the left hand show when determining

Ampere forces

1. direction of field induction force;
2. direction of current;
3. direction of the Ampere force.

A7. A magnetic field with an induction of 10 mT acts on a conductor in which the current is 50 A with a force of 50 mN. Find the length of the conductor if the field induction lines and the current are mutually perpendicular.

1. 1m; 2) 0.1 m; 3) 0.01 m; 4) 0.001 m.

A8. The chandelier swings after one push. What type of vibration is this?

1. free 2 forced 3. Self-oscillations 4. Elastic oscillations

A9 .A body of mass m on a thread of length l oscillates with a period T. What will be the period of oscillation of a body of mass 2m on a thread of length 2l?

1. ½ T 2. 2T 3. 4T 4. ¼ T 5. T

A10 . The speed of sound in air is 330 m/s. What is the wavelength of light at an oscillation frequency of 100 Hz?

1. 33km 2. 33cm 3. 3.3m 4. 0.3m

A11. What is the resonant frequency ν 0 in a circuit of a coil with an inductance of 4H and a capacitor with an electrical capacity of 9F?

1. 72πHz 2. 12πHz 3. 1/12πHz 4. 6Hz

A12 . The boy heard thunder 5 seconds after the lightning flash. The speed of sound in air is 340m/s. At what distance from the boy did the lightning flash?

A. 1700m B. 850m C. 136m D. 68m

A13. Determine the period of free electromagnetic oscillations if the oscillatory circuit contains a coil with an inductance of 4 μH and a capacitor with a capacity of 9 pF.

A14. What type of conductivity do semiconductor materials with donor impurities have?

1. Mainly electronic. 2. Mainly hole-type.

3. Equally electronic and hole. 4. Ionic.

IN 1. Establish correspondence between physicalquantities and units of measurement

AT 2. Particle of mass m carrying charge q , moves in a uniform magnetic field with induction B circumferential radius R with speed v. What happens to the orbital radius, orbital period, and kinetic energy of the particle as the magnetic field induction increases?

For each position in the first column, select the corresponding position in the second and write down the selected numbers in the table under the corresponding letters

C1. In a coil consisting of 75 turns, the magnetic flux is 4.8∙10-3 Vb. How long does it take for this flux to disappear for an average induced emf of 0.74 V to arise in the coil?

Option 4

A1. What is observed in Oersted's experiment?

1. a current-carrying conductor acts on electric charges;
2. the magnetic needle turns near the current-carrying conductor;
3. magnetic needle turns a charged conductor

A2. A moving electric charge creates:

1. only electric field;
2. both electric field and magnetic field;
3. only magnetic field.

A4. In a uniform magnetic field with an induction of 0.82 T, a conductor 1.28 m long is located perpendicular to the lines of magnetic induction. Determine the force acting on the conductor if the current in it is 18 A.

1)18.89 N; 2) 188.9 N; 3) 1.899N; 4) 0.1889 N.

A6. Induction current occurs in any closed conductive circuit if:

1. The circuit is in a uniform magnetic field;
2. The circuit moves forward in a uniform magnetic field;
3. The magnetic flux passing through the circuit changes.

A7. A straight conductor 0.5 m long, located perpendicular to the field lines with an induction of 0.02 T, is acted upon by a force of 0.15 N. Find the strength of the current flowing through the conductor.

1)0.15 A; 2)1.5 A; 3) 15 A; 4) 150 A.

A8 . What type of oscillations is observed when a load suspended on a thread deviates from its equilibrium position?

1. free 2. Forced

3. Self-oscillations 4. Elastic vibrations

A9. Determine the frequency of the waves emitted by the system if it contains a coil with an inductance of 9 H and a capacitor with an electrical capacity of 4 F.

1. 72πHz 2. 12πHz

3.6Hz 4.1/12πHz

A10. Determine at what frequency you need to tune an oscillatory circuit containing a 4 μH inductor and a 9 Pf capacitor.

1. 4*10 -8 s 2. 3*10 -18 s 3. 3.768*10 -8 s 4. 37.68*10 -18 s

A11. Determine the period of natural oscillations of the circuit if it is tuned to a frequency of 500 kHz.

1. 1μs 2. 1ks 3. 2μs 4. 2ks

A12. The boy heard thunder 2.5 seconds after the lightning flash. The speed of sound in air is 340m/s. At what distance from the boy did the lightning flash?

1. 1700m 2. 850m 3. 136m 4. 68m

A13. The number of oscillations per unit time is called...

1. frequency 2. period 3. Phase 4. Cyclic frequency

A14. How and why does the electrical resistance of metals change with increasing temperature?

1. Increases due to an increase in the speed of electron movement.

2. Decreases due to an increase in the speed of electron movement.

3. Increases due to an increase in the amplitude of vibrations of positive ions of the crystal lattice.

4. Decreases due to an increase in the amplitude of vibrations of positive ions of the crystal lattice

IN 1. Establish correspondence between physicalquantities and formulas by which these quantities are determined

AT 2. Particle of mass m carrying charge q , moves in a uniform magnetic field with induction B circumferential radius R with speed v U. What happens to the orbital radius, orbital period, and kinetic energy of the particle as the particle's mass decreases?

For each position in the first column, select the corresponding position in the second and write down the selected numbers in the table under the corresponding letters

C1. A coil with a diameter of 4 cm is in an alternating magnetic field,the lines of force of which are parallel to the axis of the coil. When the field induction changed by 1 T for 6.28 s, an EMF of 2 V arose in the coil. How many turns does the coil have?

, methodologist of CMC Zel UO

To answer the KIM Unified State Exam questions on this topic, you need to repeat the concepts:

Interaction of magnet poles,

Interaction of currents,

Magnetic induction vector, properties of magnetic field lines,

Application of the gimlet rule to determine the direction of magnetic induction of the field of direct and circular current,

Ampere power,

Lorentz force

Left hand rule for determining the direction of the Ampere force, Lorentz force,

Movement of charged particles in a magnetic field.

In the materials of the Unified State Exam KIM, there are often test tasks for determining the direction of the Ampere force and the Lorentz force, and in some cases the direction of the magnetic induction vector is specified implicitly (the poles of the magnet are depicted). A series of tasks is popular in which a frame with current is in a magnetic field and it is required to determine how the Ampere force acts on each side of the frame, as a result of which the frame rotates, shifts, stretches, contracts (you must choose the correct answer). A traditional series of tasks is to analyze formulas at a qualitative level, in which it is required to draw a conclusion about the nature of the change in one physical quantity depending on the multiple change in others.

The task appears under number A15.

1. A permanent strip magnet was brought to the magnetic needle (the north pole is darkened, see figure), which can rotate around a vertical axis perpendicular to the plane of the drawing. In this case the arrow

2. Straight conductor length L with current I placed in a uniform magnetic field perpendicular to the induction lines IN . How will the Ampere force acting on a conductor change if its length is increased by 2 times, and the current strength in the conductor is reduced by 4 times?

3. Proton p, flying into the gap between the poles of the electromagnet, has a speed perpendicular to the magnetic field induction vector, directed vertically (see figure). Where is the Lorentz force acting on it directed?

4. Straight conductor length L with current I placed in a uniform magnetic field, the direction of the induction lines IN which is perpendicular to the direction of the current. If the current strength is reduced by 2 times, and the magnetic field induction is increased by 4 times, then the Ampere force acting on the conductor

5. A particle with a negative charge q flew into the gap between the poles of an electromagnet, having a speed directed horizontally and perpendicular to the magnetic field induction vector (see figure). Where is the Lorentz force acting on it directed?

6. The figure shows a cylindrical conductor through which electric current flows. The direction of the current is indicated by the arrow. What is the direction of the magnetic induction vector at point C?

7. The figure shows a coil of wire through which electric current flows in the direction indicated by the arrow. The coil is located in a vertical plane. At the center of the coil, the magnetic field induction vector of the current is directed

8. In the circuit in the figure, all the conductors are thin, lie in the same plane, parallel to each other, the distances between adjacent conductors are the same, I is the current strength. Ampere force acting on conductor No. 3 in this case:

9. The angle between the current-carrying conductor and the direction of the magnetic induction vector of the magnetic field increases from 30° to 90°. Ampere force in this case:

10. The Lorentz force acting on an electron moving in a magnetic field at a speed of 107 m/s in a circle in a uniform magnetic field B = 0.5 T is equal to:

1. (B1). Particle with mass m, carrying charge q IN circumferential radius R with speed u. What happens to the orbital radius, orbital period, and kinetic energy of the particle as its speed increases?

to the table

(Answer 131)

2 IN 1). Particle with mass m, carrying charge q, moves in a uniform magnetic field with induction IN circumferential radius R with speed u. What happens to the orbital radius, orbital period, and kinetic energy of the particle as the magnetic field induction increases?

For each position in the first column, select the corresponding position in the second and write down to the table selected numbers under the corresponding letters.

(Answer 223)

3. (B4). Straight conductor length l= 0.1 m, through which the current flows, is in a uniform magnetic field with induction B = 0.4 T and is located at an angle of 90° to the vector. What is the current strength if the force acting on the conductor from the magnetic field is 0.2 N?

Option 13

C1. An electric circuit consists of a galvanic element ε, a light bulb and an inductor L connected in series. Describe the phenomena that occur when the switch is opened.

meter. If the switch is then opened, then when the magnetic flux disappears in the coil, an extra opening current I will arise.

ψ = Li,

dL dt = dL di dtdi .

ε si = − L + dL di .

ε si = − L dt di .

10. When power is supplied to the circuit shown in Fig. 13.1.3 in the circuit, the current value will increase from zero to the nominal value over a certain period of time due to the phenomenon of self-induction. The resulting extracurrents, in accordance with Lenz’s rule, are always directed in the opposite direction, i.e. they interfere with the cause that causes them. They prevent the increase

for some time.

ε + εsi = iR,

L dt di +iR = ε.

i(t) = R ε − cons te− RL t .

const = R ε .

16. From the equation, in particular, it follows that when the switch is opened (Fig. 13.1.1), the current strength will decrease according to an exponential law. In the first moments after opening the circuit, the induced emf and the self-induction emf will add up and give a short-term surge in current strength, i.e. the light bulb will briefly increase its brightness (Fig. 13.1.4).

Rice. 13.1.4. Dependence of current strength in a circuit with inductance on time

C2. A skier with a mass m = 60 kg starts from rest from a springboard with a height of H = 40 m; at the moment of takeoff, his speed is horizontal. In the process of moving along the springboard, the friction force performed work AT = 5.25 kJ. Determine the skier's flight range in the horizontal direction if the landing point is h = 45 m below the lift-off level from the springboard. Ignore air resistance.

Rice. 13.2 Skier on a springboard

1. The law of conservation of energy when a skier moves along a springboard:

5. The amount of heat imparted to the gas:

Q = A + U = 37.5 J;

C4. The electrical circuit consists of a source with ε = 21 V with internal resistance r = 1 Ohm and two resistors: R1 = 50 Ohm and R2 = 30 Ohm. The voltmeter's own resistance is Rv = 320 Ohm, the ammeter's resistance is RA = 5 Ohm. Determine instrument readings.

C5. A particle with mass m = 10 − 7 kg, carrying a charge q = 10 − 5 C moves uniformly along a circle of radius R = 2 cm in a magnetic field with induction B = 2 T. The center of the circle is located on the main optical lens at a distance d = 15 cm from it. The focal length of the lens is F = 10 cm. At what speed does the image of the particle move in the lens?

3. For the image, the angular velocity will remain unchanged, but the radius of the circle will double, therefore:

vx = ω 2R = 8 m s;

C6. On a plate with reflection coefficient ρ of incident light, N identical photons fall perpendicularly every second, and the force of light pressure F is applied. What is the wavelength of the incident light?

p = St ε f (1+ ρ ) ; pS = N hc λ (1+ ρ ) ; pS = F; F = N hc λ (1+ ρ ) ; 2. Incident light length:

λ = Nhc (1 + ρ); F

Rice. 14.1.1. Self-induction phenomenon

Rice. 14.1.2. Self-induction

Option 14

C1. An electric circuit consists of a galvanic element ε, a light bulb and an inductor L connected in series. Describe the phenomena that occur when the switch is closed.

1. The phenomenon of electromagnetic induction is observed in all cases of changes in the magnetic flux through the circuit. In particular, an induced emf can be generated in the circuit itself when the current value changes in it, which leads to the appearance of additional currents. This phenomenon is called self-induction, and additionally arising currents are called

are generated by extra-currents or self-induction currents.

2. The phenomenon of self-induction can be studied using an installation, the schematic diagram of which is shown in Fig. 14.1.2. A coil L with a large number of turns, through a rheostat r and a switch k, is connected to a source of emf ε. Additionally, a galvanometer G is connected to the coil. When the switch is short-circuited at point A, the current will branch, with a current of magnitude i flowing through the coil, and current i1 through the galvanometer. If the switch is then opened, then when the magnetic field disappears in the coil

current, an extra opening current I will occur.

3. According to Lenz's law, the extra current will prevent the magnetic flux from decreasing, i.e. will be directed towards the decreasing current, but through the galvanometer the extra current will pass in the direction opposite to the original one, which will lead to the galvanometer needle throwing in the opposite direction. If the coil is equipped with an iron core, the amount of extra current increases. Instead of a galvanometer, in this case you can turn on an incandescent light bulb, which is actually specified in the problem conditions; when a self-induction current occurs, the light bulb will flash brightly.

4. It is known that the magnetic flux coupled to the coil is proportional to the magnitude of the current flowing through it

ψ = Li,

the proportionality factor L is called the circuit inductance. The dimension of inductance is determined by the equation:

L = d i ψ , [ L] = Wb A = Gn(henry) .

5. Let us obtain the equation for the self-inductive emf ε si for the coil:

6. In the general case, inductance, along with the geometry of the coil in media, can depend on the current strength, i.e. L = f (i), this can be taken into account when differentiating

dL dt = dL di dtdi .

7. The self-induction emf, taking into account the last relationship, will be represented by the following equation:

ε si = − L + dL di .

8. If the inductance does not depend on the magnitude of the current, the equation simplifies

ε si = − L dt di .

9. Thus, the self-induction emf is proportional to the rate of change in the current value.

10. When power is applied to the circuit,

The current value shown in Fig. 14.1.3 in the circuit will increase from zero to the nominal value over a certain period of time due to the phenomenon of self-induction. The resulting extracurrents, in accordance with Lenz’s rule, are always directed in the opposite direction, i.e. they interfere with the cause that causes them. They prevent the current from increasing in the circuit. In a given

case, when the key is closed, the light Rice. 13.1.3. Closing and opening currents will not flare up immediately, but its intensity will increase over some time.

11. When the switch is connected to position 1, extra currents will prevent the increase in current in the circuit, and in position 2, on the contrary, extra currents will slow down the decrease in the main current. For simplicity of analysis, we will assume that the resistance R included in the circuit characterizes the circuit resistance, the internal resistance of the source and the active resistance of the coil L. Ohm’s law in this case will take the form:

ε + εsi = iR,

where ε is the source emf, ε si is the self-induction emf, i is the instantaneous value of the current, which is a function of time. Let us substitute the self-induction EMF equation into Ohm’s law:

L dt di +iR = ε.

12. Let us divide the variables in the differential equation:

and integrate, considering L a constant value: L ∫ ε − di iR = ∫ dt ,

R L ln(ε − iR) = t + const .

13. It can be seen that the general solution of the differential equation can be represented in the form:

i(t) = R ε − cons te− RL t .

14. We determine the integration constant from the initial conditions. At t =0

V the moment the power is supplied, the current in the circuit is zero i(t) = 0. Substituting the zero value of the current, we get:

const = R ε .

15. The solution to equation i(t) will take the final form:

16. From the equation, in particular, it follows that when the key is closed (Fig. 13.1.1), the current strength will increase exponentially.

C2. After the impact at point A, the box slides up the inclined plane with an initial speed v0 = 5 m/s. At point B the box is lifted off the inclined plane. At what distance S from the inclined plane will the box fall? The coefficient of friction between the box and the plane is μ = 0.2. The length of the inclined plane AB = L = 0.5 m, the angle of inclination of the plane α = 300. Neglect air resistance.

1. When moving from the initial position, the initially reported box

Rice. 14.2. Flight box kinetic energy is converted into work against force

friction, kinetic energy at point B and increase in potential energy box:

2. From point B the boxes will move along a parabolic trajectory:

3. Distance from the inclined plane to the point of incidence: x(τ ) = vB cosατ ≈ 4 0.87 0.57 ≈ 1.98 m;

C3. An ideal monatomic gas in an amount of ν = 2 mol was first cooled, reducing the pressure by 2 times, and then heated to the initial temperature T1 = 360 K. How much heat did the gas receive in section 2 - 3?

4. Gas work in section 2 → 3:

A2 → 3 = p(V3 − V2 ) = ν R(T3 − T2 ) ≈ 2992J; 5. Heat received by gas:

Q = U2 → 3 + A2 → 3 ≈ 7478J;

C4. The electrical circuit consists of an EMF source with ε = 21 V with internal resistance r = 1 Ohm, resistors R1 = 50 Ohm, R2 = 30 Ohm, a voltmeter with its own resistance RV = 320 Ohm and an ammeter with resistance RA = 5 Ohm. Determine instrument readings.

1. Load resistance:

RV,A = RV + RA = 325 Ohm; R1,2 = R1 + R2 = 80 Ohm; V ≈ 20.4 V;

C5. A particle with mass m = 10 − 7 kg and charge q = 10 − 5 C moves with a constant speed v = 6 m/s in a circle in a magnetic field with induction B = 1.5 T. The center of the circle is located on the main optical axis of the collecting lens, and the plane of the circle is perpendicular to the main optical axis and is located at a distance d = 15 cm from it. The focal length of the lens is F = 10 cm. Along what circle of radius does the image of the particle in the lens move?

1. Radius of particle motion:

C6. Light with a wavelength λ = 600 nm falls perpendicularly onto a plate of area S = 4 cm2, which reflects 70% and absorbs 30% of the incident light. Luminous flux power N = 120 W. How much pressure does light exert on the plate?