A charged particle of mass m is a carrier. Exam solutions

Option 1

A1. What explains the interaction of two parallel DC conductors?

1. interaction of electric charges;
2. the action 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 the other conductor.

A2. What particle is affected by the magnetic field?

1. on a moving charged;
2. on a moving uncharged;
3. on the resting charged;
4. to a resting uncharged one.

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

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

A6. Electromagnetic induction is:

1. a phenomenon that characterizes the effect of a magnetic field on a moving charge;
2. the phenomenon of the appearance in a closed circuit of an electric current when the magnetic flux changes;
3. a phenomenon that characterizes the effect of a magnetic field on a conductor with a current.

A7. Children swing on a swing. What kind of vibration is it?

1. free 2. forced 3. self-oscillations

A8. A body of mass m on a thread of length l vibrates 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 3.4T 4. ¼ T

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

1.147km 2.147cm 3.14.7m 4.147m

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 cannon shot. The speed of sound in the air is 340m / s. How far is the obstacle from the boy?

A12. Determine the free period electromagnetic waves, if the oscillating 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 3.368 * 10 -8 s 4.37.68 * 10 -18 s

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

1. self-oscillating system 2. oscillating 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 movement of electrons.

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

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

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

IN 1.

VALUES		UNITS
	inductance		tesla (T)
	magnetic flux		henry (gn)
	magnetic induction		weber (wb)
			volts (V)

IN 2. Particle of mass m carrying charge q B on a circle radius R with speed v ... What happens to the radius of the orbit, the period of revolution and the kinetic energy of the particle with an increase in the speed of motion?

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

Option 2

A1. The rotation of the magnetic needle near the conductor with current is explained by the fact that it is acted upon by:

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

A2.

1. only an electric field;
2. only the 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 vector of magnetic induction. What is the force acting on the conductor from the side of the magnetic field if the current in the conductor is 2 A?

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

A6. Lorentz force acts

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

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

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

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

1. free 2. forced

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

1.1000Hz 2.100Hz 3.10Hz 4.10000Hz 5.1Hz

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

1.4 * 10 -8 s 2.4 * 10 -18 s 3.368 * 10 -8 s 4.37.68 * 10 -3 s

A11 ... Determine the frequency of the emitted waves by a system containing a 9H coil and a 4F capacitor.

1.72πHz 2.12πHz 3.36Hz 4.6Hz 5.1 / 12πHz

A12. Which of the characteristics of a light wave is used to determine its color?

1.wavelength 2.frequency

3. In phase 4. In amplitude

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

1. free 2. forced

3. Self-oscillations 4. Elastic oscillations

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

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

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

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

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

IN 1. Establish a correspondence between physical

VALUES		UNITS
	Force acting on a conductor with current from the side of the magnetic field
	Magnetic field energy
	Force acting on an electric charge moving in a magnetic field.

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

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

C1. At what angle to the lines of force of a magnetic field with an induction of 0.5 T 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 EMF of induction equal to 0.35 V is excited at its ends? (resistivity of copper 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 at resting electric charges;
2. only on moving electric charges;
3. both moving and resting electric charges.

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

1. 18 H; 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 the field induction force;
2. direction of current;
3. the direction of the Ampere force.

A7. A magnetic field with an induction of 10 mT acts on a conductor in which the current strength is 50 A, with a strength of 50 mN. Find the length of the conductor if the lines of induction of the field 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 it?

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

A9 .A body of mass m on a thread of length l vibrates with a period T. What will be the period of oscillation of a body with a mass of 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 a frequency of 100Hz?

1.33km 2.33cm 3.3m 4.3m

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

1.72πHz 2.12πHz 3.1 / 12πHz 4.6Hz

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

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

A13. Determine the period of free electromagnetic oscillations, if the oscillating 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. Mostly electronic. 2. Mostly perforated.

3. Equally electron and hole. 4. Ionic.

IN 1. Establish a correspondence between physicalquantities and units of their measurement

VALUES		UNITS
	amperage		weber (wb)
	magnetic flux		ampere (A)
	EMF induction		tesla (T)
			volts (V)

IN 2. A particle of mass m carrying a charge q , moves in a uniform magnetic field with induction B on a circle radius R with speed v. What will happen to the orbital radius, orbital period and kinetic energy of a particle with increasing magnetic induction?

For each position of the first column, select the corresponding position of 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 Wb. How long should this flux disappear for an average induction EMF of 0.74 V to appear in the coil?

Option 4

A1. What is observed in Oersted's experience?

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

A2. A moving electric charge creates:

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

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

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

A6. Induction current occurs in any closed conducting loop if:

1. The contour is in a uniform magnetic field;
2. The contour moves translationally in a uniform magnetic field;
3. The magnetic flux permeating the circuit changes.

A7. On a straight conductor 0.5 m long, located perpendicular to the field lines of the field with an induction of 0.02 T, a force of 0.15 N acts. Find the current flowing through the conductor.

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

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

1. free 2. compelled

3. Self-oscillations 4. Elastic oscillations

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

1.72πHz 2.12πHz

3.6Hz 4.1 / 12πHz

A10. Determine to what frequency you need to tune the oscillating circuit containing a coil with an inductance of 4 μH and a capacitor with a capacity of 9 Pf.

1.4 * 10 -8 s 2.3 * 10 -18 s 3.368 * 10 -8 s 4.37.68 * 10 -18 s

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

1.1ms 2.1s 3.2ms 4.2s

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

1.1700m 2. 850m 3. 136m 4.68m

A13. The number of oscillations per unit of 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 the increase in the speed of movement of electrons.

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

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

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

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

VALUES		UNITS
	EMF of induction in moving conductors
	force acting on an electric charge moving in a magnetic field
	magnetic flux

IN 2. A particle of mass m carrying a charge q , moves in a uniform magnetic field with induction B on a circle radius R with speed v U. What will happen to the orbital radius, orbital period and kinetic energy of the particle when the particle's mass decreases?

For each position of the first column, select the corresponding position of 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 placed in an alternating magnetic field, lines of force which are parallel to the axis of the coil. When the field induction changes by 1 T for 6.28 s, an EMF of 2 V appears in the coil. How many turns does the coil have?

Example ... A particle of mass m, carrying a charge q, flies into a homogeneous magnetic field perpendicular to the lines of the vector V(fig. 10). Determine the radius of the circle, the period and the 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 with the centrifugal force

we obtain for the particle radius the equality

Particle orbital period

. (3.3.3)

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

(3.3.3 ΄).

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

Consider the motion of a particle moving at an angle< 90° к направлению линий вектора V(fig. 11). Determine the step of the turn of the helix h. Speed v has two components, one of which v çç = v cosβ, is parallel V, the other v ^ = v sin β - perpendicular to the lines of magnetic induction V.

When the particle moves along the lines V the magnetic component of the force is equal to zero, therefore, the particle moves along the field uniformly with the speed

v çç = v cosβ.

Spiral turn pitch

h = v çç T = v T cosβ.

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

(3.3.4)

On a conductor element with a current Id l Ampere's 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 element of the conductor and the magnetic induction.

For a conductor of finite length, you need 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 are directed along the current.

Example ... A conductor in the form of a semicircle with a radius of R = 5 cm (Fig. 12) is placed in a uniform magnetic field, the lines of force of which are directed away from us (shown 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 is B = 1 μT.

Solution ... Let us use formula (3.3.6), taking into account that under the integral there is a vector product, and hence, ultimately, a vector quantity. It is convenient to find the sum of vectors by projecting vectors - terms on the coordinate axes 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-hand side is equal to zero, since the sum of the projections d F is equal to zero, as follows from the figure: because of the symmetry of the picture, each positive projection corresponds to a negative one of the same value. Then the required force is equal only to the second integral

F = ∫ dF у = ∫ dF cosβ,

where β is the angle between vectors d F and the axis ОΥ, and the element of the length of the conductor can be represented as dl = R cos β. Since the angle is measured from the OΥ axis to the left and right, the integration limits 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 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) with a current in a magnetic field forces act. Who strive to turn it this way. To the magnetic moment R m of the frame coincided with the direction of the magnetic induction. In this case, the torque M acting on a circuit with an area S with a 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. are called stable balance, and the position with α = 180 0 - unstable balance.

The elementary work of the magnetic field when the frame is rotated at an angle α

Option 13

C1. The electrical circuit consists of a series-connected galvanic cell ε, a light bulb and an inductor L. Describe the phenomena that occur when the key is opened.

1. The phenomenon of electromagnetic induction
tions is observed in all cases of change
of the magnetic flux through the circuit.
In particular, the EMF induction can generate
change in the circuit itself when changing
the amount of current in it, which leads to
the appearance of additional currents. it	Rice. 13.1.1. Self-induction phenomenon
the phenomenon was called self-induction
tion, and additionally arising currents
are called extracurrents or currents
self-induction.
2. Investigate the phenomenon of self-induction
tion can be done at the installation, in principle
whose schematic is shown in Fig.
13.12. Coil L with a large number of vit-
kov, through rheostat r and switch k
are connected to the source of EMF ε. Before-
In addition, a gal-
vanometer G. When short-circuited
switch at point A, the current will branch,
and a current of value i will flow
through the coil, and the current i1 through the galvanized	Rice. 13.1.2. Self-induction

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

ψ = Li,

	εsi = -				(Li) = - L

dL dt = dL di dtdi.

ε si = - L + dL di.

ε si = - L dt di.

10. When power is applied to the circuit shown in Figure 13.1.3 in the circuit, the current value will increase from zero value to the nominal value for a certain period of time due to the phenomenon of self-induction. The emerging extracurrents in accordance with the Lenz rule are always directed oppositely, i.e. they interfere with the cause that causes them. They prevent an increase in

for some time.

ε + εsi = iR,

L dt di + iR = ε.

Ldi = (ε - iR) dt,
						(ε −iR)
and integrate, assuming L to be a constant:
		L∫			= ∫ dt,
			ε −iR
		ln (ε - iR)		T + const.

i (t) = R ε - cons te - RL t.

const = R ε.

i (t) =
			- eR.

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

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

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

Rice. 13.2 Skier on the trampoline

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

mgH =

A T;

v 0 =

2 gH -

v 0 =

2. Kinematics of horizontal flight:

gτ 2

S = v0 τ = 75m;

C3. In a vertical sealed cy-

lindre under a piston with a mass of m = 10 kg and

area s = 20 cm2 is the ideal

ny monatomic gas. Originally

the piston was at a height of h = 20 cm

from the bottom of the cylinder, and after heating

the piston has risen to a height of H = 25 cm.

How much heat was given to the gas

during heating? External pressure

p0 = 105 Pa.

1. Gas pressure during heating

Rice. 13.3. Ideal gas under the piston

mg + p S = p S;

p1 = p2 = 1.5 105 Pa;

P0 S = p2 S;

2. Work done by heating:

A = p1 V = p1 S (H - h) = 15 J;

3. From the equations of state of an ideal gas:

= ν RT;

T = pV 1;

pV2 = ν RT2;

T = pV 2;

4. Change in the internal energy of the gas:

ν R T = 3 p (V - V)

22.5 J;

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 an internal resistance r = 1 Ohm and two resistors: R1 = 50 Ohm and R2 = 30 Ohm. The intrinsic resistance of the voltmeter is Rv = 320 Ohm, the resistance of the ammeter is RA = 5 Ohm. Determine the readings of the instruments.

Resistance of the whole circuit:

R Σ =

(R 1 + R 2) R 3

R 4;

R 1 + R 2 + R 3

R Σ =

5 = 69 Ohm

The strength of the current flowing through the am-

21 = 0.3 A;

I A =

RΣ + r

Voltmeter readings:

Rice. 13.4. Electrical diagram

(R 1 + R 2) R 3

0.3 64 = 19.2 B;

A R 1 + R 2 + R 3

C5. A particle with a mass of m = 10 - 7 kg, carrying a charge q = 10 - 5 C, moves uniformly around a circle of radius R = 2 cm in a magnetic field with induction B = 2 T. The center of the circle is 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 particle image in the lens move?

Particle velocity and angular velocity

QvB; v =

10− 5 2 2 10− 2

≈ 4

10− 7

10− 2

Lens magnification:

1 ; f =

30 cm; Γ = 2;

d - F

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

vx = ω 2R = 8 ms;

C6. On a plate with a reflection coefficient ρ of the incident light, N identical photons are incident perpendicularly every second, and the force of the light pressure F disappears. 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. Length of incident light:

λ = Nhc (1 + ρ); F

Rice. 14.1.1. Self-induction phenomenon

Rice. 14.1.2. Self-induction

Option 14

C1. The electrical circuit consists of a series-connected galvanic cell ε, a light bulb and an inductor L. Describe the phenomena that occur when the key is closed.

1. The phenomenon of electromagnetic induction is observed in all cases of changes in the magnetic flux through the circuit. In particular, the induction 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 the additionally arising currents are called

are produced by extracurrents or self-induction currents.

2. The phenomenon of self-induction can be investigated using a setup, a schematic diagram of which is shown in Fig. 14.1.2. Coil L with a large number of turns, through rheostat r and switch k, are connected to the source of EMF ε. In addition, a galvanometer G is connected to the coil. When the switch is shorted at point A, the current will branch, and the current of value i will flow through the coil, and the current i1 through the galvanometer. If you then open the switch, then when the magnetic field disappears in the coil

extracurrent opening I will occur.

3. According to Lenz's law, the extracurrent will prevent a decrease in the magnetic flux, i.e. will be directed in the direction of the decreasing current, but through the galvanometer the extra current will pass in the direction opposite to the initial one, which will lead to the throw of the galvanometer arrow in reverse direction... If the coil is equipped with an iron core, then 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 statement, 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 value of the current flowing through it

ψ = Li,

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

L = d i ψ, [L] = Wb A = Gn (henry).

5. We obtain the EMF equation of self-induction ε si for the coil:

	εsi = -				(Li) = - L

6. In the general case, the 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 EMF of self-induction, taking into account the last relation, will be represented by the following equation:

ε si = - L + dL di.

8. If the inductance is independent of the magnitude of the current, the equation is simplified

ε si = - L dt di.

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

10. When power is applied to the circuit,

shown in Figure 14.1.3 in the circuit, the current value will increase from zero value to the nominal value for a certain period of time due to the phenomenon of self-induction. The emerging extracurrents in accordance with the Lenz rule are always directed oppositely, i.e. they interfere with the cause that causes them. They prevent an increase in the current in the circuit. In a given

case, when the key is closed, the light Rice. 13.1.3. Making and breaking currents will not flare up immediately, but its heat will build up over a period of time.

11. When the switch is connected to position 1, extra currents will prevent an increase in the 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 resistance of the circuit, the internal resistance of the source and the active resistance of the coil L. In this case, Ohm's law will take the form:

ε + εsi = iR,

where ε is the EMF of the source, ε si is the EMF of self-induction, i is the instantaneous value of the current value, 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:

	Ldi = (ε - iR) dt,
		(ε −iR)

and integrate, assuming L to be a constant: L ∫ ε - di iR = ∫ dt,

R L ln (ε - iR) = t + const.

13. It is seen that, common decision differential equation can be represented as:

i (t) = R ε - cons te - RL t.

14. The integration constant is determined from initial conditions... At t = 0

v the moment of power supply, 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 of the equation i (t) will take the final form:

i (t) =
			- eR.

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

C2. After the impact at point A, the box slides up an inclined plane with an initial speed v0 = 5 m / s. At point B, the boxes are torn off the inclined plane. At what distance S from the inclined plane will the boxes 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 originally reported box

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

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

mv 0 2	Mv B 2	+ μ mgLcosα + mgLcosα; v0 2 = vB 2 + 2gLcosε (μ + 1);
	v B =	v0 2 - 2gLcosα (μ + 1) = 25 - 2 10 0.5 0.87 1.2 4

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

	x (t) = vB cosα t;		y (t) = h + vB sin α t -
	y (τ) = 0; h = Lcosα;
gτ 2	- vB sin ατ - Lcosα = 0; 5τ			- 2τ - 0.435 = 0;			- 0.4τ - 0.087
	τ = 0.2 +	0.04 + 0.087 ≈ 0.57c;

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 a factor of 2, and then heated to the initial temperature T1 = 360 K. What amount of heat did the gas receive in section 2 - 3?

1. Gas temperature in state 2:

= ν RT;

T 2 =

p 1 V = ν RT;

2 = 180K;

2. Change in the internal energy of the gas

on section 2 → 3:

→3

ν R (T - T);

Figure 14.3. Gas state change

U2 → 3 = 1.5

2 8.31 180 ≈ 4487 J;

3. Points 2 and 3 lie on the same isobar, therefore:

pV = ν RT;

ν RT2

= ν RT 3;

pV3 = ν RT3;

4. Gas work in section 2 → 3:

A2 → 3 = p (V3 - V2) = ν R (T3 - T2) ≈ 2992 J; 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 an 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 a resistance RA = 5 Ohm. Determine the readings of the instruments.

1. Load resistance:

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

C5. A particle with a mass of m = 10 - 7 kg and a charge of q = 10 - 5 C moves at a constant speed of v = 6 m / s in a circular magnetic field with an induction of 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 of d = 15 cm from it. The focal length of the lens is F = 10 cm. At what radius does the particle image in the lens move?

1. The radius of motion of a particle:

QvB; R =

2. Lens magnification:

; f =

30 cm; Γ = 2;

d - F

3. Image radius:

R * = 2R =

2mv =

2 10− 7 6

≈ 0.08m;

10− 5 1,5

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

1. Light pressure on the plate:						120 (1+ 0,7)
		(1 + ρ) =			+ ρ) =				≈ 1,7 10	−3
							−4