A particle of mass m carrying charge q is moving. A magnetic field
, methodologist UMC Zel UO
To answer the questions of KIM USE on this topic, it is necessary to repeat the concepts:
The interaction of the poles of magnets,
The interaction of currents,
Magnetic induction vector, properties of lines of force magnetic field,
Application of the gimlet rule to determine the direction of the magnetic induction of the field of direct and circular current,
ampere power,
Lorentz force,
Left hand rule for determining the direction of the Ampère force, Lorentz force,
Movement of charged particles in a magnetic field.
In the materials of the KIM USE, there are often test tasks to determine the direction of the Ampère force and the Lorentz force, and in some cases the direction of the magnetic induction vector is given implicitly (the poles of the magnet are shown). 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, shrinks (you must choose the correct answer). A traditional series of tasks for the analysis of 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 of others.
The task is found under the number A15.
1. A permanent bar magnet was brought to the magnetic needle (the north pole is dark, see the figure), which can rotate around a vertical axis perpendicular to the plane of the drawing. While the arrow
2. Straight conductor length L with current I placed in a uniform magnetic field perpendicular to the lines of induction V . How will the Ampere force acting on the conductor change if its length is doubled and the current 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?
4. Straight conductor length L with current I placed in a uniform magnetic field, the direction of the induction lines V 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 Ampère force acting on the conductor
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will increase by 2 times |
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will decrease by 4 times |
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will decrease by 2 times |
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Will not change |
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?
6. The figure shows a cylindrical conductor through which an electric current flows. The direction of the current is indicated by an arrow. How is the magnetic induction vector directed at point C?
7. The figure shows a coil of wire through which an 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 current magnetic field induction vector is directed
8. In the diagram 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. The Ampere force acting on conductor No. 3 in this case:
9. The angle between the conductor with current and the direction of the magnetic induction vector of the magnetic field increases from 30° to 90°. The ampere force is:
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1) increases by 2 times |
2) decreases by 2 times |
3) does not change |
4) decreases to 0 |
10. The Lorentz force acting on an electron moving in a magnetic field at a speed of 107 m / s along a circle in a uniform magnetic field B \u003d 0.5 T is equal to:
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4)8 10-11 N |
1. (B1). Particle mass m, charge carrier q V around the circumference of the radius R with speed u. What will happen to the radius of the orbit, the period of revolution and the kinetic energy of the particle with an increase in the speed of movement?
to the table
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physical quantities |
their changes |
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orbit radius |
will increase |
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circulation period |
decrease |
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kinetic energy |
Will not change |
(Answer 131)
2 IN 1). particle mass m, which carries a charge q, moves in a uniform magnetic field with induction V around the circumference of the radius R with speed u. What will happen to the radius of the orbit, the period of revolution and the kinetic energy of the particle with an increase in the induction of the magnetic field?
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.
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physical quantities |
their changes |
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orbit radius |
will increase |
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circulation period |
decrease |
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kinetic energy |
Will not change |
(Answer 223)
3. (B4). Straight conductor length l\u003d 0.1 m, through which the current flows, is in a uniform magnetic field with induction B \u003d 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 1
A1. What explains the interaction of two parallel conductors with direct current?
- interaction of electric charges;
- action electric field one conductor with current to the current in another conductor;
- the effect of the magnetic field of one conductor on the current in another conductor.
A2. Which particle is affected by the magnetic field?
- on a moving charged;
- on a moving uncharged;
- to a charged one at rest;
- to an uncharged one at rest.
A4. A straight conductor 10 cm long is placed in a uniform magnetic field with an induction of 4 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 side of the magnetic field, if the current strength in the conductor is 3 A?
- 1.2 N; 2) 0.6 N; 3) 2.4 N.
A6. Electromagnetic induction is:
- a phenomenon characterizing the effect of a magnetic field on a moving charge;
- closed circuit occurrence phenomenon electric current when changing the magnetic flux;
- a phenomenon that characterizes the effect of a magnetic field on a current-carrying conductor.
A7. Children swing on swings. What kind of oscillation 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. 4T 4. ¼ T
A9. The speed of sound in water is 1470m/s. What is the length of a sound wave with an oscillation period of 0.01 s?
1. 147km 2. 1.47cm 3. 14.7m 4. 0.147m
A10 . What is the number of oscillations in 2πs called?
1st frequency 2nd period 3rd phase 4th cycle frequency
A11. The boy heard an echo 10 seconds after the cannon 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 oscillatory circuit contains a coil with an inductance of 1 μH and a capacitor with a capacitance of 36pF.
1. 40ns 2. 3*10 -18 s 3. 3.768*10 -8 s 4. 37.68*10 -18 s
A13. Protozoa oscillatory system containing a capacitor and an inductor is called ...
1. self-oscillatory system 2. oscillatory system
3. Oscillating circuit 4. Oscillating plant
A14. How and why does the electrical resistance of semiconductors change with increasing temperature?
1. Decreases due to an increase in the speed of electrons.
2. Increases due to an increase in the amplitude of oscillations of the positive ions of the crystal lattice.
3. Decreases due to an increase in the concentration of free charge carriers.
4. Increases due to an increase in the concentration of free electric charge carriers.
IN 1.
VALUES | UNITS | ||
inductance | tesla (Tl) |
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magnetic flux | henry (Hn) |
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magnetic field induction | weber (Wb) |
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volt (V) |
IN 2. Particle of mass m , carrying charge q B around the circumference of the radius R with speed v . What will happen to the radius of the orbit, the period of revolution and the kinetic energy of the particle with an increase in the speed of movement?
C1. In a coil with an inductance of 0.4 H, an EMF of self-induction of 20 V occurred. 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 current-carrying conductor is explained by the fact that it is affected by:
- magnetic field created by charges moving in a conductor;
- electric field created by the charges of the conductor;
- electric field created by the moving charges of a conductor.
A2.
- only electric field;
- only magnetic field.
A4. A straight conductor 5 cm long is located 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 side of the magnetic field, if the current strength in the conductor is 2 A?
- 0.25 N; 2) 0.5 N; 3) 1.5 N.
A6. The Lorentz force works
- on an uncharged particle in a magnetic field;
- on a charged particle resting in a magnetic field;
- on a charged particle moving along the lines of magnetic field induction.
A7. For a 2m square frame 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 space under study?
- Tl; 2) 2 T; 3) 3T.
A8. What type of oscillation occurs when the 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 33cm?
1. 1000Hz 2. 100Hz 3. 10Hz 4. 10000Hz 5. 0.1Hz
A10 Determine the period of free electromagnetic oscillations if the oscillatory circuit contains a capacitor with a capacitance of 1 μF and an inductance coil of 36H.
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 the emitted waves by a system containing a coil with an inductance of 9H and a capacitor with an electrical capacity of 4F.
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. Continuous oscillations that occur 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 water solution Salt NaCl is a conductor?
1. Salt in water breaks down into charged Na ions+ and Cl - .
2. After the salt dissolves, the NaCl molecules transfer a charge
3. In solution, electrons are detached from the NaCl molecule and charge is transferred.
4. When interacting with salt, water molecules decompose into hydrogen and oxygen ions
IN 1. Establish a correspondence between physical
VALUES | UNITS | ||
The force acting on a conductor with current from the magnetic field | |||
Magnetic field energy | |||
The force acting on an electric charge moving in a magnetic field. | |||
Moves in a uniform magnetic field with induction B around the circumference of the radius R with speed v. What will happen to the radius of the orbit, the period of revolution and the kinetic energy of the particle with an increase in the charge of the particle?
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 magnetic field lines 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, EMF induction equal to 0.35 V? (copper resistivity ρ= 0.017 Ohm∙mm 2 /m)
Option 3
A1. Magnetic fields are created:
- both stationary and moving electric charges;
- immobile electric charges;
- moving electric charges.
A2. The magnetic field has an effect:
- only on electrical charges at rest;
- only on moving electric charges;
- both moving and resting electric charges.
A4. What force acts from a uniform magnetic field with an induction of 30 mT on a rectilinear conductor 50 cm long located in the field, through which a current of 12 A flows? The wire forms a right angle with the direction of the magnetic induction vector of the field.
- 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
- direction of field induction force;
- current direction;
- direction of Ampere's 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 force of 50 mN. Find the length of the conductor if the field induction lines and the current are mutually perpendicular.
- 1m; 2) 0.1 m; 3) 0.01 m; 4) 0.001 m.
A8. The chandelier swings after one push. What type of oscillation 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 330m/s. What is the wavelength of light at a 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 did the lightning flash from the boy?
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 capacitance of 9pF.
A14. What type of conductivity do semiconductor materials with donor impurities have?
1. Mostly electronic. 2. Mostly holey.
3. In equally electronic and hole. 4. Ionic.
IN 1. Establish a correspondence between physicalquantities and units of their measurement
VALUES | UNITS | ||
current strength | weber (Wb) |
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magnetic flux | ampere (A) |
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EMF induction | tesla (Tl) |
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volt (V) |
IN 2. A particle of mass m that carries a charge q , moves in a uniform magnetic field with induction B around the circumference of the radius R with speed v. What will happen to the radius of the orbit, the period of revolution and the kinetic energy of the particle with an increase in the induction of the magnetic field?
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 flow disappear in order for the coil to have an average induction emf of 0.74 V?
Option 4
A1. What is observed in Oersted's experiment?
- a conductor with current acts on electric charges;
- the magnetic needle turns near the conductor with current;
- magnetic needle turns charged conductor
A2. A moving electric charge creates:
- only electric field;
- both electric field and magnetic field;
- 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. Determinant of 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. An inductive current occurs in any closed conducting circuit if:
- The circuit is in a uniform magnetic field;
- The circuit moves forward in a uniform magnetic field;
- The magnetic flux penetrating 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 subjected to 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 oscillation is observed when a load suspended on a thread deviates from the 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 9H and a capacitor with an electric capacitance of 4F.
1. 72πHz 2. 12πHz
3. 6Hz 4. 1/12πHz
A10. Determine at what frequency you need to tune an oscillatory circuit containing a coil with an inductance of 4 μH and a capacitor with a capacitance of 9Pf.
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. 1us 2. 1ks 3. 2us 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 did the lightning flash from the boy?
1. 1700m 2. 850m 3. 136m 4. 68m
A13. The number of oscillations per unit of time is called..
1st frequency 2nd period 3rd phase 4th cycle 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 electrons.
2. Decreases due to an increase in the speed 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 the 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 that carries a charge q , moves in a uniform magnetic field with induction B around the circumference of the radius R with speed v U. What happens to the radius of the orbit, the period of revolution and the kinetic energy of the particle as the mass of the particle 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,whose lines of force 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 appeared in the coil. How many turns does the coil have.
Option 13
C1. Electrical circuit consists of a galvanic cell ε, a light bulb and an inductor L connected in series. Describe the phenomena that occur when the key is opened.
1. The phenomenon of electromagnetic induction |
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tion is observed in all cases of change |
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magnetic flux through the loop. |
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In particular, the induction EMF can gener- |
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change in the circuit itself when changing |
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current in it, which leads to |
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appearance of additional currents. This |
Rice. 13.1.1. The phenomenon of self-induction |
The phenomenon is called self-induction |
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tions, and additionally arising currents |
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called extra currents or currents |
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self-induction. |
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2. Investigate the phenomenon of self-induction |
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tions can be installed at the installation, in principle |
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whose scheme is shown in fig. |
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13.12. Coil L with a large number of vit- |
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kov, through rheostat r and switch k |
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connected to the source of EMF ε. Before- |
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In addition, a gal- |
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vanometer G. If the trans- |
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switch at point A, the current will branch, |
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moreover, a current of value i will flow |
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through the coil, and the current i1 through the galvanic |
Rice. 13.1.2. self induction |
meter. If the switch is then opened, then when the magnetic flux disappears in the coil, an extra current of opening I will occur.
ψ = Li ,
εsi = − |
(Li) = −L |
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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 will increase from zero to nominal over 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 the increase
some time.
ε + εsi = iR ,
L dt di +iR = ε.
Ldi = (ε − iR) dt, |
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(ε −iR ) |
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and integrate assuming L to be a constant: |
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L∫ |
= ∫ dt , |
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ε −iR |
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log(ε − iR) |
T + const . |
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i(t) = R ε − cons te − RL t .
const = Rε .
i(t) = |
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− eR . |
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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 the EMF of self-induction 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. The dependence of the 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 H = 40 m, at the moment of separation his speed is horizontal. In the process of moving along the springboard, the friction force did the work AT = 5.25 kJ. Determine the range of the skier's flight in the horizontal direction if the landing point was h = 45 m below the level of separation from the springboard. Air resistance is ignored.
Rice. 13.2 Skier on a ski jump
1. The law of conservation of energy when a skier moves on a springboard:
mgH= |
A T ; |
v 0 = |
2 gH |
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v 0 = |
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2. Kinematics of level flight: |
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gτ 2 |
S = v0 τ = 75m; |
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C3. In a vertical sealed qi- |
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lindre under the piston mass m = 10 kg and |
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area s \u003d 20 cm2 is an ideal |
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ny monatomic gas. Initially |
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the piston was at a height h = 20 cm |
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from the bottom of the cylinder, and after heating |
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the piston has risen to a height H = 25 cm. |
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How much heat was imparted to the gas |
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during heating? External pressure |
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p0 = 105 Pa. |
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1. Gas pressure during heating - |
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Rice. 13.3. Ideal gas under a piston |
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mg + pS = pS; |
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p1 = p2 = 1.5 105 Pa; |
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P0 S = p2 S; |
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2. Work done when heated: |
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A = p1 V = p1 S(H − h) = 15 J; |
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3. From the equations of state of an ideal gas: |
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= νRT; |
T = pV 1 ; |
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pV2 = vRT2 ; |
T = pV 2 ; |
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4. Change in the internal energy of the gas: |
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ν R T = 3 p(V − V ) |
22.5 J; |
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5. The amount of heat reported 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 Rv = 320 ohms, the resistance of the ammeter RA = 5 ohms. Determine instrument readings.
Whole circuit resistance: |
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RΣ = |
(R 1 + R 2 ) R 3 |
R4; |
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R1 + R2 + R3 |
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RΣ = |
5 = 69 ohm |
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The strength of the current flowing through the am- |
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21 = 0.3 A; |
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I A = |
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RΣ + r |
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Voltmeter readings: |
Rice. 13.4. Wiring diagram |
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(R 1 + R 2 ) R 3 |
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0.3 64 = 19.2 B; |
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A R 1 + R 2 + R 3 |
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C5. A particle with a 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. How fast does the particle image move in the lens?
speed and angular velocity particle motion |
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QvB; v= |
10− 5 2 2 10− 2 |
≈ 4 |
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10− 7 |
10− 2 |
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Lens magnification: |
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one ; f= |
30 cm; Γ = 2; |
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d − F |
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3. For the image, the angular velocity will remain unchanged, and the radius of the circle will double, therefore:
vx = ω 2R = 8 m s;
C6. On a plate with a reflection coefficient ρ of the incident light, N identical photons fall perpendicularly every second, and the force of light pressure F prevails. 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. The phenomenon of self-induction
Rice. 14.1.2. self induction
Option 14
C1. An electrical circuit consists of a galvanic cell ε, a light bulb and an inductor L connected in series. Describe the phenomena that occur when the key is closed.
1. I am a phenomenon electromagnetic induction 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 additionally arising currents are called
are driven by extra currents or self-induction currents.
2. It is possible to study the phenomenon of self-induction on the installation, the schematic diagram of which is shown in fig. 14.1.2. Coil L with a large number of turns, through a rheostat r and a switch k are connected to an EMF source 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 i will flow through the coil, and the current i1 through the galvanometer. If the switch is then opened, then when the magnetic field disappears in the coil,
current, an extra current of 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 towards the decreasing current, but the extra current will pass through the galvanometer in the direction opposite to the original one, which will lead to the throw of the galvanometer needle in the opposite direction. If the coil is provided with an iron core, then the magnitude of the extra current increases. Instead of a galvanometer, in this case, you can turn on an incandescent light bulb, which is actually set in the condition of the problem; 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 inductance of the circuit. The dimension of inductance is determined by the equation:
L \u003d d i ψ , [ L] \u003d Wb A \u003d Hn (henry) .
5. We obtain the equation for the EMF of self-induction ε si for the coil:
εsi = − |
(Li) = −L |
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6. In the general case, the inductance, along with the geometry of the coil in media, can depend on the strength of the current, i.e. L \u003d 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 does not depend on the magnitude of the current, the equation simplifies
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 will increase from zero to nominal over 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 the increase in 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 incandescence will increase over time.
11. When the switch is connected to position 1, extra currents will prevent an 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 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. Ohm's law in this case 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, which is a function of time. Let us substitute the self-induction EMF equation into Ohm's law:
L dt di +iR = ε.
12. We divide the variables in the differential equation:
Ldi = (ε − iR) dt, |
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(ε −iR ) |
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and integrate assuming L to be constant: L ∫ ε − di iR = ∫ dt ,
R L ln(ε − iR) = t + const .
13. It can be seen that the general solution differential equation can be represented as:
i(t) = R ε − cons te − RL t .
14. Let us determine the integration constant from the initial conditions. At t =0
v the moment of power supply, the current in the circuit is equal to zero i(t) = 0. Substituting the zero value of the current, we obtain:
const = Rε .
15. The solution of the equation i(t) will take the final form:
i(t) = |
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− eR . |
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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 lifts off the inclined plane. At what distance S from the inclined plane will the box fall? The friction coefficient of the box on the plane μ = 0.2. The length of the inclined plane AB \u003d L \u003d 0.5 m, the angle of inclination of the plane α \u003d 300. Ignore 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 the force
friction, the kinetic energy at point B and the increase in the potential energy of the box:
mv 0 2 |
Mv B 2 |
+ μ mgLcosα + mgLcosα ; v0 2 = vB 2 + 2gLcosε (μ + 1) ; |
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v B = |
v0 2 − 2gLcosα (μ + 1) = 25 − 2 10 0.5 0.87 1.2 4 |
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2. From point B, the box will move along a parabolic trajectory:
x(t) = vB cosα t; |
y(t) = h + vB sin α t − |
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y(τ ) = 0; h = Lcosα ; |
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gτ 2 |
− vB sin ατ − Lcosα = 0; 5τ |
− 2τ − 0.435 = 0; |
− 0.4τ − 0.087 |
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τ = 0.2 + |
0.04 + 0.087 ≈ 0.57c; |
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3. Distance from the inclined plane to the point of fall: x(τ) = vB cosατ ≈ 4 0.87 0.57 ≈ 1.98m;
C3. An ideal monatomic gas in an amount of ν = 2 mol was first cooled by 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?
1. Gas temperature in state 2: |
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= νRT; |
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T2= |
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p 1 V = ν RT ; |
2=180K; |
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2. Change in the internal energy of the gas |
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in section 2 → 3: |
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→3 |
νR(T − T); |
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Fig.14.3. Changing the state of the gas |
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U2 → 3 = 1.5 |
2 8.31 180 ≈ 4487 J; |
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3. Points 2 and 3 lie on the same isobar, therefore: |
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pV = vRT ; |
vRT2 |
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= ν RT 3 ; |
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pV3 = vRT3 ; |
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4. Gas operation 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 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 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 B;
C5. A particle with a mass m = 10 − 7 kg and a charge q = 10 − 5 C moves with a constant speed v = 6 m/s along a circle in a magnetic field with an induction B = 1.5 T. The center of the circle is on the main optical axis of the converging lens, and the plane of the circle is perpendicular to the main optical axis and is at a distance d = 15 cm from it. The focal length of the lens is F = 10 cm. On a circle of what radius does the particle image move in the lens?
1. Radius of particle motion:
QvB; R= |
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2. Lens magnification: |
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; f= |
30 cm; Γ = 2; |
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d − F |
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3. Image Radius: |
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R* = 2R = |
2mv= |
2 10− 7 6 |
≈ 0.08m; |
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10− 5 1,5 |
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C6. On a plate with an area S = 4 cm2, which reflects 70% and absorbs 30% of the incident light, light with a wavelength λ = 600 nm is incident perpendicularly. Luminous flux power N = 120 W. How much pressure does the light exert on the plate?
1. Light pressure on the plate: |
120 (1+ 0,7) |
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(1 + p) = |
+ ρ) = |
≈ 1,7 10 |
−3 |
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−4 |
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Example . A particle of mass m, carrying a charge q, flies into a uniform 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 speed 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
we obtain for the radius of the particle the equality
Particle orbital period
.
(3.3.3)
The circular frequency ω is the 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 particles, the period and frequency depend only on the induction of the magnetic field.
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Consider the motion of a particle moving at an angle< 90° к направлению линий вектора V(Fig. 11). Let us determine the pitch 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 is perpendicular to the lines of magnetic induction V.
When a particle moves along lines V the magnetic component of the force is zero, so the particle moves uniformly along the field with a speed
vçç = v cosβ.
Helix pitch
h = v çç T = v T cosβ.
Substituting the expression for T from formula (1.3.3), we obtain:
(3.3.4)
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Per conductor element with current Id l Ampère 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 Ampère force, as well as 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 half ring 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 strength of the current flowing through the conductor is I \u003d 2 A, and the magnetic field induction B \u003d 1 μT.
Solution . We use formula (3.3.6), taking into account that under the integral is vector product, and hence, 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.
According to the left hand rule, we find the force vectors d F acting on each element of the conductor (Fig. 12).
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The first integral on the right side is equal to zero, because 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 desired force is only equal to the second integral
F = ∫ dF y = ∫ dF cosβ,
where β is the angle between the vectors d F and axis ОΥ, and the length element of the conductor can be represented as dl = R cos β. Since the angle is measured from the ОΥ axis to the left and to the right, the integration limits will be the values - 90 0 and 90 0 . Substituting dl into dF and solving the second integral, we get
F= 
Numerical calculation gives: F = 2 2 A 10 -6 T 0.05 m = 2 10 -7 N.
Answer: F = 2 10 -7 N.
Ampère'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 currents.
on the frame ( circuit) forces act with current in a magnetic field. Who seek to turn her so. To make the magnetic moment R m frame coincided with the direction of magnetic induction. At the same time, the torque M, acting on the circuit 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 in which the angle α = 0 0 . called stable balance, and the position with α = 180 0 - unstable balance.
Elementary work of the magnetic field when the frame is rotated through an angle α
