distance l, equal to 15 cm.
Topic 2. Superposition principle for fields created by point charges
11. At the vertices of a regular hexagon in vacuum there are three positive and three negative charges. Find the tension electric field in the center of the hexagon with various combinations in the arrangement of these charges. Hexagon side a = 3 cm, the magnitude of each charge q
1.5 nC.
12. In a uniform field with intensity E 0 \u003d 40 kV / m is the charge q \u003d 27 nC. Find the strength E of the resulting field at a distance r = 9 cm from the charge at the points: a) lying on the line of force passing through the charge; b) lying on a straight line passing through the charge perpendicular to the lines of force.
13. Point charges q 1 \u003d 30 nC and q 2 \u003d - 20 nC are in
dielectric medium with ε = 2.5 at a distance d = 20 cm from each other. Determine the electric field strength E at a point remote from the first charge at a distance r 1 \u003d 30 cm, and from the second - at r 2 \u003d 15 cm.
14. The rhombus is made up of two equilateral triangles co
side a = 0.2 m. At the vertices at sharp corners placed charges q 1 = q 2 = 6 10−8 C. A charge q 3 = is placed at the vertex of one obtuse angle
= −8 10 −8 Cl. Find the electric field strength E at the fourth vertex. The charges are in a vacuum.
15. Charges of the same magnitude but different signs q 1 = q 2 =
1.8 10 −8 C are located at two vertices of an equilateral triangle with side a = 0.2 m. Find the electric field strength at the third vertex of the triangle. The charges are in a vacuum.
16. At three vertices of a square with a side a = 0.4 m in
in a dielectric medium with ε = 1.6 there are charges q 1 = q 2 = q 3 = 5 10−6 C. Find the tension E at the fourth vertex.
17. Charges q 1 \u003d 7.5 nC and q 2 \u003d -14.7 nC are located in vacuum at a distance d \u003d 5 cm from each other. Find the electric field strength at a point at a distance r 1 \u003d 3 cm from the positive charge and r 2 \u003d 4 cm from the negative charge.
18. Two point charges q 1 = 2q and q 2 = − 3 q are at distance d from each other. Find the position of the point where the field strength E is zero.
19. At two opposite vertices of a square with a side
a \u003d 0.3 m in a dielectric medium with ε \u003d 1.5 there are charges of q 1 \u003d q 2 \u003d 2 10−7 C. Find the strength E and the potential of the electric field ϕ at the other two vertices of the square.
20. Find the electric field strength E at a point lying in the middle between point charges q 1 = 8 10–9 C and q 2 = = 6 10–9 C, located in vacuum at a distance r = 12 cm, in case a) charges of the same name; b) opposite charges.
Topic 3. The principle of superposition for fields created by a distributed charge
21. Thin rod length l \u003d 20 cm carries a uniformly distributed charge q \u003d 0.1 μC. Determine the strength E of the electric field created by a distributed charge in vacuum
in point A lying on the axis of the rod at a distance a = 20 cm from its end.
22. Thin rod length l = 20 cm uniformly charged with
linear density τ = 0.1 μC/m. Determine the strength E of the electric field created by a distributed charge in a dielectric medium with ε = 1.9 at point A, lying on a straight line perpendicular to the axis of the rod and passing through its center, at a distance a = 20 cm from the center of the rod.
23. A thin ring carries a distributed charge q = 0.2 μC. Determine the strength E of the electric field created by the distributed charge in vacuum at point A, equidistant from all points of the ring at a distance r = 20 cm. Ring radius R = 10 cm.
24. An infinite thin rod bounded on one side carries a uniformly distributed charge with a linear
density τ = 0.5 µC/m. Determine the strength E of the electric field created by the distributed charge in vacuum at point A, which lies on the axis of the rod at a distance a = 20 cm from its beginning.
25. A charge is uniformly distributed with a linear density τ = 0.2 μC/m along a thin ring with a radius R = 20 cm. Determine
the maximum value of the electric field strength E, created by a distributed charge in a dielectric medium with ε = 2, on the axis of the ring.
26. Straight thin wire length l = 1 m carries a uniformly distributed charge. Calculate the linear charge density τ if the field strength E in vacuum at point A, which lies on a straight line perpendicular to the axis of the rod and passing through its middle, at a distance a = 0.5 m from its middle, is equal to E = 200 V/m.
27. The distance between two thin endless rods parallel to each other, d = 16 cm. Rods
are uniformly charged with a linear density τ = 15 nC/m and are in a dielectric medium with ε = 2.2. Determine the strength E of the electric field created by distributed charges at point A, remote at a distance r \u003d 10 cm from both rods.
28. Thin rod length l \u003d 10 cm is uniformly charged with a linear density τ \u003d 0.4 μC. Determine the strength E of the electric field created by the distributed charge in vacuum at point A, lying on a straight line perpendicular to the axis of the rod and passing through one of its ends, at a distance a = 8 cm from this end.
29. Along a thin semicircle of radius R = 10 cm evenly
the charge is distributed with a linear density τ = 1 μC/m. Determine the strength E of the electric field created by the distributed charge in vacuum at point A, which coincides with the center of the ring.
30. Two-thirds of a thin ring with a radius R = 10 cm carry a charge uniformly distributed with a linear density τ = 0.2 μC / m. Determine the strength E of the electric field created by the distributed charge in vacuum at the point O, which coincides with the center of the ring.
Topic 4. Gauss's theorem
concentric |
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radius R and 2R in vacuum, |
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evenly |
distributed |
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surface densities σ1 = σ2 = σ. (rice. |
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2R 31). Using |
Gauss theorem, |
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dependence of the electric field strength E (r) on the distance for regions I, II, III. Plot E(r) .
32. See the condition of problem 31. Take σ1 = σ, σ2 = − σ. |
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33. See |
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Take σ1 = −4 σ, σ2 = σ. |
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34. See |
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Take σ1 = −2 σ, σ2 = σ. |
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35. Ha of two infinite parallel |
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planes, |
located |
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evenly |
distributed |
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surface densities σ1 = 2σ and σ2 = σ |
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(Fig. 32). Using the Gauss theorem and the principle |
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superposition of electric fields, find the expression E(x) of the electric field strength for areas I, II, III. Build
graph E(x). |
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36. See |
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chi 35. Take σ1 = −4 σ, σ2 = 2σ. |
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37. See |
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σ 2 σ |
chi 35. Take σ1 = σ, σ2 = − σ. |
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coaxial |
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endless |
cylinders |
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III II |
radii R and 2R located in |
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evenly |
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distributed |
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superficial |
densities |
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σ1 = −2 σ, and |
= σ (Fig. 33). |
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Using the Gauss theorem, find |
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distance dependence E(r) of the electric field strength for
39. 1 = − σ, σ2 = σ.
40. See problem 38. Take σ 1 = − σ, σ2 = 2σ.
Topic 5. Potential and potential difference. The work of the forces of the electrostatic field
41. Two point charges q 1 = 6 μC and q 2 = 3 μC are in a dielectric medium with ε = 3.3 at a distance d = 60 cm from each other.
What work must be done by external forces to reduce the distance between the charges by half?
42. Thin disk radius r is uniformly charged with surface density σ. Find the potential of the electric field in vacuum at a point lying on the axis of the disk at a distance a from it.
43. How much work must be done to transfer the charge q =
= 6 nC from a point at a distance a 1 \u003d 0.5 m from the surface of the ball, to a point located at a distance a 2 \u003d 0.1 m from
its surface? Ball radius R = 5 cm, ball potential ϕ = 200 V.
44. Eight identical drops of mercury charged to the potential ϕ 1 = 10 V, merge into one. What is the potential ϕ of the resulting drop?
45. Thin rod length l = 50 cm bent into a ring. He
uniformly charged with a linear charge density τ = 800 nC/m and is in a medium with a dielectric constant with ε = 1.4. Determine the potential ϕ at a point located on the axis of the ring at a distance d = 10 cm from its center.
46. The field in vacuum is formed by a point dipole with an electric moment p = 200 pC·m. Determine the potential difference U two points of the field located symmetrically with respect to the dipole on its axis at a distance r = 40 cm from the center of the dipole.
47. The electric field is formed in vacuum infinitely
long charged filament, the linear charge density of which is τ = 20 pC/m. Determine the potential difference of two points of the field spaced from the thread at a distance of r 1 = 8 cm and r 2 = 12 cm.
48. Two parallel charged planes, surface
whose charge densities σ1 = 2 μC/m2 and σ2 = −0.8 μC/m2 are located in a dielectric medium with ε = 3 at a distance d = 0.6 cm from each other. Determine the potential difference U between the planes.
49. A thin square frame is placed in a vacuum and
uniformly charged with a linear charge density τ = 200 pC/m. Determine the potential ϕ of the field at the intersection point of the diagonals.
50. Two electrical charge q 1 = q and q 2 = −2 q are located at a distance l = 6a from each other. Find the locus of points on the plane in which these charges lie, where the potential of the electric field they create is zero.
Topic 6. Movement of charged bodies in an electrostatic field
51. How much will the kinetic energy of a charged ball with mass m \u003d 1 g and charge q 1 \u003d 1 nC change when it moves in vacuum under the action of the field of a point charge q 2 \u003d 1 μC from a point remote r 1 \u003d 3 cm from this charge in point located at r 2 =
= 10 cm from him? What is the final speed of the ball if the initial speed is v 0 = 0.5 m/s?
52. An electron with a speed υ 0 \u003d 1.6 106 m / s flew into an electric field perpendicular to the velocity with a strength E
= 90 V/cm. How far from the point of entry will the electron travel when
its speed will make an angle α = 45° with the initial direction?
53. An electron with energy K = 400 eV (at infinity) moves
in vacuum along the field line towards the surface of a metal charged sphere with a radius R \u003d 10 cm. Determine the minimum distance a that an electron will approach the surface of a sphere if its charge q \u003d - 10 nC.
54. An electron passing through a plane air capacitor
from one plate to another, acquired a speed υ = 105 m/s. The distance between the plates d = 8 mm. Find: 1) potential difference U between the plates; 2) surface charge density σ on the plates.
55. An infinite plane is in a vacuum and is uniformly charged with a surface density σ = − 35.4 nC/m2. Towards lines of force the electric field created by the plane moves the electron. Determine the minimum distance l minto which an electron can approach this plane if at a distance l 0 =
= 10 cm from the plane, it had a kinetic energy K = 80 eV.
56. What is the minimum speed υ min must have a proton so that it can reach the surface of a charged metal ball with a radius R = 10 cm, moving from a point located on
distance a = 30 cm from the center of the ball? Ball potential ϕ = 400 V.
57. Into a uniform electric field of intensity E =
= 200 V/m flies (along the line of force) an electron with a speed υ 0 =
= 2 mm/s. Determine the distance l , which the electron will pass to the point where its speed will be equal to half the initial one.
58. Proton with speed υ 0 = 6 105 m/s flew into a uniform electric field perpendicular to the velocity υ0 with
tension
E = 100 V/m. How far from the initial direction of motion will the electron move when its velocity υ makes an angle α = 60° with this direction? What is the potential difference between the entry point in the field and this point?
59. An electron flies into a uniform electric field in the direction opposite to the direction of the lines of force. At some point of the field with a potential ϕ1 = 100 V, the electron had a velocity υ0 = 2 Mm/s. Determine the potential ϕ2 of the field point at which the electron's velocity will be three times greater than the initial one. What path will the electron take if the electric field strength E \u003d
5 10 4 V/m?
60. An electron flies into a flat air capacitor of length
l = 5 cm with a velocity υ0 = 4 107 m/s directed parallel to the plates. The capacitor is charged to a voltage of U = 400 V. The distance between the plates is d = 1 cm. Find the displacement of the electron caused by the field of the capacitor, the direction and magnitude of its speed at the moment of departure?
Topic 7. Electricity. Capacitors. Electric field energy
61. Capacitors C 1 \u003d 10 μF and C2 \u003d 8 μF are charged to voltages U 1 \u003d 60 V and U 2 \u003d 100 V, respectively. Determine the voltage on the capacitor plates after they are connected by plates having the same charges.
62. Two flat capacitors with capacitances C 1 = 1 uF and C2 =
= 8 uF connected in parallel and charged to a potential difference U \u003d 50 V. Find the potential difference between the capacitor plates if, after disconnecting from the voltage source, the distance between the plates of the first capacitor was reduced by 2 times.
63. Flat air capacitor charged to voltage U = 180 V and disconnected from the voltage source. What will be the voltage between the plates if the distance between them is increased from d 1 \u003d 5 mm to d 2 \u003d 12 mm? Find Job A by
separation of the plates and the density w e of the energy of the electric field before and after the expansion of the plates. Plate area S = 175 cm2.
64. Two capacitors with capacitances C 1 \u003d 2 μF and C2 \u003d 5 μF are charged to voltages U 1 \u003d 100 V and U 2 \u003d 150 V, respectively.
Determine the voltage U on the capacitor plates after they are connected by plates with opposite charges.
65. A metal ball with a radius R 1 \u003d 10 cm is charged to a potential ϕ1 \u003d 150 V, it is surrounded by a concentric conductive uncharged shell with a radius R 2 \u003d 15 cm. What will the potential of the ball ϕ be equal to if the shell is grounded? Connect the ball to the shell with a conductor?
66. Flat capacitor capacitance C = 600 pF. The dielectric is glass with a permittivity ε = 6. The capacitor was charged to U = 300 V and disconnected from the voltage source. What work must be done to remove the dielectric plate from the capacitor?
67. Capacitors with capacity C 1 = 4 uF charged to U 1 =
= 600 V, and capacity C 2 \u003d 2 μF, charged to U 2 \u003d 200 V, connected by similarly charged plates. Find energy
W jumping spark.
68. Two metal balls with radii R 1 \u003d 5 cm and R 2 \u003d 10 cm have charges q 1 \u003d 40 nC and q 2 \u003d - 20 nC, respectively. Find
energy W, which is released during the discharge, if the balls are connected by a conductor.
69. A charged ball with a radius R 1 = 3 cm is brought into contact with an uncharged ball with a radius R 2 = 5 cm. After the balls were separated, the energy of the second ball turned out to be W 2 =
= 0.4 J. What charge q 1 was on the first ball before the contact?
70. Capacitors with capacitances C 1 = 1 uF, C 2 = 2 uF and C 3 =
= 3uF connected to voltage source U = 220 V. Determine the energy W of each capacitor in the case of their series and parallel connection.
Topic 8. Direct electric current. Ohm's laws. Work and current power
71. In a circuit consisting of a battery and a resistor with a resistance R \u003d 10 Ohm, turn on the voltmeter first in series, then in parallel with the resistance R. Voltmeter readings are the same in both cases. Voltmeter resistance R V
10 3 ohm. Find the internal resistance of the battery r.
72. EMF source ε \u003d 100 V, internal resistance r \u003d
= 5 ohm. A resistor is connected to the source R 1 \u003d 100 ohms. In parallel, a capacitor was connected to it with a series
connected to it by another resistor with a resistance R 2 \u003d 200 Ohms. The charge on the capacitor turned out to be q = 10−6 C. Determine the capacitance of the capacitor C.
73. From a battery whose emfε = 600 V, it is required to transfer energy over a distance l = 1 km. Power consumption Р = 5 kW. Find the minimum power loss in the network if the diameter of the copper supply wires is d = 0.5 cm.
74. At a current strength I 1 \u003d 3 A, power P 1 \u003d 18 W is released in the external battery circuit, at a current I 2 \u003d 1 A - P 2 \u003d 10 W. Determine the current strength I short circuit of the EMF source.
75. Battery EMF ε = 24 V. Greatest Strength current that the battery can give I max \u003d 10 A. Determine the maximum power Pmax that can be released in the external circuit.
76. At the end of the battery charging, the voltmeter, which is connected to its poles, shows the voltage U 1 \u003d 12 V. Charging current I 1 \u003d 4 A. At the beginning of battery discharge at current I 2
= 5 A voltmeter shows voltage U 2 \u003d 11.8 V. Determine the electromotive force ε and the internal resistance r of the battery.
77. From a generator whose EMFε = 220 V, it is required to transfer energy over a distance l = 2.5 km. Consumer power P = 10 kW. Find the minimum section of conductive copper wires d min if the power loss in the network should not exceed 5% of the consumer's power.
78. The electric motor is powered by a network with a voltage of U \u003d \u003d 220 V. What is the power of the engine and its efficiency when the current I 1 \u003d 2 A flows through its winding, if the current I 2 \u003d 5 A flows through the circuit when the armature is fully braked?
79. In a network with voltage U \u003d 100 V connected a coil with a resistance R 1 \u003d 2 kOhm and a voltmeter connected in series. Voltmeter reading U 1 = 80 V. When the coil was replaced by another, the voltmeter showed U 2 = 60 V. Determine the resistance R 2 of the other coil.
80. A battery with EMF ε and internal resistance r is closed to external resistance R. The highest power released
in the external circuit is equal to P max = 9 W. In this case, a current flows I \u003d 3 A. Find the EMF of the battery ε and its internal resistance r.
Topic 9. Kirchhoff rules
81. Two current sources (ε 1 \u003d 8 V, r 1 \u003d 2 Ohm; ε 2 \u003d 6 V, r 2 \u003d 1.6 Ohm)
and a rheostat (R = 10 ohms) are connected as shown in fig. 34. Calculate the strength of the current flowing through the rheostat.
ε1 , |
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ε2 , |
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82. Determine the current strength in the resistance R 3 (Fig. 35) and the voltage at the ends of this resistance, if ε 1 = 4 V, ε 2 = 3 V,
identical internal resistances equal to r 1 \u003d r 2 \u003d r 3 \u003d 1 Ohm, are interconnected by like poles. The resistance of the connecting wires is negligible. What are the currents flowing through the batteries?
ε 1, r 1 |
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er 1 |
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ε 2, r 2 |
ε 2, r 2 |
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Location:
1. The sum of 4 interior angles of a rhombus is 360°, just like any quadrilateral. Opposite angles of the rhombus have the same value, and always in the 1st pair of equal angles - the angles are acute, in the second - obtuse. 2 angles that are adjacent to the 1st side add up to developed angle.
Rhombuses with equal side sizes can look quite different from each other. This difference is due to the different internal angles. That is, to determine the angle of a rhombus, it is not enough to know only the length of its side.
2. To calculate the angles of a rhombus, it is enough to know the lengths of the diagonals of the rhombus. After constructing the diagonals, the rhombus is divided into 4 triangles. The diagonals of the rhombus are at right angles, that is, the triangles that are formed turn out to be rectangular.
Rhombus- a symmetrical figure, its diagonals are at the same time the axes of symmetry, which is why each internal triangle is equal to the others. The acute angles of the triangles, which are formed by the diagonals of the rhombus, are equal to ½ of the desired angles of the rhombus.
1. In a uniform electric field with a strength of 3 MV / m, the lines of force of which make an angle of 30 ° with the vertical, a ball of mass 2 g hangs on a thread, and the charge is 3.3 nC. Determine the tension in the thread.
2. The rhombus is composed of two equilateral triangles with a side whose length is 0.2 m. At the vertices at acute angles of the rhombus, identical positive charges of 6⋅10 -7 C each are placed. A negative charge of 8⋅10 -7 C is placed at the top at one of the obtuse angles. Determine the electric field strength at the fourth vertex of the rhombus. (answer in kV/m)
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3. What angle α with the vertical will be made by a thread on which a ball of mass 25 mg hangs, if the ball is placed in a horizontal homogeneous electric field with a strength of 35 V / m, giving it a charge of 7 μC?
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4. Four identical charges of 40 μC each are located at the vertices of a square with sides but= 2 m. What will be the field strength at a distance of 2 but from the center of the square on the continuation of the diagonal? (answer in kV/m)
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5. Two charged balls with masses of 0.2 g and 0.8 g, having charges of 3⋅10 -7 C and 2⋅10 -7 C, respectively, are connected by a light non-conductive thread 20 cm long and move along the line of force of a uniform electric field. The field strength is 10 4 N/C and is directed vertically downwards. Determine the acceleration of the balls and the tension in the thread (in mN).
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6. The figure shows the electric field strength vector at point C; the field is created by two point charges q A and q B . What is the approximate charge q B if the charge q A is +2 μC? Express your answer in microcoulombs (µC).
= 1.05*elStat2_6 & answer_ check
7. A dust grain with a positive charge of 10 -11 C and a mass of 10 -6 kg flew into a uniform electric field along its lines of force with an initial speed of 0.1 m / s and moved a distance of 4 cm. What was the speed of the dust grain if the intensity fields 10 5 V/m?
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8. A point charge q, placed at the origin, creates at point A (see figure) an electrostatic field with a strength of E 1 = 65 V / m. Determine the value of the field strength modulus E 2 at point C.
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