Paraffin plate fills all the space between the plane capacitor plates. Capacitor electrical capacity with paraffin 4 μF, its charge is 0.2 μl. What work should be done to pull the plate from the condenser?
Task # 6.4.56 from "Collection of tasks for preparing for entrance exams on physics UGNTU"
Given:
\\ (C_0 \u003d 4 \\) μФ, \\ (q \u003d 0.2 \\) μl, \\ (a -? \\)
The solution of the problem:
The desired work \\ (a \\) of the external force, which should be made to pull the paraffin plate from the capacitor, according to the law of energy conservation, can be determined as the difference of final \\ (W_2 \\) and the initial \\ (W_1 \\) of the condenser energy, so:
The condenser energies mentioned in this problem are advisable to find according to such formulas:
\\ [\\ left \\ (\\ begin (gathered)
(W_2) \u003d \\ FRAC (((q ^ 2))) ((2c)) \\ hfill \\\\
\\ End (Gathered) \\ Right. \\]
The final electrical capacity of the capacitor \\ (C \\) (i.e., after the extraction of the paraffin plate) is associated with the initial \\ (C_0 \\) with this relation:
Here \\ (\\ varepsilon \\) is the dielectric permeability of paraffin, equal to 2.
Then we have:
\\ [\\ left \\ (\\ begin (gathered)
(W_2) \u003d \\ FRAC (((q ^ 2) \\ varepsilon)) ((2 (C_0))) \\ hfill \\\\
(W_1) \u003d \\ FRAC (((Q ^ 2))) ((2 (C_0))) \\ hfill \\\\
\\ End (Gathered) \\ Right. \\]
The resulting expressions will substitute to the very first formula:
Excellent, the task is solved, we consider the answer:
Answer: 5 MJ.
If you do not understand the solution and you have some kind of question or you have found an error, then boldly leave the comment below.
Page 3 of 4
41. The electrostatic field is created by a ball with a radius R \u003d 8 cm, evenly charged with a bulk density ρ \u003d 10 NK / M 3. Determine the difference of potentials between the two points of this field, lying at a distance R 1 \u003d 10 cm and R 2 \u003d 15 cm from the center of the ball.
42. The electrostatic field is created by a ball with a radius R \u003d 10 cm, evenly charged with a bulk density ρ \u003d 20 NK / M 3. Determine the potential difference between points underlying the ball at distances R 1 \u003d 2 cm and R 2 \u003d 8 cm from its center.
43. The electrostatic field is created by an infinite cylinder with a radius of 8 mm, evenly charged with a linear density τ \u003d 10 nkl / m. Determine the difference of potentials between the two points of this field, lying at a distance R 1 \u003d 2 mm and R 2 \u003d 7 mm from the surface of this cylinder.
44. In a homogeneous electrostatic field, the infinite plane-parallel glass plate (ε \u003d 7) is placed in a homogeneous electrostatic field of tension E 0 \u003d 700 V / m. Determine: 1) the tension of the electrostatic field inside the plate; 2) electrical displacement inside the plate; 3) polarity glass; 4) Surface density of linked charges on glass.
45. The space between the plates of the flat capacitor is filled with paraffin (ε \u003d 2). Distance between the plates D = 8.85 mm. What potential difference should be applied to the plates so that the surface density of the associated charges on the paraffin was 0.1 nkl / cm 2?
46. \u200b\u200bThe distance between the plates of the flat condenser is D \u003d 5 mm. After charging the capacitor, until the potential difference U \u003d 500 V between the plates of the capacitor was tightened with a glass plate (ε \u003d 7). Determine: 1) the dielectric susceptibility of the glass; 2) Surface density of linked charges on a glass plate.
47. Determine the surface density of linked charges on a mica plate (ε = 7) thickness d.\u003d 1 mm, serving a flat capacitor insulator if the potential difference between the plates of Con U = 300 V.
48. There are two layers of dielectric - a mica plate (ε 1 \u003d 7) with a thickness d 1 \u003d 1 mm and paraffin (ε 2 \u003d 2) with a thickness d 2 \u003d 0.5 mm. Determine: 1) the tensions of electrostatic fields in the dielectric layers; 2) electrical displacement if the potential difference between the condenser's plates U \u003d 500 V.
49. The distance between the plates of the flat condenser is D \u003d 1 cm, the potential difference U \u003d 200 V. Determine the surface density σ` of the associated charges of the ebonite plate (ε \u003d 3) placed on the bottom plate of the capacitor. Plate thickness d 2 \u003d 8 mm.
50. Free charges are evenly distributed with a bulk density ρ \u003d 5 NL / M 3 over a ball with a radius R \u003d 10 cm from a homogeneous isotropic dielectric with permeability ε \u003d 5. Determine the intensity of the electrostatic field at distances R 1 \u003d 5 cm and R 2 \u003d 15 cm From the center of the ball.
51. Distance between flat condenser plates d.\u003d 5 mm, potential difference U.\u003d 1.2 kV. Determine: 1) Surface charge density on capacitor plates; 2) Surface density of charges on a dielectric, if it is known that the dielectric perception of the dielectric, filling the space between the plates, x \u003d 1.
52. The space between the plates of the flat capacitor is filled with glass (ε \u003d 7). The distance between the plates d \u003d 5 mm, the difference of potentials u \u003d 1 square. Determine: 1) field strength in glass; 2) surface charge density on condenser plates; 3) Surface density of linked charges on glass.
53. Determine the distance between the plates of the flat condenser if the difference of potentials U \u003d 150 V is applied between them, and the area of \u200b\u200beach plate S \u003d 100 cm 2, its charge Q \u003d 10 ND. The dielectric serves as a mica (ε \u003d 7).
54. The difference between the potentials U 1 \u003d 500 V. Plates S \u003d 200 cm 2, the distance between them d \u003d 1.5 mm was applied to the plates of the flat air capacitor. After disconnecting the capacitor from the voltage source in space between the plates, paraffin (ε \u003d 2) was made. Determine the difference in potentials U 2 between plates after making dielectric. Also determine the capacitance of the capacitor C 1 and C 2 to and after making a dielectric.
55. The difference in the potentials U 1 \u003d 500 V. Plates s \u003d 200 cm 2, the distance between them D \u003d 1.5 mm was applied to the plates of the flat air capacitor. With the power supply turned on, paraffin (ε \u003d 2) was made in the space between the plates). Determine the difference in potentials U 2 between plates after making dielectric. Also determine the capacitance of the capacitor C 1 and C 2 to and after making a dielectric.
56. Determine the container of the coaxial cable with a length of 10 m, if the radius of its central core R 1 \u003d 1 cm, the radius of the shell R 2 \u003d 1.5 cm, and the insulating material serves as a rubber (ε \u003d 2.5).
57. Determine the intensity of the electrostatic field at a distance d \u003d 1 cm from the axis of the coaxial cable, if the radius of its central veins R 1 \u003d 0.5 cm, and the radius of the shell R 2 \u003d 1.5 cm. The potential difference between the central residential and the shell U \u003d 1 square
58. The spherical condenser consists of two concentric spheres R radius R 1 = 5 cm and R 2 \u003d 5.5 cm. The space between the condenser occupation is filled with oil (ε \u003d 2.2). Determine: 1) the capacity of this capacitor; 2) The ball of which radius placed in the oil has the same container.
59. Determine the intensity of the electrostatic field at a distance x \u003d 2 cm from the center of the air spherical capacitor formed by two balls (the inner radius R 1 \u003d cm, the outer - R 2 \u003d 3 cm), between which the difference of potentials u \u003d 1 kV is applied.
60. Two flat air capacitors of the same capacity are connected in parallel and charged to the potential difference. U \u003d.300 V. Determine the difference in the potentials of this system if the space between the plates of one of the capacitors is filled with mica (ε = 7).
Task 1. Conductor C 1 capacitor, charged to the difference in potentials U, was connected in parallel to the ends of the system of two successively connected uncharged capacitors, the capacles of which from 2 and C 3. Which charge will leak through the connecting wires?
Decision.Initially, the charge of the first capacitor was equal to Q \u003d C 1 U. After the connection, this charge was redistributed between the capacitors in such a way that the voltages on the first capacitor and the connected battery would be the same. We have:
q 1 + Q 2 \u003d Q, ,
where Q 1 is the charge on the first condenser after the connection, and the Q 2 is the charge on the connected battery. Solving these two equations, we find Q 1 and the woven chargeΔq \u003d Q - Q 1 =
Task 2. To the source with E.D.S. U was connected successively two air capacitor, each Capacity C. Then one of the capacitors were filled with a homogeneous dielectric with permeability ε. How many times did the electric field strength decreased in this condenser? Which charge will pass through the source?
Decision.Find at first the woven charge. The charge of the condenser before filling the dielectric is equal, and the charge after filling
Hence the flowing charge is Δq \u003d Q 2 - Q 1.
The field strength is first equal, where D is the distance between the plates. After the administration of the dielectric, it becomes equal
From here.
Answer: ,.
Task 3.The dielectric with dielectric constant ε fills the space between the plane condenser plates. The capacitance of the capacitor is C. The condenser is charged to the potential difference U and disconnected from the voltage source. Then the dielectric is slowly removed from the condenser. What work should be done at the same time?
Decision: Since the condenser is disconnected from the voltage source, the charge on its plates does not change. Energy stored by the capacitor is equal to
where C is a capacitance of a condenser with a dielectric. After the dielectric is removed, the capacitor capacitance decreases in ε times. Hence,
i.e., the energy stored by a capacitor will increase in ε times. To increase the energy, it is necessary to work to remove the dielectric, the value of which is:
The fact that to remove a dielectric should be done, clear from general considerations: there is an attraction between the charge induced on the dielectric and the charge of the plate, against the forces of which the external work is performed when the dielectric is removed from the condenser.
Task 4. The space between the plane condenser plates is filled with two dielectric layers 1 and 2 with the thicknesses D 1 and D 2 and with permeability of ε 1 and ε 2. The connection of each plane is S. Find: a) capacitance of the capacitor; b) density σ / related charges at the border Section of dielectric layers, if the voltage on the condenser is equal to U and the electric field is directed from the layer 1 to the layer 2.
Figure 3.15. To task 4.
Decision.Let the capacitor charge be q. (Fig.3.15). Then the electrical induction in it is equal to d \u003d q / s, and the electric field strengths are described by expressions:
The potential difference between the plates is equal \u003d E 1 D 1 + E 2 D 2. In turn, the capacitor capacitor C \u003d Q / U, so:
Polarized in layers will find with the help of the formula:
and the surface density of the associated charge, therefore
Answer: ,.
Task 5. Flat condenser, the area of \u200b\u200beach plate of which S.\u003d 400cm 2, filled with two dielectric layers. The border between them is parallel to the plates. The first layer - pressspan (ε 1 \u003d 2) thickness L 1 \u003d 0.2 cm; The second layer - glass (ε 2 \u003d 7) thickness L 2 \u003d 0.3cm. The capacitor is charged to the potential difference U \u003d 600 V. Find the energy of the condenser.
Decision: The energy of the condenser can be found by the formula :. We define a pre-electrical capacity, where Q \u003d σs is the capacitor charge.
Since in a flat capacitor within each dielectric, the field is uniformly, then U \u003d E 1 L 1 + E 2 L 2. Field voltage in each dielectric layer:
Then the electric capacity of the condenser
a condenser energy
Task 6. There is a flat air capacitor, the area of \u200b\u200beach plating of which is S. What work against the electric forces should be made to slowly increase the distance between the plates from x 1 to x 2, if it is maintained unchanged: a) the charge of the capacitor Q;
b) voltage on the condenser U?
Decision.a) Initially, the energy of the capacitor was equal. After increasing the distance, the energy is equal. The perfect work is equal to A \u003d W 2 - W 1,
b) if the voltage on the condenser is supported constant, then with an increase in the distance between the plates through the source proceeds
At the same time the battery makes negative work A 1 \u003d -ΔQU. Therefore, the energy balance in this case will be recorded as:
By deciding this equation, we will find the job A:
Conclusions:Electrical capacity - is an important characteristic Properties of conductors and capacitors characterizing the ability to accumulate charge.
Control questions second level (Collection of tasks)
1. Find electrical capacity from a secluded metal ball with radius R \u003d 1 cm.
2. To determine the electrical capacity from the metal sphere with a radius R \u003d 2 cm, immersed in water.
3. Determine the electrical capacity from the ground, taking it for the ball with a radius R \u003d 6400 km.
4. Two metal balls with radius R 1 \u003d 2 cm and R 2 \u003d 6 cm are connected by a conductor, the container of which can be neglected. The balls were reported Q \u003d 1 ND. Find the surface density of σ charges on the balls. [σ 1 \u003d 49.8 nkl / m 2; σ 2 \u003d 16.6 nkl / m 2]
5. The ball with a radius R 1 \u003d 6 cm is charged to the potential φ 1 \u003d 300 V, and the ball with a radius R 2 \u003d 4 cm - to the potential φ 2 \u003d 500 V. Determine the potential of the balls after they were connected to the metal conductor. Capacity connecting conductors neglected.
6. Two concentric metal spheres radius R 1 \u003d 2 cm and R 2 \u003d 2.1 cm form a spherical capacitor. Determine its electrical capacity if the space between the spheres is filled with paraffin.
7. The metal ball with a radius of 5 cm is surrounded by a ball layer of the dielectric (ε \u003d 7) with a thickness of 1 cm and is placed concentral in the metal sphere with an internal radius of 7 cm. What is the container of such a capacitor?
8. On one of the plates of a flat capacitor with a container with charge + Q, and on another charge + 4Q. Determine the potential difference between the capacitor plates.
9. Two identical flat air capacitors with a capacity C \u003d 100 PF each are connected to the battery sequentially. Determine how much the battery capacity changes if the space between the plates of one of the capacitors is filled with paraffin. [Will increase by 16.7 PF]
10. Between the plates of the flat capacitor, the area of \u200b\u200bwhich S is placed a layered dielectric consisting of n layers of a substance with a dielectric constant ε 1 and from n layers of substance with dielectric constant ε 2. The layers alternate and each has a thickness d. Find the capacitor capacitance. [ε 0 ε 1 ε 2 s / dn (ε 1 + ε 2)]
11. The space between the plates of the flat capacitor is filled with a dielectric, the dielectric permeability of which linearly changes from the value of ε 1 in one plate to the value of ε 2 ˂ε 1 in another. The distance between the plates D, the plates area is S. Find the capacity of such a capacitor. [ε 0 (ε 1 -ε 2) S / D Ln (ε 1 / ε 2)]
12. In the space between the plates of the flat capacitor, there is a homogeneous flow of electrons, which creates a uniform volumetric charge. The distance between the plates is d. The potential of one of the plates is φ 0. With what value of the volume density of the charge ρ potential and field strength in another plate are zero? [ρ \u003d -2ε 0 φ 0 / d 2]
13. Two capacitors with a capacity of 1 \u003d 5 μF and from 2 \u003d 8 μF are connected sequentially and attached to the battery with EDC 80 V. Determine the charges Q 1 and Q 2 capacitors and the difference in potentials U 1 and U 2 between their plates.
14. Two identical flat air capacitor are connected in series in the battery, which is connected to a current source with EDC 12 V. Determine how much the voltage on one of the capacitors will change if the other is immersed in transformer oil (ε \u003d 2.2).
15. Capacitor with a capacity of 1 \u003d 0.6 μF was charged to voltage U 1 \u003d 300 V and connected in parallel with the second capacitor with a capacity of 2 \u003d 0.4 μF, charged to the voltage U 2 \u003d 150 V. Find the value of the charge, flowed from Plates of the first condenser on the second.
16. Capacitor with a capacity C, \u003d 0.2 μF was charged to voltage U 1 \u003d 320 V. After it was connected in parallel with the second capacitor, charged to the voltage U 2 \u003d 450 B, the voltage on it changed to U \u003d 400 V . Calculate the container with 2 second capacitor.
17. The space between the plates of the plane capacitor is filled with two layers of the dielectric: glass thickness d 1 \u003d 0.2 cm and a layer of paraffin thick d 2 \u003d 0.3 cm. The potential difference between the plates U \u003d 300 V. Determine the intensity of the field and the drop in the potential in Each of the layers.
18. The capacitor with a capacity of 20 μF is charged to a voltage of 400 V. It is connected to it with a capacitor with a capacity of 1 μF, as a result of which the latter is charged. Then, turning off this capacitor, charge the second capacitor with the same tank (1 μF), the third, etc., then the capacitors are connected in series. What maximum voltage Can I get in this way?
19. A flat capacitor whose plates are located horizontally, half off with a liquid dielectric. What part of the kanalogic capacitor should be pouring the liquid during the vertical location of the plates so that the containers in both cases are the same? Dielectric constant fluid ε.
20. Four identical metal plates are located in the air at equal distances of the variety of each other. The area of \u200b\u200beach plate is equal to S. The extreme plates are interconnected, the medium plates are connected to the battery, the EDC of which is equal to. Find charges of medium plates. It is not possible to assume that the distance to the neighbor plates is not enough compared to their dimensions.
21. At an arranged horizontally uncharged flat capacitor, the lower plate is fixed, and the upper suspension is suspended to the skewer of the scales. Scales are in equilibrium, with a distance between the plates d \u003d 1 mm. Which mass of the weight should be put on the second cup of weights to keep the balance at the same distance between the plates, if the condenser is charged to the voltage u \u003d 1000 V? The area of \u200b\u200bthe plates of the condenser S \u003d 50 cm 2.
22. One condenser plate is fixed motionless, the second is suspended to the spring with the stiffness coefficient k. Plates S. on how much the spring is lengthened if the plates report are equal, but opposite by the charge sign Q? The field between the plates is considered homogeneous. [ΔL \u003d Q 2 / 2ε 0 ks]
23. One condenser plate is fixed motionless at the bottom of a wide vessel with a liquid dielectric (the dielectric permeability of its ε, density ρ). The second, having a view of a height of H, floats over it, immersed by 1/4 of its volume if the plates are not charged. What potential difference should be attached to the plates so that the upper plate immersed half? The initial distance between the plates of the capacitor H. The field between the plates is considered homogeneous.
24. Flat air capacitor with a plate S \u003d 5 cm 2 is connected to the battery, the EDC of which \u003d 300 V. Determine the operation of the external forces on the sliding of the plates from D 1 \u003d 1 mm to D 2 - 3 mm if the plate is turned off before expiding from the battery.
25. Flat air capacitor with an area of \u200b\u200bplate S \u003d 5 cm 2 is connected to the battery, the EDC of which \u003d 300 V. Determine the operation of the external forces on the sliding of the plates from D 1 \u003d - 1 mm to D 2 \u003d 3 mm if the plates in the exposure process remain connected to the battery. [-0.13 μJ]
26. Metal ball with a radius R \u003d 2 cm carries the charge Q \u003d 30 ND. The ball is surrounded by a layer of paraffin thick d \u003d 3 cm. Determine the energy of the electric field enclosed in the dielectric layer.
27. The flat condenser is located in an external homogeneous electric field with tension E, the direction of which coincides with the direction of the field in the condenser. The charges q and -q are uniformly distributed over the plates. What work should be done to turn the capacitor, changing the plate plates? Distance between plates d. Influence of gravity to neglect. [A \u003d 2QDE]
28. A large thin conductive plate, the area of \u200b\u200bwhich is S, and the thickness d, is placed in a homogeneous electric field with echo E, perpendicular to the plate. What amount of heat is highlighted in the conductor if the field instantly turn off?
29. Two flat capacitors with a capacity of judging, connected in parallel and charged to voltage U, disconnect from the source. Plates of one of the capacitors can move freely towards each other. Find their speed at the time when the gap between the capacitor plates decreases twice. The mass of each plate is equal to M. Heavyness to neglected.
30. Two metal balls with radius R 1 \u003d 5 cm and R 2 \u003d 10 cm have charges Q 1 \u003d 40 NNC and Q 2 \u003d -20 NNL, respectively. Find the energy that is distinguished by discharge if the balls are connected by the conductor.
Third-level control questions (tests)
1. Which of the expressions below is the determination of the capacitor's electrical capacity?
4. The energy of the electric field is determined by the expression:
6. Which of the expressions below is the determination of the electric field energy density?
| but); | b) r e \u003d; | c) R e \u003d; | d) R E \u003d. |
7. Determine the potential difference between the edges of the first capacitor, if the potential difference between the folds of the third capacitor is U.
1. U 2. 3U 3. U / 3 4. 0 5.
8. Determine the charge of the first capacitor if the third one is 3Q charge?
1. Q 2. 2Q 3. 3Q 4. 0 5. Q / 3
9. How does the capacitance change the capacitor if he has a dielectric with dielectric constant E?
1) will decrease in e times. 2) will increase in e times. 3) will remain the same.
4) will be equal to zero.
10. What is the capacitance of the capacitors depicted battery?
1) 0.5C 2) C 3) 2C 4) 1.5C 5) 2.5C
3.1. The space between the plates of the flat capacitor is filled with glass (E \u003d 7). The distance between the plates D \u003d 5 mm, the potential difference U \u003d 500 V. determine the energy of the polarized glass plate, if its area S \u003d 50 cm 2.
3.2. A flat air capacitor with a capacity C \u003d 10 PF is charged to the potential difference U \u003d 1 square. After disconnecting the capacitor from the voltage source, the distance between the capacitor plates was doubled. Determine: 1) the potential difference on the capacitor plates after their slides; 2) the work of external forces on the sliding of the plates.
3.3. The potential difference between the condenser plates U \u003d 200 V. The area of \u200b\u200beach plate S \u003d 100 cm 2, the distance between the plates d \u003d 1 mm, the space between them is filled with paraffin (E \u003d 2). Determine the strength of attraction of plates to each other.
3.4. The flat condenser with the size of the plates 25 * 25 cm 2 and the distance between them d 1 \u003d 0.5 mm is charged to the difference in the potentials U 1 \u003d 10 V and disconnected from the source. What will be the difference of potentials u 2 if the plates are pushing to the distance d 2 \u003d 5 mm?
3.5. A flat air capacitor with a container with a current source that supports the potential difference between the plates, equal to U. What kind of charge will be held through the source when filling such a dielectric with dielectric constant E? [(E-1) Cu]
3.6. How will the energy connected to the source of a constant voltage of a flat capacitor change in an increase in the distance between its plates 2 times and the administration between the dielectric plates with E \u003d 4?
[will increase by 2 times]
3.7. Flat air capacitor is charged to some potential difference. The capacitor was placed a dielectric plate filling the entire space between the plates. After that, for the restoration of the previous potential difference had to increase the plate charge three times. Determine the dielectric permeability e plate.
3.8. A plane-parallel plate made of solid dielectric with dielectric constant e so that air gaps remained between the plates of the flat air capacitor with the dielectric constant. How will the strength of attraction of plates change to each other? If the condenser is charged and disconnected from the current source? [Will not change]
3.9. The flat air capacitor is charged to the difference in potentials U and disconnected from the current source. Determine the difference in potentials if the distance between the condenser plays is increased in n times. [NU]
