Chapter 24. Capacitance and Dielectrics Lecture 1. Dr. Armen Kocharian

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1 Chapter 24 Capacitance and Dielectrics Lecture 1 Dr. Armen Kocharian

2 Capacitors Capacitors are devices that store electric charge Examples of where capacitors are used include: radio receivers filters in power supplies energy-storing devices in electronic flashes

3 Definition of Capacitance The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors C Q = ΔV The SI unit of capacitance is the farad (F)

4 Makeup of a Capacitor A capacitor consists of two conductors These conductors are called plates When the conductor is charged, the plates carry charges of equal magnitude and opposite directions A potential difference exists between the plates due to the charge

5 C=Q/V ab Constant +Q conductor a some random path conductor b -Q If Q doubles (triples, quadruples...), the field doubles (triples, quadruples...) Then V ab also doubles (triples, quadruples...) But C=Q/V ab remains the same

6 More About Capacitance Capacitance will always be a positive quantity The capacitance of a given capacitor is constant The capacitance is a measure of the capacitor s ability to store charge The farad is a large unit, typically you will see microfarads (μf) and picofarads (pf)

7 Parallel Plate Capacitor Each plate is connected to a terminal of the battery If the capacitor is initially uncharged, the battery establishes an electric field in the connecting wires

8 Parallel Plate Capacitor, cont This field applies a force on electrons in the wire just outside of the plates The force causes the electrons to move onto the negative plate This continues until equilibrium is achieved The plate, the wire and the terminal are all at the same potential At this point, there is no field present in the wire and the movement of the electrons ceases

9 Parallel Plate Capacitor, final The plate is now negatively charged A similar process occurs at the other plate, electrons moving away from the plate and leaving it positively charged In its final configuration, the potential difference across the capacitor plates is the same as that between the terminals of the battery

10 Capacitance Isolated Sphere Assume a spherical charged conductor Assume V = 0 at infinity C Q Q R = = = = ΔV k Q/ R k e 4πε R Note, this is independent of the charge and the potential difference e o

11 Capacitance Parallel Plates The charge density on the plates is σ = Q/A A is the area of each plate, which are equal Q is the charge on each plate, equal with opposite signs The electric field is uniform between the plates and zero elsewhere

12 Capacitance Parallel Plates, cont. The capacitance is proportional to the area of its plates and inversely proportional to the distance between the plates C Q Q Q εoa = = = = ΔV Ed Qd/ ε A d o

13 Parallel Plate Assumptions The assumption that the electric field is uniform is valid in the central region, but not at the ends of the plates If the separation between the plates is small compared with the length of the plates, the effect of the non-uniform field can be ignored

14 Energy in a Capacitor Overview Consider the circuit to be a system Before the switch is closed, the energy is stored as chemical energy in the battery When the switch is closed, the energy is transformed from chemical to electric potential energy

15 Energy in a Capacitor Overview, cont The electric potential energy is related to the separation of the positive and negative charges on the plates A capacitor can be described as a device that stores energy as well as charge

16 Capacitance of a Cylindrical Capacitor, From Gauss s Law, the field between the cylinders is E = 2k e λ / r ΔV = -2k e λ ln (b/a) The capacitance becomes C Q = = ΔV 2k ln b/ a e ( )

17 Capacitance of a Spherical Capacitor b dr V = = b Va Edr r kq e = kq = 2 e kq e r r a a a b a The potential difference will be 1 1 Δ V = keq b a The capacitance will be Q 1 ab C = = = ΔV 1 1 ke b a ke a b b ( ) b

18 Circuit Symbols A circuit diagram is a simplified representation of an actual circuit Circuit symbols are used to represent the various elements Lines are used to represent wires The battery s positive terminal is indicated by the longer line

19 Connecting capacitors together Two ways of connecting capacitors together: V V a V b a V b in parallel in series

20 Capacitors in Parallel When capacitors are first connected in the circuit, electrons are transferred from the left plates through the battery to the right plate, leaving the left plate positively charged and the right plate negatively charged

21 Capacitors in Parallel, 2 The flow of charges ceases when the voltage across the capacitors equals that of the battery The capacitors reach their maximum charge when the flow of charge ceases The total charge is equal to the sum of the charges on the capacitors Q total = Q 1 + Q 2 The potential difference across the capacitors is the same And each is equal to the voltage of the battery

22 Capacitors in Parallel, 3 The capacitors can be replaced with one capacitor with a capacitance of C eq The equivalent capacitor must have exactly the same external effect on the circuit as the original capacitors

23 Capacitors in Parallel, final C eq = C 1 + C 2 + The equivalent capacitance of a parallel combination of capacitors is greater than any of the individual capacitors Essentially, the areas are combined

24 Equivalent Capacitance, Example The 1.0-μF and 3.0-μF capacitors are in parallel as are the 6.0-μF and 2.0-μF capacitors These parallel combinations are in series with the capacitors next to them The series combinations are in parallel and the final equivalent capacitance can be found

25 Capacitors in parallel The potential difference across the two capacitors is the same Q 1 = C 1 V ab and Q 2 = C 2 V ab Therefore, Q=Q 1 +Q 2 = (C 1 + C 2 ) V ab This is equivalent to equivalent capacitance

26 Capacitors in Series When a battery is connected to the circuit, electrons are transferred from the left plate of C 1 to the right plate of C 2 through the battery

27 Capacitors in Series, 2 As this negative charge accumulates on the right plate of C 2, an equivalent amount of negative charge is removed from the left plate of C 2, leaving it with an excess positive charge All of the right plates gain charges of Q and all the left plates have charges of +Q

28 Capacitors in Series, 3 An equivalent capacitor can be found that performs the same function as the series combination The potential differences add up to the battery voltage

29 Capacitors in Series, final Q = Q 1 + Q 2 + ΔV = V 1 + V = + + C C C eq 1 2 The equivalent capacitance of a series combination is always less than any individual capacitor in the combination

30 Capacitors in series These two plates are connected The two connected plates effectively form a single conductor Thus, the two connected plates have equal and opposite charge

31 Capacitors in series (cont.) V a Q -Q Q -Q V b Remember, definition: Thus, this is entirely equivalent to V a Q -Q V b C eq equivalent capacitance

32 For more than two capacitors in parallel or in serees the results generalize to

33 Problem-Solving Hints Be careful with the choice of units In SI, capacitance is in farads, distance is in meters and the potential differences are in volts Electric fields can be in V/m or N/C When two or more capacitors are connected in parallel, the potential differences across them are the same The charge on each capacitor is proportional to its capacitance The capacitors add directly to give the equivalent capacitance

34 Problem-Solving Hints, cont When two or more capacitors are connected in series, they carry the same charge, but the potential differences across them are not the same The capacitances add as reciprocals and the equivalent capacitance is always less than the smallest individual capacitor

35 Energy Stored in a Capacitor Assume the capacitor is being charged and, at some point, has a charge q on it The work needed to transfer a charge from one plate to the other is q dw =Δ Vdq = dq C The total work required is W Q 2 C Q q dq 0 C 2 = =

36 Energy, cont The work done in charging the capacitor appears as electric potential energy U: 2 Q 1 1 ( ) 2 U = = QΔ V = C ΔV 2C 2 2 This applies to a capacitor of any geometry The energy stored increases as the charge increases and as the potential difference increases In practice, there is a maximum voltage before discharge occurs between the plates

37 Energy, final The energy can be considered to be stored in the electric field For a parallel-plate capacitor, the energy can be expressed in terms of the field as U = ½ (ε o Ad)E 2 It can also be expressed in terms of the energy density (energy per unit volume) u E = ½ ε o E 2

38 Example C 1 C 2 Find the equivalent capacitance of this network. C 3 The trick here is to take it one step at a time C 1 and C 3 are in series. So this circuit is equivalent to C 4 C 3 Then, this is equivalent to C eq

39 C 3 C 4 C 1 C 2 Another example Find the equivalent capacitance of this network. Again, take it in steps. C 1 and C 2 are in series. So this is equivalent to C 3 C 4 C 5

40 C 3 C 4 C 5 Now this looks a little different than what we have seen. But it is just three capacitors in parallel. We can redraw it as C 3 C 4 C 5 which is equivalent to C eq

41 Energy stored in a capacitor A capacitor stores potential energy By conservation of energy, the stored energy is equal to the work done in charging up the capacitor Our goal now is to calculate this work, and thus the amount of energy stored in the capacitor

42 Once the capacitor is charged Let q and v be the charge and potential of the capacitor at some instant while it is being charged q<q and v<v, but still v=q/c If we want to increase the charge from q q+dq, we need to do an amount of work dw The total work done in charging up the capacitor is Potential energy stored in the capacitor is

43 Energy in the electric field If a capacitor is charged, there is an electric field between the two conductors We can think of the energy of the capacitor as being stored in the electric field For a parallel plate capacitor, ignoring edge effects, the volume over which the field is active is Axd

44 Then, the energy per unit volume (energy density) is But the capacitance and electric field are given by Putting it all together: This is the energy density (energy per unit volume) associated with an electric field Derived it for parallel plate capacitor, but valid in general

45 Problem Capacitors C 1 = 6 μf, C 2 = 3 μf and ΔV= 20 V are given. Capacitor C 1 are first charged by closing switch S 1. Switch S 1 then is opened and the charged capacitor is connected to uncharged capacitor C 2 by closing switch S 2 (C 1 >C 2 ) Find the initial charge acquired by C 1 and the final charge on each capacitor.

46 Example Q 1i = Q initial charge of C 1 = C 1 V Q 1f = final charge of C 1 Q 2f = final charge of C 2 Charge Q total = Q 1i + Q 2i C Q = Δ V Q = = 120 μc Q 20.0 After we close the switches, this charge Will distribute itself partially on C 1 and partiallyon C 2, but with Q total = Q 1f + Q 2f Q1+ Q2 = Q Δ V Q C 1 = ΔV Q = C Q = 120 μc Q 1 2 Δ V = Q C 120 Q Q = C C ( 3.00)( 120 Q ) = ( 6.00) Q 1 2 Q = = 40.0 μc 9.00 Q = 120 μ C 40.0 μ C = 80.0 μ C

47 Example C 1 and C 2 (C 1 >C 2 ) are both charged to potential V, but with opposite polarity. They are removed from the battery, and are connected as shown. Then we close the two switches Find V ab after the switches have been closed Q 1i = initial charge of C 1 = C 1 V Q 2i = initial charge of C 2 = - C 2 V Charge Q total = Q 1i + Q 2i = (C 1 -C 2 )V - + After we close the switches, this charge will distribute itself partially on C 1 and partially on C 2, but with Q total = Q 1f + Q 2f

48 +Q 1f -Q 1f Q total = Q 1i + Q 2i = (C 1 -C 2 )V=Q 1f + Q 2f +Q 2f -Q 2f Q 1f = C 1 V ab Q 2f = C 2 V ab Q 1f + Q 2f = (C 1 + C 2 ) V ab Then, equating the two boxed equations

49 Now calculate the energy before and after E before = ½ C 1 V 2 + ½ C 2 V 2 = ½ (C 1 + C 2 ) V 2 E after = ½ C eq V ab, where C eq is the equivalent capacitance of the circuit after the switches have been closed C 1 and C 2 are in parallel C eq = C 1 + C 2 E after = ½ (C 1 + C 2 ) V ab What happens to conservation of energy???? It turns out that some of the energy is radiated as electromagnetic waves!!

50 Some Uses of Capacitors Defibrillators When fibrillation occurs, the heart produces a rapid, irregular pattern of beats A fast discharge of electrical energy through the heart can return the organ to its normal beat pattern In general, capacitors act as energy reservoirs that can be slowly charged and then discharged quickly to provide large amounts of energy in a short pulse

51 Capacitors with Dielectrics A dielectric is a nonconducting material that, when placed between the plates of a capacitor, increases the capacitance Dielectrics include rubber, plastic, and waxed paper For a parallel-plate capacitor, C = κc o = κε o (A/d) The capacitance is multiplied by the factor κ when the dielectric completely fills the region between the plates

52 Dielectrics, cont In theory, d could be made very small to create a very large capacitance In practice, there is a limit to d d is limited by the electric discharge that could occur though the dielectric medium separating the plates For a given d, the maximum voltage that can be applied to a capacitor without causing a discharge depends on the dielectric strength of the material

53 Dielectrics, final Dielectrics provide the following advantages: Increase in capacitance Increase the maximum operating voltage Possible mechanical support between the plates This allows the plates to be close together without touching This decreases d and increases C

54

55 Types of Capacitors Tubular Metallic foil may be interlaced with thin sheets of paper or Mylar The layers are rolled into a cylinder to form a small package for the capacitor

56 Types of Capacitors Oil Filled Common for highvoltage capacitors A number of interwoven metallic plates are immersed in silicon oil

57 Types of Capacitors Electrolytic Used to store large amounts of charge at relatively low voltages The electrolyte is a solution that conducts electricity by virtue of motion of ions contained in the solution

58 Types of Capacitors Variable Variable capacitors consist of two interwoven sets of metallic plates One plate is fixed and the other is movable These capacitors generally vary between 10 and 500 pf Used in radio tuning circuits

59 Capacitor types Capacitors are often classified by the materials used between electrodes Some types are air, paper, plastic film, mica, ceramic, electrolyte, and tantalum Often you can tell them apart by the packaging Plastic Film Capacitor Ceramic Capacitor Tantalum Capacitor Electrolyte Capacitor

60 Electric Dipole An electric dipole consists of two charges of equal magnitude and opposite signs The charges are separated by 2a The electric dipole moment (p) is directed along the line joining the charges from q to +q

61 Electric Dipole, 2 The electric dipole moment has a magnitude of p = 2aq Assume the dipole is placed in a uniform external field, E E is external to the dipole; it is not the field produced by the dipole Assume the dipole makes an angle θ with the field

62 Electric Dipole, 3 Each charge has a force of F = Eq acting on it The net force on the dipole is zero The forces produce a net torque on the dipole

63 Electric Dipole, final The magnitude of the torque is: τ = 2Fa sin θ = pe sin θ The torque can also be expressed as the cross product of the moment and the field: τ = p x E The potential energy can be expressed as a function of the orientation of the dipole with the field: U f U i = pe(cos θ i cosθ f ) U = - pe cos θ = - p E

64 Polar vs. Nonpolar Molecules Molecules are said to be polarized when a separation exists between the average position of the negative charges and the average position of the positive charges Polar molecules are those in which this condition is always present Molecules without a permanent polarization are called nonpolar molecules

65 Water Molecules A water molecule is an example of a polar molecule The center of the negative charge is near the center of the oxygen atom The x is the center of the positive charge distribution

66 Polar Molecules and Dipoles The average positions of the positive and negative charges act as point charges Therefore, polar molecules can be modeled as electric dipoles

67 Induced Polarization A symmetrical molecule has no permanent polarization (a) Polarization can be induced by placing the molecule in an electric field (b) Induced polarization is the effect that predominates in most materials used as dielectrics in capacitors

68 Dielectrics An Atomic View The molecules that make up the dielectric are modeled as dipoles The molecules are randomly oriented in the absence of an electric field

69 Dielectrics An Atomic View, 2 An external electric field is applied This produces a torque on the molecules The molecules partially align with the electric field

70 Dielectrics An Atomic View, 3 The degree of alignment of the molecules with the field depends on temperature and the magnitude of the field In general, the alignment increases with decreasing temperature the alignment increases with increasing field strength

71 Dielectrics An Atomic View, 4 If the molecules of the dielectric are nonpolar molecules, the electric field produces some charge separation This produces an induced dipole moment The effect is then the same as if the molecules were polar

72 Dielectrics An Atomic View, final An external field can polarize the dielectric whether the molecules are polar or nonpolar The charged edges of the dielectric act as a second pair of plates producing an induced electric field in the direction opposite the original electric field

73 Induced Charge and Field The electric field due to the plates is directed to the right and it polarizes the dielectric The net effect on the dielectric is an induced surface charge that results in an induced electric field If the dielectric were replaced with a conductor, the net field between the plates would be zero

74 Geometry of Some Capacitors

75 Chapter 26 Capacitance and Dielectrics

76 Quick Quiz 26.1 A capacitor stores charge Q at a potential difference ΔV. If the voltage applied by a battery to the capacitor is doubled to 2ΔV: (a) the capacitance falls to half its initial value and the charge remains the same (b) the capacitance and the charge both fall to half their initial values (c) the capacitance and the charge both double (d) the capacitance remains the same and the charge doubles

77 Quick Quiz 26.1 Answer: (d). The capacitance is a property of the physical system and does not vary with applied voltage. According to Equation 26.1, if the voltage is doubled, the charge is doubled.

78 Quick Quiz 26.2 Many computer keyboard buttons are constructed of capacitors, as shown in the figure below. When a key is pushed down, the soft insulator between the movable plate and the fixed plate is compressed. When the key is pressed, the capacitance (a) increases (b) decreases (c) changes in a way that we cannot determine because the complicated electric circuit connected to the keyboard button may cause a change in ΔV.

79 Quick Quiz 26.2 Answer: (a). When the key is pressed, the plate separation is decreased and the capacitance increases. Capacitance depends only on how a capacitor is constructed and not on the external circuit.

80 Quick Quiz 26.3 Two capacitors are identical. They can be connected in series or in parallel. If you want the smallest equivalent capacitance for the combination, you should connect them in (a) series (b) parallel (c) Either combination has the same capacitance.

81 Quick Quiz 26.3 Answer: (a). When connecting capacitors in series, the inverses of the capacitances add, resulting in a smaller overall equivalent capacitance.

82 Quick Quiz 26.4 Consider the two capacitors in question 3 again. Each capacitor is charged to a voltage of 10 V. If you want the largest combined potential difference across the combination, you should connect them in (a) series (b) parallel (c) Either combination has the same potential difference.

83 Quick Quiz 26.4 Answer: (a). When capacitors are connected in series, the voltages add, for a total of 20 V in this case. If they are combined in parallel, the voltage across the combination is still 10 V.

84 Quick Quiz 26.5 You have three capacitors and a battery. In which of the following combinations of the three capacitors will the maximum possible energy be stored when the combination is attached to the battery? (a) series (b) parallel (c) Both combinations will store the same amount of energy.

85 Quick Quiz 26.5 Answer: (b). For a given voltage, the energy stored in a capacitor is proportional to C: U = C(ΔV) 2 /2. Thus, you want to maximize the equivalent capacitance. You do this by connecting the three capacitors in parallel, so that the capacitances add.

86 Quick Quiz 26.6 You charge a parallel-plate capacitor, remove it from the battery, and prevent the wires connected to the plates from touching each other. When you pull the plates apart to a larger separation, do the following quantities increase, decrease, or stay the same? (a) C; (b) Q; (c) E between the plates; (d) ΔV ; (e) energy stored in the capacitor.

87 Quick Quiz 26.6 Answer: (a) C decreases (Eq. 26.3). (b) Q stays the same because there is no place for the charge to flow. (c) E remains constant (see Eq and the paragraph following it). (d) ΔV increases because ΔV = Q/C, Q is constant (part b), and C decreases (part a). (e) The energy stored in the capacitor is proportional to both Q and ΔV (Eq ) and thus increases. The additional energy comes from the work you do in pulling the two plates apart.

88 Quick Quiz 26.7 Repeat Quick Quiz 26.6, but this time answer the questions for the situation in which the battery remains connected to the capacitor while you pull the plates apart.

89 Quick Quiz 26.7 Answer: (a) C decreases (Eq. 26.3). (b) Q decreases. The battery supplies a constant potential difference ΔV; thus, charge must flow out of the capacitor if C = Q /ΔV is to decrease. (c) E decreases because the charge density on the plates decreases. (d) ΔV remains constant because of the presence of the battery. (e) The energy stored in the capacitor decreases (Eq ).

90 Quick Quiz 26.8 If you have ever tried to hang a picture or a mirror, you know it can be difficult to locate a wooden stud in which to anchor your nail or screw. A carpenter s stud-finder is basically a capacitor with its plates arranged side by side instead of facing one another, as shown in the figure below. When the device is moved over a stud, the capacitance will: (a) increase (b) decrease

91 Quick Quiz 26.8 Answer: (a). The dielectric constant of wood (and of all other insulating materials, for that matter) is greater than 1; therefore, the capacitance increases (Eq ). This increase is sensed by the stud-finder's special circuitry, which causes an indicator on the device to light up.

92 Quick Quiz 26.9 A fully charged parallel-plate capacitor remains connected to a battery while you slide a dielectric between the plates. Do the following quantities increase, decrease, or stay the same? (a) C; (b) Q; (c) E between the plates; (d) ΔV.

93 Quick Quiz 26.9 Answer: (a) C increases (Eq ). (b) Q increases. Because the battery maintains a constant ΔV, Q must increase if C increases. (c) E between the plates remains constant because ΔV = Ed and neither ΔV nor d changes. The electric field due to the charges on the plates increases because more charge has flowed onto the plates. The induced surface charges on the dielectric create a field that opposes the increase in the field caused by the greater number of charges on the plates (see Section 26.7). (d) The battery maintains a constant ΔV.

94 The positive charge is the end view of a positively charged glass rod. A negatively charged particle moves in a circular arc around the glass rod. Is the work done on the charged particle by the rod s electric field positive, negative or zero? 1. Positive 2. Negative 3. Zero

95 The positive charge is the end view of a positively charged glass rod. A negatively charged particle moves in a circular arc around the glass rod. Is the work done on the charged particle by the rod s electric field positive, negative or zero? 1. Positive 2. Negative 3. Zero

96 Rank in order, from largest to smallest, the potential energies U a to U d of these four pairs of charges. Each + symbol represents the same amount of charge. 1. U a = U b > U c = U d 2. U a = U c > U b = U d 3. U b = U d > U a = U c 4. U d > U b = U c > U a 5. U d > U c > U b > U a

97 Rank in order, from largest to smallest, the potential energies U a to U d of these four pairs of charges. Each + symbol represents the same amount of charge. 1. U a = U b > U c = U d 2. U a = U c > U b = U d 3. U b = U d > U a = U c 4. U d > U b = U c > U a 5. U d > U c > U b > U a

98 A proton is released from rest at point B, where the potential is 0 V. Afterward, the proton 1. moves toward A with an increasing speed. 2. moves toward A with a steady speed. 3. remains at rest at B. 4. moves toward C with a steady speed. 5. moves toward C with an increasing speed.

99 A proton is released from rest at point B, where the potential is 0 V. Afterward, the proton 1. moves toward A with an increasing speed. 2. moves toward A with a steady speed. 3. remains at rest at B. 4. moves toward C with a steady speed. 5. moves toward C with an increasing speed.

100 Rank in order, from largest to smallest, the potentials V a to V e at the points a to e. 1. V a = V b = V c = V d = V e 2. V a = V b > V c > V d = V e 3. V d = V e > V c > V a = V b 4. V b = V c = V e > V a = V d 5. V a = V b = V d = V e > V c

101 Rank in order, from largest to smallest, the potentials V a to V e at the points a to e. 1. V a = V b = V c = V d = V e 2. V a = V b > V c > V d = V e 3. V d = V e > V c > V a = V b 4. V b = V c = V e > V a = V d 5. V a = V b = V d = V e > V c

102 Rank in order, from largest to smallest, the potential differences V 12, V 13, and V 23 between points 1 and 2, points 1 and 3, and points 2 and V 12 > V 13 = V V 13 > V 12 > V V 13 > V 23 > V V 13 = V 23 > V V 23 > V 12 > V 13

103 Rank in order, from largest to smallest, the potential differences V 12, V 13, and V 23 between points 1 and 2, points 1 and 3, and points 2 and V 12 > V 13 = V V 13 > V 12 > V V 13 > V 23 > V V 13 = V 23 > V V 23 > V 12 > V 13

104 Chapter 29 Reading Quiz

105 What are the units of potential difference? 1. Amperes 2. Potentiometers 3. Farads 4. Volts 5. Henrys

106 What are the units of potential difference? 1. Amperes 2. Potentiometers 3. Farads 4. Volts 5. Henrys

107 New units of the electric field were introduced in this chapter. They are: 1. V/C. 2. N/C. 3. V/m. 4. J/m W/m.

108 New units of the electric field were introduced in this chapter. They are: 1. V/C. 2. N/C. 3. V/m. 4. J/m W/m.

109 The electric potential inside a capacitor 1. is constant. 2. increases linearly from the negative to the positive plate. 3. decreases linearly from the negative to the positive plate. 4. decreases inversely with distance from the negative plate. 5. decreases inversely with the square of the distance from the negative plate.

110 The electric potential inside a capacitor 1. is constant. 2. increases linearly from the negative to the positive plate. 3. decreases linearly from the negative to the positive plate. 4. decreases inversely with distance from the negative plate. 5. decreases inversely with the square of the distance from the negative plate.

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