Capacitors and Capacitance

Transcription

1 [International campus Lab] Objective Investigate the force between the charged plates of a parallel plate capacitor and determine the capacitance of the capacitor. Determine the effect of a dielectric between the plates of a parallel plate capacitor. Theory Reference Young & Freedman, University Physics (14 th ed.), Pearson, (p.10~14) 24.4 Dielectrics (p.21~2) 24.2 Capacitors in Series and Parallel (p.14~17) 22.3 Gauss s Law (p.753~757) 22.4 Applications of Gauss s Law Ex.22. (p.70~71) Any two conductors separated by an insulator (or a vacuum) form a capacitor (Fig.1). Each conductor initially has zero net charge and electrons are transferred from one conductor to the other; this is called charging the capacitor. Then the two conductors have charges with equal magnitude and opposite sign. When we say that a charge QQ is stored on the capacitor, we mean that the conductor at higher potential has charge +QQ and the conductor at lower potential has charge QQ. 1. A capacitor is a device that stores electric potential energy and electric charge. Capacitors have a tremendous number of practical applications in devices such as electronic flash units for photography, microphones, and radio receivers. The electric field at any point in the region between the conductors is proportional to the magnitude QQ of charge on each conductor. It follows that the potential difference VV between the conductors is also proportional to QQ. If we double the magnitude of charge on each conductor, the charge density at each point doubles, the electric field at each point doubles, and the potential difference between conductors doubles; however, the ratio of charge to potential difference does not change. This ratio is called the capacitance CC of the capacitor: CC = QQ VV (1) Fig 1 Any two conductors aa and bb insulated from each other form a capacitor. The greater the capacitance CC of a capacitor, the greater the magnitude QQ of charge on either conductor for a given potential difference VV and hence the greater the amount of 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page 1 / 13

2 stored energy. Thus capacitance is a measure of the ability of a capacitor to store energy. We will see that the value of the capacitance depends only on the shapes and sizes of the conductors and on the nature of the insulating material between them. The SI unit of capacitance is called farad (F). 1 F = 1 farad = 1 C/V = 1 coulomb/volt The simplest form of capacitor consists of two parallel conducting plates, each with area AA, separated by a distance that is small in comparison with their dimension (Fig. 2). We call this arrangement a parallel-plate capacitor. When the plates are charged, because opposite charges attract, most of the charge accumulates at the opposing faces of the plates and the electric field is almost completely localized in the region between the plates. A small amount of charge resides on the outer surfaces of the plates, and there is some fringing of the field at the edges (Fig. 3a). But if the plates are very large in comparison to the distance between them, the amount of charge on the outer surface is negligibly small, and the fringing can be neglected except near the edges. In this case we can assume that the field is uniform in the interior region between the plates, as shown in Fig. 3b, and the charges are distributed uniformly over the opposing surfaces. If the surface charge densities of each plate are +σσ and σσ, we can find the electric field in the region between the plates using following equation known as Gauss s law. Φ EE = EE = EE AA = QQ encl εε 0 (2) We consider a cylindrical Gaussian surface SS 1 with flat ends of area AA (Fig. 3b). The upper end of surface is within the positive plate. Since the field is zero within the volume of any solid conductor under electrostatic conditions, there is no electric flux through this end. The electric field between the plates is perpendicular to the lower end, so on that end, EE is equal to EE and the flux is EEAA ; this is positive, since EE is directed out of the Gaussian surface. There is no flux through the side walls of the cylinder, since these walls are parallel to EE. So the total flux integral in Gauss s law is EEAA. The net charge enclosed by the cylinder is σσaa, so equation (2) yields EEAA = σσaa εε 0 ; we then have EE = σσ εε 0 (3) The Gaussian surface SS 4 yields the same result. Surfaces SS 2 and SS 3 yield EE = 0. The magnitude of surface charge density σσ is equal to the magnitude of the total charge QQ on each plate divided by the area AA of the plate, or σσ = QQ/AA, so equation (3) can be expressed as Fig 2 A charged parallel-plate capacitor. EE = σσ εε 0 = QQ εε 0 AA (4) The field is uniform and the distance between the plates is, so the potential difference between the two plates is VV = EEEE = 1 εε 0 QQQQ AA (5) From equations (1) and (5), we see that the capacitance CC of a parallel-plate capacitor in vacuum is Fig 3 Electric field between oppositely charged parallel plates. CC = QQ VV = εε AA 0 () 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page 2 / 13

3 The capacitance depends only on the geometry of the capacitor; it is directly proportional to the area AA of each plate and inversely proportional to their separation. εε 0 is a universal constant. εε 0 = C 2 /N m (oooo F/m) If the fields due to the charges on each plate are EE aa and EE bb in Fig. 3b, the field between the plates is EE = EE aa + EE bb and EE aa = EE bb since the plates have the same magnitude but opposite sign of total charge QQ. So the field with the charge QQ on the upper plate is EE aa = (1 2)EE = σσ 2εε 0 = QQ 2εε 0 AA and the magnitude of the electrostatic force exerted on the charge QQ on the lower plate by the upper plate is given by The original capacitance CC 0 is given by CC 0 = QQ VV 0, and the capacitance CC with the dielectric present is CC = QQ VV. The charge QQ is same in both cases, and VV is less than VV 0, so we conclude that the capacitance CC with the dielectric present is greater than CC 0. When the space between plates is completely filled by the electric, the ratio of CC to CC 0 is called the dielectric constant KK of the material: KK = CC CC 0 (9) The dielectric constant KK is a pure number. Because CC is always greater than CC 0, KK is always greater than unity. For air at ordinary temperatures and pressures, KK = FF = ( QQ)EE aa = QQ2 2εε 0 AA == εε 0AAVV (7) The negative sign of this equation means that the attractive force is exerted on the lower plate. From equations (5), () and (7), we can express the capacitance CC of a capacitor as a function of the force FF exerted on the lower plate. CC = 2FFFF VV 2 () Material KK Material KK Vacuum 1 Mica 3~ Air (1 atm) Glass 5~10 Air (100 atm) Germanium 1 Teflon 2.1 Glycerin 42.5 Polyethylene 2.2 Water 0.4 When the charge is constant, QQ = CC 0 VV 0 = CCCC and CC CC 0 = VV. In this case, equation (9) can be rewritten as VV 0 VV = VV 0 KK (10) 2. Dielectrics Most capacitors have a dielectric between their conducting plates. Placing a solid dielectric between the plates of a capacitor serves several functions, one of which is that the capacitance of a capacitor of given dimensions is greater when there is a dielectric material between the plates than when there is vacuum. When a dielectric material is inserted between the plates while the charge is kept constant, the potential difference between the plates decrease by a factor KK. Therefore the field between the plates must decrease by the same factor. EE = EE 0 KK (11) Consider a charged capacitor with magnitude of charge QQ on each plate and potential difference VV 0. When we insert an uncharged sheet of dielectric between the plates, the potential difference decreases to a smaller value VV. When we remove the dielectric, the potential difference returns to its original value VV 0. Fig 4 A common type of capacitor uses dielectric sheet to separate the conductors. 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page 3 / 13

4 Since the electric-field magnitude is smaller when the dielectric is present, the surface charge density must be smaller as well. The surface charge on the conducting plates does not change, but an induced charge of the opposite sign appears on each surface of the dielectric (Fig. 5). Using these expressions in equation (11) and rearranging the result, we find σσ ii = σσ 1 1 (13) KK The dielectric was originally electrically neutral and is still neutral; the induced surface charges arise as a result of redistribution of positive and negative charge within the dielectric material, a phenomenon called polarization. The induced surface charge is directly proportional to the electric-field magnitude EE in the material for many common dielectrics except special cases, so KK is a constant. We can derive a relationship between this induced surface charge and the charge on the plates. Let s denote the magnitude of the charge per unit area induced on the surfaces of the dielectric by σσ ii. The magnitude of the surface charge density on the capacitor plates is σσ. Then the net surface charge on each side of the capacitor has magnitude (σσ σσ ii ). Now we can use equation (4) to express the field without and with dielectric respectively, then we have This equation shows that when KK is very large, σσ ii is nearly as large as σσ, in this case, σσ ii nearly cancel σσ, and the field and potential difference are much smaller than their value in vacuum, in other words, a capacitor can stores more electric charges for a given VV when there is a dielectric material between the plates than when there is vacuum. The product KKεε 0 is called the permittivity of the dielectric, denoted by εε: εε = KKεε 0 (14) In terms of εε we can express the electric field within the dielectric as EE = σσ (15) εε EE 0 = σσ εε 0 EE = σσ σσ ii εε 0 (12) The capacitance when the dielectric is present is given by CC = KKCC 0 = KKεε 0 AA = εε AA (1) In empty space, where KK = 1, εε = εε 0, equation (1) reduce to equation (4) for a parallel-plate capacitor in vacuum. For this reason, εε 0 is sometimes called the permittivity of vacuum. Equation (1) shows that extremely high capacitances CC can be obtained with plates that have a large surface area AA and are separated by a small distance by a dielectric with a large value of KK. Fig 5 (a) Electric field of magnitude EE 0 with vacuum between charged plates. (b) Introduction of a dielectric of dielectric constant KK. (c) The induced surface charges and their field. (d) Resultant field of magnitude EE 0 /KK. For a given charge density σσ, the induced charges on the dielectric s surfaces reduce the electric field between the plates. 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page 4 / 13

5 Equipment 1. List Item(s) Qty. Description Capacitor Apparatus 1 Constructs variable parallel plate capacitors. High Voltage Power Supply (Power cord included) 1 Produces high voltage up to 25kV. Patch Cords (High Voltage) (with safety shrouded banana plugs) 2 Carry high voltage power. Capacitor Plates Set 1 set Constructs parallel plate capacitors. Plate 1: rr = 53 mm Plate 2: rr = 75 mm Dielectrics 1 set Increase the capacitance a capacitor. Acryl: rr = 75 mm, = 3 mm, KK = 2.5 Glass: rr = 75 mm, = 3 mm, KK = 5. Electronic Balance (DC adaptor included) 1 Measures mass of an object with a precision to 0.01 g. Discharger 1 Releases stored electric charge from capacitor plates. Bubble Level 1 Checks the level of a surface. 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page 5 / 13

6 2. Details (1) Capacitor Apparatus (2) High voltage Power Supply The Apparatus forms a parallel plate capacitor using various sizes of plate electrodes. The micrometer mechanism allows the spacing between the plates to be adjusted in small increments, thus permitting the system to be tuned. The spindle of the micrometer has one thread per millimeter, and thus one completer revolution moves the spindle through a distance 1 mm. The thimble has 100 graduations, each being 0.01 mm. Thus, turning the thimble through one division (1/100 turn) moves the spindle axially 0.01 mm. The high voltage power supply provides very high voltage up to 25,000 V. The current available from the power supply is too low to cause any permanent damage. However, the voltage is high enough to cause a distinctly unpleasant sensation. Do not touch any connectors or connected conductors while the power supply is turned on. (3) Electronic Balance 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page / 13

7 Procedure (2) Place the electronic balance on the platform. To prevent electric shock, keep these instructions in mind. Although the current available from the power supply is too low to cause any permanent damage, the voltage on the capacitor plates is high enough to cause a distinctly unpleasant sensation if you touch them. 1. Do not touch any plates or connectors while the power supply is turned on. 2. Prior to touching the plates or cables, you should be sure to discharge the charges stored in the apparatus by turning off the grounded power supply. 3. After turning off the power supply, you should always double check if the charges remain on the plates. Use the discharger to release any residual charge on the plates by touching them at the same time. (3) Mount the capacitor plates (rr = 53mm) in the apparatus. 1 Lift up the moving arm for easy installation of upper plate. 2 Set the micrometer about 10 mm position. 3 Insert the upper plate into the holder of the moving arm. 4 Place the lower plate on the balance. 4. If an arc occurs, as indicated by a sizzling noise, turn off the power supply immediately and make sure the plates are exactly parallel. It also could occur due to high humidity. If it does, change your experimental conditions i.e. increase the separation of the plates or decrease the voltage. Lower Plate Upper Plate (1) Level the capacitor apparatus. Using a bubble level as a reference, level the apparatus by adjusting the leveling feet of the platform. 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page 7 / 13

8 (4) Connect the power supply to the plates. () Turn on the electronic balance. DO NOT turn on the power supply until you finish setup. The electronic balance is a precision instrument. Subjecting it to impact could cause it to fail. Treat it with care. Zero the balance by pressing [ 영점 ] or [ 용기 ] button. The balance is sensitive to very small forces and vibrations. Avoid touching the apparatus including connected patch cords while making your measurements. Note (5) Level the electrode plates. For zeroing the balance, press [ 영점 ] and stand by until the [ZERO] mark lights up on the display. 1 Lower the moving arm until the separation of the plates becomes about 5 mm. 2 Adjust the level of the lower plate using the level feet of the balance. Make sure the plates are exactly parallel. If the initial value is relatively high, [ 영점 ] button will not work. In this case, you can zero it using [ 용기 ] button. ([TARE] mark will light up on the display.) If you have any problem zeroing the balance, turn the power off and then on again. Note If any of [CT], [%], [PCS], [CHK], or [ANI] symbols lights up on the display, press [ 모드 ] repeatedly until all symbols disappear. 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page / 13

9 (7) Adjust the plate separation to 0 mm. Using the micrometer, adjust the plate separation to 0 mm. This is indicated by the mass reading increasing suenly. Record the micrometer reading as a reference point = 0 mm. (Make sure the plates are exactly parallel.) Do not touch any plates or connectors while the power supply is turned on. Although the current from the power supply is too low to cause any permanent damage, the voltage on the capacitor plates is high enough to cause a distinctly unpleasant sensation if you touch them. If an arc occurs, turn off the power supply immediately. 1. Make sure the plates are exactly parallel. 2. If the balance malfunctions, turn the power off and then on again. () Adjust the plate separation to mm. Using the micrometer, slowly increase the plate separation to mm. An arc could occur due to high humidity. In this case, change your experimental conditions, i.e. increase the separation of the plates or decrease the voltages. (11) Calculate the force FF = mmmm exerted on the lower plates. This micrometer features 0~25 mm measuring range. Do not rotate the thimble more than the maximum scale setting. (12) Using equation (), calculate the measured value of the capacitance CC of your capacitor. (9) Set the potential difference VV across the parallel plates capacitor to 4 kv. With the voltage adjust knob set at zero, switch on the power supply and gradually increase the voltage until the voltmeter shows 4 kv. CC = 2FFFF VV 2 () (13) Using equation (), calculate the theoretical value of the capacitance CC of your capacitor. CC = QQ VV = εε AA 0 () (10) Record the mass mm. The balance will show negative values because an attractive force is exerted on the lower plate. (14) Compare two values above by finding the percent error. 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page 9 / 13

10 (15) Repeat the steps (9)-(14) for each of following VV. Record the values obtained from three independent measurements with each voltage and find the average values. VV mm FF = mmmm CC = 2FFFF VV 2 (kv) (kg) (N) 4 (17) Repeat step (1) for the different area AA of the plates. 10 Mount the capacitor plates of radius rr = 75 mm. The area rr (m) AA = ππrr 2 (m 2 ) CC theory = εε 0 AA AA of the rr = 75 mm plate is twice as large as that of the rr = 53 mm plate Verify the capacitance is always constant if the shape and size of the capacitor does not change. Before touching the plates or cables, you should be sure to discharge the charges stored in the apparatus by turning off the power supply and touching the discharger. (1) Repeat your experiments for each of following. Set VV = kv and record the values obtained from three independent measurements with each separation and find the average values. VV (kv) mm (kg) FF = mmmm (N) CC = 2FFFF VV VV (kv) mm (kg) FF = mmmm (N) CC = 2FFFF VV rr (m) AA = ππrr 2 (m 2 ) CC theory = εε 0 AA rr (m) AA = ππrr 2 (m 2 ) CC theory = εε 0 AA Plot a graph of CC as ordinates against the matching values of 1/ as abscissa and verify CC is proportional to 1/. 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page 10 / 13

11 (1) Compare the results of step (1) and (17). Note Find the relationship between CC and AA. (19) Repeat your experiments using dielectrics. If a dielectric with thickness = 3mm is inserted between the plates separated by a distance = mm, then this capacitor is equivalent to two capacitors in series as shown below. Mount rr = 75mm capacitor plates and adjust the plate separation to mm. Insert acryl or glass dielectric having thickness 3mm between the plates. Set VV = kv. Record the values obtained from three independent measurements with each dielectric and find the average values. Before touching the plates or cables, you should be sure to discharge the charges stored in the apparatus by turning off the power supply and touching the discharger. Handle the glass dielectric with care. It is very fragile. Suppose the capacitances of each capacitor are CC 1 and CC 2, the dielectric constants are KK 1 and KK 2, and each is separated by a distance = 1 = 2 = (1 2), then 1 CC = 1 CC CC 2 or CC = CC 1CC 2 CC 1 + CC 2 CC 1 = KK 1 εε 0 AA and CC 2 = KK 2 εε 0 AA CC = KK 1KK 2 KK 1 + KK 2 εε 0 AA Dielectric VV mm FF = mmmm CC substance (kv) (kg) (N) only air acryl + air glass + air Dielectric rr AA = ππrr 2 CC theory substance (m) (m 2 ) only air acryl + air glass + air * Reference: - Acryl: KK = Glass: KK = 5. 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page 11 / 13

12 Result & Discussion Your TA will inform you of the guidelines for writing the laboratory report during the lecture. End of Lab Checklist Please put your equipment in order as shown below. Delete your data files and empty the trash can from the lab computer. Turn off the Computer. With the voltage control knob set at zero, turn off the Power Supply and unplug the power cable. Turn off the Electronic Balance and unplug the dc adaptor. Place the Electronic Balance on the platform of the capacitor apparatus. Put the Capacitor Plates, Dielectric Plates, Patch Cords, Bubble Level, and Discharger in the container. Handle the glass Dielectric Plate with care. It is very fragile. 5 Songdogwahak-ro, Yeonsu-gu, Incheon 2193, KOREA ( ) Page 12 / 13