1 Coulomb s law, Capacitance and Dielectrics

Transcription

1 1 Coulomb s law, Capacitance and Dielectrics This exercise serves as an illustration of the contents of the chapters 2.1, 2.54 and 4 of the textbook: Coulomb s law, Capacitance and Dielectrics. Coulomb s law concerns forces of interaction between static electric charges. You will inevitably gain some practical experience with the dielectric properties of materials and with the concept of capacitance. Choose among the following: Coulomb s law (1.1) F(r) and F(q), Dependence of distance and charge Choose between one or more among (1.1.3): Charges with opposite sign Dependence of distance II Superposition Image charge And / or: Capacitance and dielectrics (1.2) Capacitance of a plate capacitor (1.2.1) Dielectrics (1.2.2) All exercises consist of several parts that can be chosen (almost) independently. There is no demand of any particular number of sub-sections that has to be completed to pass the exercises, but it is demanded that serious work is done with the sections chosen throughout the time allocated for the exercises. In the following pages the main text will be on the pages to the right, while small boxes with technical advice and additional information will be found on pages to the left. As a preparation before the exercises, please read all pages to the right already at home. 1-1

2 UNILAB Digital Coulomb Meter: Use and Properties Before use of the digital coulomb meter the charge collecting plate should be mounted in the positive contact terminal and the negative point should be neutralized (grounded connected to earth). Charge will now be measured by contact to the charge collecting plate, i.e. by establishing contact between the object on which it is desired to measure the charge and the charge collecting plate. The coulomb meter has in input-capacitance of 1 µf. A simpel charge measurement can only be performed on objects with a capacitance, which is below 0.01 µf = pf. This is because the coulomb meter will only be able to move almost all the charge to itself if the capacitance of the object of investigation is much smaller than the capacitance of the meter itself. As a result of the charge measurement only a negligible charge remains on the object after measurement. The display of the coulomb meter includes a " '' up to 999 nc. Above this value the fourth digit is a "1'' and the polarity is no longer shown. Please leave the coulomb meter with the switch in the "off'' position to optimise life time of the battery. Specifications Interval: ± 1999 nc in steps of 1 nc. Internal capacitance: 1 µf. Internal voltage supply: PP3 battery (9 V). For additional information, see "UNILAB Notes for use No. 97''. 1-2

3 1.1 Coulomb s law Apparatus, Coulomb s law 1. van de Graaff generator 2. coulomb meter 3. charged spheres 4. "transport''-sphere 5. scales 6. ruler 7. grounding unit 8. discharge voltmeter (historic) 9. metal plate med handle (not shown) Numbers refer to the images Theory Coulomb s law states that the force between two charged (point) particles is given by the following expression (here the force from Q on q): F 1 qq r =, (1.1) 2 4πε r 0 r in which q and Q are the charges of the particles, r q and r Q are the position vectors and r = r q - r Q. q is (of course) exerting the force F on Q. The purpose of the exercise is to measure the magnitude (and direction) of the force and to determine the dependence of distance between the charges and of the magnitude and sign of the charges. Another just as important goal is to provide the participants with a sense of how charges can be handled and measured. 1-3

4 About transportation of charges Note: During all measurements in this exercise it is important that both the scales and the coulomb meter are electrically grounded (the scales has a contact point for ground in the rear)). "Ground'' is an object with a capacitance that is so large that one can add or remove charge to or from the object without changing its electrical potential. Charges from the electrostatic van de Graaff generator (1) can be transferred to the spheres (3) used for measurements for example by use of the two "transport" spheres that are mounted on (intended) nonconductive plexiglass rods (4). When the transport spheres are being charged using the van de Graaff generator the charges are moving on the surfaces, charges are re-distributed and a new distribution of charge is obtained when the spheres are moved away from the van de Graaff generator. This means that the voltage of the transport spheres will not necessarily be equal to that of the van de Graaff when the spheres are moved away. In order to transfer as much charge as possible to the transport spheres you can use the small sphere in top of the van de Graaff generator and remove the transport sphere vertically up and away from the van de Graaff generator. Hold the plastic rod at the end farthest away from the sphere and bring the two spheres in contact. Remember to discharge the spheres and zero the scales (5) before initiation of a new experiment. The spheres can be discharged by touching the surface of the sphere (briefly) with the grounded spiral-shaped thread (with a handle). Make sure the discharging is performed well away from other conductive materials (or mirror charges may play a game on you). Coulomb meter, sensitivity Because of the limited sensitivity of the coulomb meter it can be an advantage to repeat the charging and measurements of charge (e.g.) 10 times, because this makes use of the integrating power of the coulomb meter. In this way the accumulated charge is summed (integrated) whereby 10 times the individual charging is collected. In this way the uncertainty on measurement may be reduced. Method of division One way in which you can control the amount of charge on the upper measurement sphere is by first charging the sphere as much as possible using the van de Graaff generator. After having done the first measurement of the force, the charge on this sphere can be divided to half by sharing the charge with the transport sphere. If the transport sphere is now discharged between each division of charge, you will produce a series of divisions with a factor of 2 for the charge on the upper sphere. 1-4

5 1.1.1 Measurements First, we wish to measure the dependence of distance as described in the expression for the force; therefore we chose to keep q and Q constant. For the charges q and Q obtained from the van de Graaff generator (see the box "About transportation of charges") the weight ("force") is measured as a function of distance r between the spheres. The charge remaining in the transport spheres are measured individually using the digital coulomb meter after charging the measurement spheres (c in the figure to the left). Secondly, we wish to measure the dependence of the amount of charge on the spheres, and the distance between the spheres is now chosen to one specific setting. Also the charge, Q, on the lower measurement sphere, the one standing on the scales, can remain constant. The charge, q, on the upper sphere should now be varied systematically. One way to do this is by acquiring charge from a third sphere (a "transport sphere"). This transport sphere can be charged from the van de Graaff generator again and again. During the transportation it should just be kept isolated, so that it will not inadvertently discharge. By letting the transport sphere get into contact with the sphere on which we wish to vary or increase the charge, the charge will distribute itself evenly between the two. Following completion of each measurement of the force between the two spheres in the setup the charge q (on the upper sphere) is measured with the coulomb meter. During measurements with the coulomb meter the charge is removed from the object that is being measured, and the object is therefore completely discharged during the measurement. After each measurement of q the experiment is repeated with a new value of q, e.g. by recharging the sphere using the transport sphere (see the box about "Method of division"). Finally use the coulomb meter to measure the amount of charge Q that was on the lower sphere during the complete experiment Calculations to be done using the Data Your task is to determine the force as a function of distance F(r) and to determine the uncertainty of the measurement. In the experiments the magnitude of the force that acts between the charges is measured as a "weight". Plot the weight as read off the scales and determine from the plot the dependence of this force on the distance r. In order to test the hypothesis of, say, Coulomb s law it may be advantageous to do the plot in several different ways, some of which will make comparison to a given model simpler than others. Determine the slope of your graphs and the uncertainty on these slopes and try using the table value of the vacuum permittivity ε 0 = 8, Nm 2 C -2 to give an independent evaluation of the magnitude of the product of the two charges 1-5

6 qq and the uncertainty on this product (note: if you were having problems with stability of the charge on the spheres during the experiment with dependence on distance this will not be possible.). (If the above was possible) Compare the product of the charges as determined by the calculation above with the ones measured individually using the coulomb meter. How do the uncertainties vary on your graphical displays? Do you see deviations from what you expected with respect to the behaviour of the graphs? If you find such deviations then try to consider possible explanations of these (discuss both the magnitude and trend of the observed deviations). 1-6

7 1.1.3 Tasks Charges with opposite signs How can you use the van de Graaff generator to obtain charges with opposite sign on the transport sphere? Image charge Investigate the force between a charge and its image as a function of the magnitude of the charge and of the distance to a conductive surface. This can be done by using a grounded metal plate (the square aluminium plates supplied) as a substitute for the upper sphere. The experiments are done as described above. Try also to charge the spheres and afterwards insert the grounded metal plate between the spheres. Hold the metal plate in a fixed position. Now, remove the upper sphere while keeping an eye on the display of the scales. Explain the result. Dependence of distance II Try to perform the experiment with force dependence of distance using charges of opposite sign. Plot the results in the same diagram (or a copy including the original data from your previous experiment), but of course in a different quadrant. The principle of Superposition It is supposed to be well known that forces obey the principle of superposition. We will try to show experimentally that this is indeed the case. For a selected distance between the spheres the principle of superposition can be investigated by first examining the interaction between a sphere on the scales and the two free spheres individually hereafter the interaction between the lower sphere and the pair of spheres (or you could do this in the opposite order; this will simplify positioning the single upper sphere at the same position it had when being one of a pair). Remember carefully measuring and recording all relevant distances in your notes of the experiments. Evaluate based on the inevitable least distance between the two spheres in the pair the magnitude of the projection of the involved vectors and thereby how much the force from the pair is expected to be smaller than for the sum of the individual spheres if positioned exactly above the lower sphere. Compare your calculated result with what you have just measured and evaluate all the associated uncertainties. 1-8

8 UNILAB Digital Coulomb Meter: Use and Properties Before use of the digital coulomb meter the charge collecting plate should be mounted in the positive contact terminal and the negative point should be neutralized (grounded connected to earth). Charge will now be measured by contact to the charge collecting plate, i.e. by establishing contact between the object on which it is desired to measure the charge and the charge collecting plate. The coulomb meter has in input-capacitance of 1 µf. A simpel charge measurement can only be performed on objects with a capacitance, which is below 0.01 µf = pf. This is because the coulomb meter will only be able to move almost all the charge to itself if the capacitance of the object of investigation is much smaller than the capacitance of the meter itself. As a result of the charge measurement only a negligible charge remains on the object after measurement. The display of the coulomb meter includes a " '' up to 999 nc. Above this value the fourth digit is a "1'' and the polarity is no longer shown. Please leave the coulomb meter with the switch in the "off'' position to optimise life time of the battery. Specifications Interval: ± 1999 nc in steps of 1 nc. Internal capacitance: 1 µf. Internal voltage supply: PP3 battery (9 V). For additional information, see "UNILAB Notes for use No. 97''. 1-9

9 1.2 Capacitance and dielectric materials Apparatus, capacitance 1. coulomb meter 2. capacitor plates 3. voltage supply 4. voltmeter, universal instrument 5. smaller plates 6. dielektrikum (trovidur) The numbers refer to the image The capacitance of a simple plate capacitor is given by A Q = ε ε and C =, d U C r 0 where ε r is the relative permittivity (dielectric constant) of the dielectric material. During the exercise we will study the dependence of the capacitance on some of the geometric parameters, i.e. of the area and the distance between the plates of the capacitor, and the dependence of the properties of the material between the plates, the dielectric properties of this material. Finally, we will investigate the effect of using different dielectric materials (air and trovidur) present in different geometries between the plates of the capacitor. 1-10

10 Charging of a capacitance For this experiment a small voltage divider is used (small gray box with a potentiometer) for a variable output voltage (0 100 V). The capacitance is charged by connecting the upper plate to the output of the voltage divider. Hereafter the connector is moved from the voltage divider and the charge Q on the capacitor is measured by now connecting the upper plate of the capacitance to the coulomb meter. This is simplest done by holding the connecting wire with a plastic rod and by using the same wire for charging and measuring the charge. Note that during measurement the whole charge is transferred to the coulomb meter and the capacitance is therefore discharged to the voltage U = 0 V. Be careful! Any touching of the wire or the capacitor plates by hands will partly or completely discharge the capacitor and give erroneous measurements of the charge. Also please note: For all measurements in this exercise it is important that the voltage divider, the capacitor and the coulomb meter are all grounded. Coulomb meter, sensitivity Because of the limited sensitivity of the coulomb meter it can be an advantage to repeat the charging and measurements of charge (e.g.) 10 times, because this makes use of the integrating power of the coulomb meter. In this way the accumulated charge is summed (integrated) whereby 10 times the individual charging is collected. In this way the uncertainty on measurement may be reduced. Discharge/measurement of charge Always keep other charged objects far away from an object you want to discharge. Remenber that any charge measurement involves a discharge. Cause of potential problem? See e.g. mirror charge. 1-11

11 1.2.1 Measurements of capacitance Measure the capacitance of the plate capacitor by measuring the charging voltage over the capacitor and the resulting charge on the capacitor. Start by using the larger of the two sets of plate capacitors. The plate with four legs should be put on the table and it should be connected to common ground (earth) and ground on the voltage divider. Put the other plate on top of the first using the small plexiglass distance pieces. Charge the capacitor to the voltage U (see the box "Charging of a Capacitance"). Repeat the experiment using 4 different values of the voltage (between 25 and 100 V). Voltage is measured using a voltmeter directly on the voltage divider (the small gray V-box). The dependence of distance of the Capacitance between the plates is examined using the highest voltage and measuring the accumulated charge as described above for four different distances. The dependence of area is examined by the smaller plates (as before integrate over a suitable number of chargings) using the highest available voltage and the smallest distance used above. Repeat this measurement with the largest distance used above Measurements with dielectric media Repeat the first part of the experiment (1.2.1.), but with the material trovidur between the plates instead of air and calculate the constant of dielectricity for the material(s) used (red trovidur, 5 mm thick, ε r 2,5 3,5). If time allows repeat the experiment using a ceramic plate (15 mm tyk, ε r 5). The plates of dielectric material are coated with an electrically conductive layer to eliminate the effect of any small pockets of air between the dielectricum and the metal plates. a. The volumen between the plates completely filled with the material trovidur: trovidur 1-12

12 Your Report See the general description in the Introduction. It is good practice during all excercises to keep a laboratory jounal log book, in which each team make sketches of circuits used, notes the types (and makes of) instruments used, and draw the experimental setup and take note of the data recorded. This can be done more or less simultaneously with working through the exercises and will enable you to remember afterwards what was done and why. Usually such notes are useful for remembering what was done and absolutely essential for being able to produce a good report of the exercise. 1-13

13 b. Try to experiment with combinations of air and trovidur between the plates. use the distance pieces to avoid collaps of the structure: trovidur c. Try an experiment with half a plate of trovidur. Does it matter where the plate is positioned? trovidur d. Calculate the capacitances for experiments b. and c. from the capacitance in a. Hint: Try a fictive division of the capacitances in b. and c. by parting the system along a suitably chosen field line or equipotential surface. Make use of the fact that certain entities (Q or U) are conserved under such a division. Remember that for parallel connections of capacitances the following is truer: C = C C and for serial connections you can use that: 1 C 1 = C C Use of Data Determine from the measured relation between U and Q in the first part of the experiments the magnitude of the capacitance C and the uncertainty on this value. Compare with the theoretical value. Explain the deviation that (probably) is most prominent for bigger distances between the plates. Find the relative constant of dielectricity for trovidur (exp. a. above) and calculate the capacitance for the geometries b. and c. above. Compare with measured values and discuss discrepances. 1-14