Coulomb s law with Cobra3

Coulomb s law with Cobra3 LEP Related Topics Electric field, electric field strenght, electric flux, electrostatic induction, electric constant, surface charge density, dielectric displacement, electrostatic potential. Principle Two conducting spheres are held at a distance measured by a slide device and get charged electrically with known voltage symmetrically against earth. The force exerted by the charge on the spheres is measured. The amount of charge is measured by connecting a charged sphere to a known capacitance and reading out the voltage on the capacitor with an electrometer amplifier. Equipment Cobra3 Basic Unit, UB 12150.50 1 Powergraph software 14525.61 1 Measuring module Newton 12110.00 1 Newton sensor 12110.01 1 Insulating Bar for Force ensor 12110.02 1 Plug with socket and crosshole, 2 07206.01 1 Conductor ball, d = 40 mm 06237.00 2 Insulating stem 06021.00 1 Power supply, 12 V- 12151.99 1 High voltage supply unit, 0-10 kv 13670.93 1 Connecting cord, 30 kv, l = 1000 mm 07367.00 2 Connecting cord, l = 100 mm, green-yellow 07359.15 1 Connecting cord, l = 250 mm, blue 07360.04 1 Optical profile bench l = 60 cm 08283.00 1 Base f. opt. profile-bench, adjust. 08284.00 2 lide mount f. opt. pr.-bench, h = 30 mm 08286.01 1 lide mount f. opt. pr.-bench, h = 80 cm 08286.02 1 lide device, horizontal 08713.00 1 Electrometer Amplifier 13621.00 1 Data cable for cobra probes 12150.07 1 Connecting plug 39170.00 1 Capacitor 10 nf/ 250 V, G1 39105.14 1 PC, Windows 95 or higher Tasks 1. Measure the force between the two charged balls as a function of the applied voltage. 2. Measure the force between the two charged balls as a function of their distance. 3. Measure the charge on the balls as a function of applied voltage and distance thus determining the capacitance of the assembly. et-up and procedure et up the gear as seen in Fig.1. It is essential that the high voltage cables are in as big a distance from each other as possible since not the force between the cables, but the one between the spheres is to be measured. The cable leading to the sphere on the force sensor should hang freely and exert no force on the force sensor in the direction of interest especially none that change during measurement. Maybe you can put the high voltage supply on something sufficiently high to make the cables hang free. Remember that humid air may insulate only 1 kv per mm. If a spark between the spheres discharges them, you have to lower the voltage or enlarge the distance between them. Fig. 1: Experimental set-up PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D-37070 Göttingen P2420415 1

LEP Coulomb s law with Cobra3 8 2 5 7 4 6 1 3 Fig. 4a: Virtual device settings for the distance 9 Fig. 2 Connect connector 8 of the electrometer amplifier (Fig. 2) to Cobra3 s Analog in 2 / 2. Put the bridge ( Connecting plug ) between 1 and 2 and the 10 nf capacitor between 1 and 9. Keep the electrometer amplifier s input earthed by connecting 9 and 2 with the blue cable if not measuring. Keep any high voltage away from the electrometer amplifier, it s input is for up to 10 Volt only! Fig. 4b: Virtual device settings for the high voltage Connect the Cobra3 unit to your UB port. tart the measure program on your computer. elect the Gauge PowerGraph. Add the Virtual device by clicking the white triangle button on the upper left. Click on the turquoise bars of the Virtual device and set two channels as manual inputs so as to record the applied voltage and the spheres distance during measurement by typing them by hand. Precision sets the number of digits behind the point. ee Fig. 3. and Fig. 4. Click the Analog in 2 / 2 and check the following settings: Fig. 3: PowerGraph Fig. 5: ettings on the Analog In 2/2 port 2 P2420415 PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D-37070 Göttingen

Coulomb s law with Cobra3 LEP Click the Newton device and put the settings as follows: Fig. 6: ettings of the Newton sensor The averaging is necessary in this case to reduce the noise on the force signal and you may try different strengths. Be sure to use the Tara button before starting the measurement. The device may have a temperature drift - so turn it on a while before measuring (or else measure the force without applied high voltage several times throughout your measurements e.g. for each new value of distance and subtract the value from the force data obtained). The ettings chart of your PowerGraph should look like this: Fig. 7: ettings of the PowerGraph The Display chart looks like Fig. 8. First doubble click <new display >. A display configuration menu will then appear and you can enter there the sort of displays you want to see during measurement. Doubble clicking the displays names will open this window again it is advisable to choose the auto range functions in combination with low value settings so no need occurs to think about what values will appear. Fig. 8: Display options a) Force measurement: Put the spheres to the desired distance, e.g. starting with 4 mm. Use the Tara method on the menu Module port. tart the measurement with the Continue button and watch the displays. The force display should not show too much fluctuation. et the high voltage on the high voltage supply to the desired value. Enter the voltage and distance values in the PowerGraph measuring menue and confirm with Return. Then record the data with the ave value button after the force reading fluctuations have calmed down. You may begin always with V2 = 0 kv and increase in steps of e.g. 1 kv. The distance steps may be some millimetres and may increase for higher distances. It may be good to stop the measurement before changing the distance after recording a series of different voltages and use Tara before continuing. Plot the force against the square of voltage (V2) 2 for a given distance, several plots for several distances (Fig. 11). Plot the force against the inverse square of distance 1/x 2 for a given voltage, several plots for several voltages (Fig. 12). b) Charge measurement tart the measurement with the Continue button again. Apply high voltage to the spheres and enter values of distance and voltage. The Display for the amplifier s voltage U2 should be zero. Disconnect the blue earth cable from the electrometer amplifier from 2. (ee Fig. 2) Transfer the charge of one sphere to the capacitor by unplugging the high voltage cable of one sphere from the high voltage supply (but not from the sphere) and plug it to the amplifier in where the blue earth cable had been immediately and unplug it again. The Display for the amplifier s voltage U 2 shows now a voltage proportional to the charge on the sphere. Record the values with the ave value button and earth the capacitor again with the blue cable. PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D-37070 Göttingen P2420415 3

LEP Coulomb s law with Cobra3 This measurement has to be carried out carefully as not to loose charge by spraying or influence it by stray capacities between your body and the set-up. It is essential only few time passes between disconnecting the sphere from high voltage to connecting it to the capacitor, but there is no need to connect it long to the capacitor. Charge spraying will make it impossible to get accurate values for voltages above 9 kv. Plot the charge as a function of the high voltage for a given distance between spheres, several plots for different distances (Fig.13). The charge q is calculated with the used 10 nf capacitance plugged to the electrometer amplifier as Coulomb s law states that a single charge has the field E 1 q 1 r r 2 4 p e 0 0 2 r 0 0 r 0 with r r x r q the distance vector between the place r q of the charge and r x the place where the field is observed and a second charge q at r x encounters the force F q' E 1 1 q'q r 2 4 p e 0 0 2 r 0 r 0 r 0 C q U 10 nf 10 10 9 F, e.g. U 2 = 5 V, q = C U 2 = 5 10-8 Coulomb. This is the amount of charge that encounters the force from the field of the other charge. With applied high voltage of V 2 = 5 kv is the capacitance of the spheres then (both spheres carry equal but opposite charges) or the amount E1x2 q 4 pe 0 1 x 2 F1x2 q' E 1x2 q'q 4 pe 0 1 x 2 along the straight line between the two charges (Fig. 10). (1) (2) C sphere 5 10 8 C 5000 V 10 11 F 10 pf. Theory and evaluation In this experiment we assume static electricity, i.e. the electric field E has # a charge density r as ist s source and there is no current, r 0. There is a potential that describes the electric field with E grad. The sets of points with the same potential are then surfaces containing the charges, the equipotential surfaces with E always perpendicular (orthogonal) to them. Conductors always resemble equipotential surfaces and E is always orthogonal on them. Else the charges could gain energy by moving from a region of higher potential to a place of lower potential or in a different picture, forces from E on the charges would have a tangential component to the conductors surface and thus would alter the charges distribution. Look at Fig. 9 to see the form of field lines. For charges far apart the equipotential surfaces are nearly spheres and so our conducting spheres can be replaced by point charges in their inner. But with spheres not far away from each other compared to their distance, the surface charge distribution on the spheres is not uniformly. The image charge of a charge in front of a conducting sphere does not lie in the middle of the sphere but between the centre and the surface. o there are higer multipole moments of charge distribution (dipole, quadrupole, ). The conducting spheres cannot be replaced simply by point charges but have to be replaced each by a point containing point charge, point dipole, point quadrupole, and these points do not lie in the centres of the spheres. The field of the higher moments decays different from 1/r 2 so the actual field is different from the field assumed in (1) the more, the closer the spheres get. The equipotential surfaces of a field of two point charges are no spheres but it is always possible to add some multipoles to the point charges that distort the field in a way that there is somewhere a equipotential surface resembling a sphere. For great distances of the spheres, their capacity can be assumed as the capacity of a single sphere, C 2pe 0 R q U (3). For small distances the real capacity has to be higher. This yields for the force according to (2) and (3): F1x2 q' E1x2 q'q 1 4pe 0 x pe R2 U 2 2 0 x 2 Fig. 9: Field lines and equipotential surfaces Fig. 10 4 P2420415 PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D-37070 Göttingen

Coulomb s law with Cobra3 LEP The obtained plots may look like this: Fig.11: The force grows linear with the square of the high voltage Fig.12: The force is not linear with 1/x 2 because of not neglectable size of the spheres compared to distance Fig.13: The capacitance of the two spheres Distance between spheres 2 mm 10 mm 20 mm slope 1.09 10-3 1.03 10-3 1.09 10-3 average 1.07 10-3 PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D-37070 Göttingen P2420415 5

LEP Coulomb s law with Cobra3 With known radius of the spheres, and measured voltage, distance and charge the dielectric constant f can be determined this way by e 0 F x2 p R 2 U 2. The charge q is calculatded with the used 10 nf capacitance: q = 10 nf U 2 and the spheres capacitance is then C q V 2 10 nf U 2 V 2 10 nf 1.07 10 3 10.7 pf and with (3) is, e 0 C 2pR if the only capacitance in the assembly is the conducting spheres. The literature value is 10.7 pf 2p 0.02 m 8.51 10 11 As Vm, 12 As e 0 8.85418781762 10 Vm. 6 P2420415 PHYWE series of publications Laboratory Experiments Physics PHYWE YTEME GMBH & Co. KG D-37070 Göttingen