General Physics II Lab EM2 Capacitance and Electrostatic Energy
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1 Purpose General Physics II Lab General Physics II Lab EM2 Capacitance and Electrostatic Energy In this experiment, you will examine the relationship between charge, voltage and capacitance of a parallel plate capacitor. Equipment and components Variable capacitor, DC power supply, electrometer with a cable and battery box, switch box, Faraday ice pail, proof plate, charging probe, capacitor (15 pf), ruler, aluminium sphere, grounding wristband,.cables: BNC to spades (unshielded), BNC to crocodile clips (shielded), leads: 4 mm plug to spade, 4 mm plug to plug (for earth grounding). Background Electrostatic energy associated with an electric field can be stored in a capacitor. The storage of such energy requires that one has to do work to move charges from one plate in the capacitor to the other. The charge, Q, on the plates and the voltage, V, between the plates are related according to the equation Q = CV, (1) where C is the capacitance which depends upon the geometry and dimensions of the capacitor. For a parallel plate capacitor with plate area A and separation d, its capacitance is ε A C =, (2) d where ε is the permittivity of the medium between the two plates. The permittivity of air is approximately equal to that of vacuum, ε ε0. The amount of the energy stored in a capacitor is given by 2 2 Q CV U = = (3) 2C 2 In this experiment, you will carry out measurements on a parallel-plate capacitor to verify the above equations. Procedure Measurement 1: Capacitance of the electrometer and cable It should be noted that whenever you make measurements of charge, voltage or capacitance, you need to consider the effect of the internal capacitance of the electrometer and that of the cables connected to it. As shown in Fig. 1, the electrometer can be thought of as an infinite impedance voltmeter in parallel with a capacitor. The capacitor C 1 represents the internal capacitance of the electrometer, plus the capacitance of the leads. The capacitance of the electrometer and cable C 1 adds to the external capacitor which is connected in parallel. Therefore, it is necessary to know the capacitance of the electrometer and cable in order to have an accurate measurement of the capacitance of an external capacitor. Revised: 24 February /9
2 Figure 1 Schematic of the electrometer 1. Connect the circuit as shown in Fig. 2. Before turning on the electrometer, check that the meter is mechanically zeroed. If not, please ask the technician-in-charge for help. 2. Turn on the electrometer and turn the FUNCTION switch to the 3V position. Flip the ZERO switch to the LOCK position. Adjust the ZERO SET control until the meter reads at center zero. 3. Turn the power supply on by selecting the 30 V range using the OUTPUT switch and turning the VARIABLE control knob clockwise. Set the voltage to 30 volts. 4. To prevent stray charges from producing erroneous readings, you should keep yourself grounded by wearing a grounding wristband. Keep wearing it until you have finished the experiment. 5. Toggle the switch (on the switch box) to a position such that the voltage, V 2 = 30 V, is applied across the capacitor of known capacitance, C 2. Record the value of C 2 in the lab report. 6. Flip the ZERO switch to the PUSH TO ZERO position. 7. Press and hold the "PUSH TO ZERO" button of the electrometer. Release the button and immediately switch connection of the capacitor from the power supply circuit to the electrometer circuit by toggling the switch. 8. Record the maximum voltage, V, indicated on the electrometer. (As the capacitor will discharge rapidly). 9. In the lab report, compute the value of the capacitance C 1 of the electrometer and cable. This value will be useful for the following measurements. Figure 2 Circuit diagram for the measurement of the capacitance of the electrometer and cable Revised: 24 February /9
3 Measurement 2: Relationship between voltage and charge 1. Connect the conductive sphere to the 1000V output (the green binding post) of the power supply, as shown in Fig. 3. NOTE: - Be careful with the High Voltage. - Keep all parts (e.g. the proof plane, the variable capacitor, ) clean to prevent leakage of charge. 2. Set the voltage of the power supply to 1000 V. Figure 3 Charging the conductive sphere 3. Connect the apparatus as shown in Fig. 4. Measure and record the radius of the parallel plates, R, in the lab report. CAUTION: Place the charged sphere and power supply as far as possible from the electrometer and capacitor so that the capacitor plates are not charged by induction or affected in any other way. You should also minimize your motion in the vicinity of the electrometer. Figure 4 Setup for the measurement of voltage and charge of a capacitor 4. Because the electrometer is grounded, momentarily pressing the "push to zero" button will remove any excess charge from both capacitor plates. 5. Set the initial separation of the two plates to be 2 mm. It is recommended to adjust and fix the position of the fixed plate such that the movable plate indicator reading on the scaled slide gives the plate separation directly. NOTE: The capacitor plates should be in parallel. If not, please ask your TA or technician for help. 6. Use the proof plane to transfer charges from the aluminium sphere to the ungrounded capacitor plate, which is connected to the red electrometer lead. The transfer of charge is carried out by simply touching the proof plane flat against the aluminium sphere, and then flat against the capacitor plate (see Appendix for more details). If the proof plane touches the sphere and capacitor in the same manner, then an equal amount of charges will be transferred each time. Revised: 24 February /9
4 7. Adjust the sensitivity of the electrometer, so that each charge transfer results in a measurable deflection of the meter needle. Make several charge transfers and record the electrometer readings in Table Double the plate separation to 4 mm and repeat steps 4-7. Measurement 3: Relationship between voltage and capacitance 1. Connect the electrometer to the parallel plate capacitor as shown in Fig. 4. Adjust the electrometer to the 10V range. 2. With an initial plate separation, d 0 = 2 mm, charge the parallel plates to 4 V by momentarily connecting the power supply output (set it at 4 V using the 30 V range output) to one of the plates with a charging probe. 3. Increase the plate separation d to 4 mm and record the reading of the electrometer in Table 2. CAUTION: Make the measurement quickly since the parallel plates discharges rapidly. 4. Repeat steps 2-3 with varying plate separation, d, in steps of 2 mm until d = 10 mm. NOTE: Keep d 0 = 2 mm for all measurements. Measurements 4: Surface charge density of a parallel plate capacitor 1. Connect the electrometer to the Faraday ice pail as shown in Fig. 5. Figure 5 Setup of Faraday ice pail 2. Connect the power supply (with black and green binding posts) across the capacitor plates and set the plates 5 cm apart. Set the power supply to 1000 V. 3. Ground the proof plane and then use it to touch the centre of the inner surface of the fixed plate of the capacitor. CAUTION: Ensure that there is no contact between the rod of the proof plane and the capacitor plates. Otherwise, the capacitor will be discharged. 4. Measure the charge on the proof plane by placing it inside the Faraday ice pail. Because of induction, the Faraday ice pail is charged and the voltage reading of the electrometer is proportional to the charge on the proof plane. CAUTION: Do not make any contact between the proof plane and the ice pail. 5. Repeat steps 4-5 to measure the charge distribution of the parallel plate as a function of the radial distance r from the centre of the plate (four additional values of r including one data at the edge of the plate). Record your results in Table 3. NOTE: Please do NOT put any mark on the capacitor plates. You can use a pencil to mark on the rod of the proof plane if need. Revised: 24 February /9
5 Appendix: Use of proof plane The proof plane is a conductive disk mounted on an insulated handle. By touching the proof plane to a surface, the proof plane will acquire the same amount of charge as the section of the touched surface. By measuring the charge on the proof plane, one can determine the charge density of the sample surface. The greater the charge on the proof plane, the greater the charge density of the sample surface. An electrometer and a Faraday ice pail are used to determine the charge on the proof plane. Figure 6 Use of the proof plane When the proof plane touches the sample surface, it becomes part of the surface. If the effect on the shape of the sample surface is significant, the measurement of the charge density will be inaccurate. Therefore, always touch the proof plane to the sample surface in such a way as to minimize distortion of the shape of the surface. Figure 6 show the recommended method for using the proof plane to sample charge on a surface. Revised: 24 February /9
6 Name Date Lab session (Day & time) Lab partner EM2 Capacitance and Electrostatic Energy Lab Report A. Answer the following question BEFORE the lab session (5 pts each) 1. In Fig. 1 the two capacitors, C 1 and C ext, are connected in parallel. Find the expression for the total capacitance of the circuit. 2. Briefly describe how the Faraday ice pail can be used to measure the charge on the proof plane (see Fig. 5). 3. For a capacitor made of two parallel conductive plates, does its charge density vary with the radial distance from the center of the plate? Revised: 24 February /9
7 B. Results and calculations (45 pts) Measurement 1: Capacitance of the electrometer and cable (10 pts) C 2 =, V = ± General Physics II Lab The capacitance C 1 of the electrometer and cable is in parallel with the test capacitor of capacitance C 2. Therefore, C 1 can be calculated according to C = CV / V C = The capacitance of the electrometer and cable is approximately 20 pf (1 pf = 1 pico Farad = Farad). The experimental result should be close to this value. Measurement 2: Relationship between voltage and charge (15 pts) Radius of the parallel plates, R = ± Plate area, A = ± (NOTE: ΔA = 2πRΔR and make sure that you know why) Table 1 Voltage versus Charge Plate separation d = 2 mm Plate separation d = 4 mm Charges Electrometer reading (V) Charge Electrometer reading (V) Q 0 ± Q 0 ± 2Q 0 ± 2Q 0 ± 3Q 0 ± 3Q 0 ± 4Q 0 ± 4Q 0 ± 5Q 0 ± 5Q 0 ± Measurement 3: Relationship between voltage and capacitance (10 pts) Table 2 Voltage versus Capacitance Plate separation d (mm) Electrometer reading (V) Revised: 24 February /9
8 Measurements 4: Surface charge density of a parallel plate capacitor (10 pts) Table 3 Charge versus the radial distance Radial distance r (mm) 0 Electrometer reading (V) C. Data analysis and questions (40 pts) Measurement 2 (24 pts): 1. (4 pts) Why is it sufficient to add charges to only one plate? 2. (20 pts) Plot the measured voltage as a function charge shown in Table 1. Plot the data with d = 2 mm and d = 4 mm on the same graph. Assume that an equal amount of charge, Q 0, is transferred for each touch of the proof plane. a. How does the potential vary with charge? b. How does the result depend on the plate separation? c. Use linear regression to fit your data. Show the fitted curve as a solid line together with the data points. Attached the graph to the lab report. Write down your results below: For d = 2 mm, slope m =, y-intercept c =. For d = 4 mm, slope m =, y-intercept c =. Revised: 24 February /9
9 d. Use Eq. (2) to determine the charge Q 0 that is transferred for each touch of the proof plane. Hint: The parallel plate capacitor is in parallel with the electrometer and cable (see V Fig. 1). Therefore, we have Q = nq0 = CV Q0 = C = Cm, where n is the n number of touches and m is the slope obtained above. Because Final results: ε A ε 0A = C1 + C2 = C1 +, we have Q0 = C1 + m. d d C 0 For d = 2 mm, Q 0 = For d = 4 mm, Q 0 = e. How much energy have you stored in the system ( C1+ C 2 ) for charge Q= Q 0? For d = 2 mm, U = For d = 4 mm, U = Measurement 3 (10 pts): 1. Plot 1/V (voltage) versus 1/d (plate separation). How does the potential vary with the separation d? Use linear regression to fit your data. Show the fitted curve as a solid line together with the data points. Attached the graph to the lab report. 2. Write down your fitting results: Slope m =, y-intercept c =. Measurement 4 (6 pts): 1. Plot Voltage (proportional to the charge density) versus the radial distance r. How does the charge density vary with position? Explain your result. Revised: 24 February /9
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