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COULOMB FORCE OBJECTIVES In this lab we will study how the force between two charged objects depends on the charges of those objects and the distance between them. APPARATUS Coulomb Balance (Figure 1) Power supply High voltage probe Voltmeter Vernier calipers INTRODUCTION If you could take one gram of protons and place them one meter away from one gram of electrons, the resulting force would equal to 1.5 x 1023 newtons-roughly the force it would take to "lift" an object from the surface of the Earth that had a mass about 1/5 that of the moon. That’s not a small force. The value comes from Coulomb’s law: kq q (1) F 12 2 R where the force, F is proportional to the two charges (q1 and q2) and inversely proportional to the square of the distance between them, R. A Coulomb Balance (Figure 1) can be used to investigate the force between charged objects. A conductive sphere is mounted on a rod, counterbalanced, and suspended from a thin torsion wire. An identical sphere is mounted on a slide assembly so it can be positioned at various distances from the suspended sphere. The torsion wire is able to measure very small changes in force between the two spheres. If such small amounts of charge produce such enormous forces, why does it take a very delicate torsion balance to measure the force between charged objects in the laboratory? In a way, the very magnitude of the forces is half the problem. The other half is that the Figure 1: The Coulomb Balance carriers of the electrical force, the tiny proton and the even tinier electron, are so small and the electrons are so mobile. Once you separate them, how do you keep them separated? The negatively charged electrons are not only drawn toward the positively charged protons; they also repel each other. Moreover, if there are any free electrons or ions between the separated charges, these free charges will move very quickly to reduce the field caused by the charge separation. So, since electrons and protons stick together with such tenacity, only relatively small charge differentials can be sustained in the laboratory. This is so much the case that, even though the electrostatic force is more than a billion-billion-billion-billion times as strong as the gravitational force, it takes a very delicate torsion balance to measure the electrical force, whereas we can measure the gravitational force by weighing an object with a spring balance To perform the experiment, both spheres are charged, and the sphere on the slide assembly is placed at fixed distances from the equilibrium position of the suspended sphere. The electrostatic force between the spheres causes the torsion wire to twist. The experimenter then twists the torsion wire to bring the balance back to its equilibrium position. The angle through which the torsion wire must be twisted to reestablish equilibrium is assumed to be directly proportional to the electrostatic force between the spheres. All the variables of the Coulomb relationship (Equation 1) can be varied and measured using the Coulomb Balance. We verify the inverse square relationship and the charge dependence. Note: The torsion balance gives a direct and reasonably accurate measurement of the Coulomb force. The most accurate determinations of Coulomb's law, however, are indirect. It can be shown mathematically that, if the inverse square law holds for the electrostatic force, the electric field inside a uniformly charged sphere must be zero everywhere. Measurements of the field inside a charged sphere have shown this to be true with remarkable accuracy. They assume that Coulomb’s law is not exactly an inverse square, kq1q2 and could be written as F 2 . Using this indirect method, it has been demonstrated R n experimentally that n ≤ 2 x 10-16. TIPS FOR ACCURATE RESULTS Experiments with the Coulomb Balance are straightforward and quite accurate, yet, as with any quantitative electrostatic experiment, frustration lurks just around the corner. A charged shirt sleeve, an open window, an excessively humid day-any of these and more can affect your experiment. However, if you carefully follow the tips listed below, you've got a good start toward a successful experiment. The table on which you set up the experiment should be made of an insulating material-wood, masonite, plastic, etc. If a metal table is used, image charges will arise in the table that will significantly affect the results. (This is also true for insulating materials, but the effect is significantly reduced.) Position the torsion balance at least two feet away from walls or other objects which could be charged or have a charge induced on them. When performing experiments, stand directly behind the balance and at a maximum comfortable distance from it. This will minimize the effects of static charges that may collect on clothing. Avoid wearing synthetic fabrics, because they tend to acquire large static charges. Short sleeve cotton clothes are best, and a grounding wire connected to the experimenter is helpful. When charging the spheres, turn the power supply on, charge the spheres, then immediately turn the supply off. The high voltage at the terminals of the supply can cause leakage currents that will affect the torsion balance. When charging the spheres, hold the charging probe near the end of the handle, so your hand is as far from the sphere as possible. If your hand is too close to the sphere, it will have a capacitive effect, increasing the charge on the sphere for a given voltage. This effect should be minimized so the charge on the spheres can be accurately reproduced when recharging during the experiment. Surface contamination on the rods that support the charged spheres can cause charge leakage. To prevent this, avoid handling these parts as much as possible and occasionally wipe them with alcohol to remove contamination. There will always be some charge leakage. Perform measurements as quickly as possible after charging, to minimize the leakage effects. Recharge the spheres before each measurement. Do not trust the digital readout on the power supply. Use the high voltage probe and voltmeter to determine the high voltage present on the charging probe. WARNING You will be dealing with high voltages (HV) of up to 5 kilovolts (kV) in this lab. With some precautions, however, the danger is minimal. First, remove all rings, watches, bracelets, etc. from your hands and arms. Secondly, when you are manipulating the charging (HV) probe, do so with your right hand, and put your left hand in your pocket so that it will not be grounded. The reason is that a small current through your heart is the most dangerous, and the most likely way to get that is to hold a high voltage in one hand, and have the other hand grounded (resting on a table, etc.) The current will travel from one hand, through your torso (and possibly heart) through the other hand, to ground. If you keep your hand in your pocket, however, there will not be a path to ground through your arm. PROCEDURE 1. Set up the Coulomb Balance as described in the previous section 2. Be sure the spheres are fully discharged (touch them with a grounded probe), and move the sliding sphere as far as possible from the suspended sphere. Set the torsion dial to 0°. Zero the torsion balance by appropriately rotating the bottom torsion wire retainer until the pendulum assembly is at its zero displacement position as indicated by the index marks. 3. Use the high voltage probe and voltmeter to set the potential on the charging probe to 5 kV. Turn on the voltmeter and make sure the high voltage probe is plugged in. The pin labeled “GND” on the HV probe should be plugged into the port labeled “CMN” on the voltmeter. The lead with the alligator clip should be connected to the grounding screw on the power supply. Leave the HV probe on the table and touch the charging probe to it to measure the voltage. 4. With the spheres still at maximum separation, charge both the spheres to a potential of 5 kV, using the charging probe. (One terminal of the power supply should be grounded.) The charge deposited by the probe is proportional to the potential. Immediately after charging the spheres, turn the power supply off to avoid high voltage leakage effects. 5. Position the sliding sphere at a separation of 20 cm. Adjust the torsion knob as necessary to balance the forces and bring the pendulum back to the zero position. Record the distance (R) and the angle (). 6. Separate the spheres to their maximum separation, recharge them to the same voltage, then reposition the sliding sphere at a separation of 20 cm. Re-measure the torsion angle and record your results again. Repeat this measurement three times, or until your result is repeatable to within ±1°. Record all your results. 7. Repeat steps 4-6 for separations of 14, 10, 7, and 5 cm. 8. Determine the functional relationship between the force, which is proportional to the torsion angle (); and the distance (R). This can be done in the following ways: Plot log versus log R. Explanation: If = bRn, where b and n are unknown constants, then log = n log R + log b. The slope of the graph of log versus log R will therefore be a straight line. Its slope will be equal to n (which we expect to be –2 from coulombs law) and its y-intercept will be equal to log b. Therefore, if the graph is a straight line, the function is determined, and you can compare its slope to the expected value. Plot versus R-2 This should be a straight line with slope kq2. Either of these methods will demonstrate that, for relatively large values of R, the force is proportional to 1/R2. For small values of R, however, this relationship does not hold. The reason for the deviation from the inverse square relationship at short distances is that the charged spheres are not simple point charges. A charged conductive sphere, if it is isolated from other electrostatic influences, acts as a point charge. The charges distribute themselves evenly on the surface of the sphere, so that the center of the charge distribution is just at the center of the sphere. However, when two charged spheres are separated by a distance that is not large compared to their size, the charges will redistribute themselves on the spheres so as to minimize the electrostatic energy. The force between the spheres will therefore be less than it would be if the charged spheres were actual point charges. A correction factor B, can be used to correct for his deviation. Simply multiply each value of by 1/B, where B 1 4a 3 3 R (2) where a equals the radius of the spheres and R is the separation between spheres. To correct your data: a) Calculate the correction factor B for each of the separations R that you used. b) Multiply each of your collected values of by l/B and record your results as corrected. b) Reconstruct your graph relating force and separation, but this time use corrected instead of . Make your new plot on the same graph as your original plot. How does the correction factor affect your results? 9. Now you will verify the relationship between the charges and the force. With the sphere separation (R) held at a constant value (choose a value between 7 and 10 cm), charge the spheres to different values and measure the resulting force. Keep the charge on one sphere constant, and vary the charge on the other. To charge the spheres, move them first to maximum separation. Then charge them both to 3kV. Make your force measurement at the distance R. Now remove the sliding sphere and recharge it to another value, while keeping the other sphere at 3kV. Measure this force, and repeat the process for charges on the sliding sphere of 2 and 5 kV (each time the charge on the not sliding sphere should be 3 kV). What is the relationship between force and charge? What does a Force vs. Charge graph look like?