Testing Coulomb s Law

Transcription

1 Coulomb s Law and electrostatics The electrostatic force is one of the fundamental forces of nature. The properties of the world around us depend on the strength of the electrostatic attraction between protons and electrons. The force was first systematically studied by Charles Coulomb in the 178s. Using very careful measurements of the forces between charged spheres he deduced Coulomb s law: The magnitude of the electrostatic force between two point electric charges is directly proportional to the product of the magnitudes of each charge and inversely proportional to the square of the distance between the charges. Moreover, he noted that equally charged bodies repelled each other, while oppositely charged bodies attracted. See Young and Freedman, Chapter 21 for further background. The force law can be summarised by the equation: F = 1 4πε q 1 q 2 r 2, (1) where q 1 and q 2 are the charges and r is their separation. The constant ε is called the permittivity of free space. It sets the order of magnitude of the electrostatic force. In SI units the accepted value for ε is C 2 /(N m 2 ). The electrostatic force has several interesting features. Firstly it is very strong - the electrostatic force between a proton and an electron is almost 4 orders of magnitude stronger than the gravitational force between them. Secondly, the force depends on the inverse of the square of the distance. The modern interpretation of this is that it is mediated by massless particles (photons). In this experiment you will confirm the inverse square dependence of the electrostatic force and determine the value of ε by measuring the force between a charged sphere and its image induced in a metal plate. Preparatory task. Estimate the magnitude of the force you will be measuring. The Capacitance of a metal sphere of radius R is 4πε. In this experiment, we will charge a sphere of R

2 radius 2cm to 1 kv. Estimate the force when two such spheres are placed with their centres 5cm apart. Do you think this is a large or small force? Think of an every day object with a similar weight. Outline of the method We will measure the force between the sphere and an earthed metal plate. The metal plate acts as a mirror, and the force generated is mathematically equivalent to the force between two oppositely charged spheres. This is illustrated in the diagram below (Fig 1). Because we only need to charge one sphere at a time, this set-up is considerably simpler than measuring the force between two charge spheres. A particular problem is that charge will easily leak from one sphere to the other if the air in the lab is at all damp. In outline, this experiment requires the following steps: 1. calibrate the force meter. 2. charge up the sphere using the 25kV supply. 3. measure the charge on the sphere using the electrometer 4. place the sphere next to the earthed metal plate 5. measure the force

3 6. move the plate and repeat measurement of the force as the distance to the plate increases/decreases. 7. plot a graph of the force as a function of distance and hence confirm the inverse square-law dependence of the electrostatic force and determine the value of the fundamental constant ε. Starting the Experiment Checking the calibration of the force meter Switch on the force meter and allow it to warm up. Familiarise yourself with the operation of the mobile-cassy read out. Once switched on, the unit will have a digital readout showing the force in milli-newton. If you press GENTLY on the charged sphere, you will see the force read-out change. Note how sensitive the unit is. Check that the force meter is measuring forces in the vertical direction, as shown by the arrows on the front. Note that you will need to frequently zero the unit so that you can see small changes in the force. Press the arrow keys, to show the properties option. Select this and chose compensate offset to set the measured force to zero. Now you can use the force meter to weigh a paper clip. The measured force varies because of vibrations in the lab. In order to obtain sufficiently accurate measurements, you will need to take 5 readings and average the results. Practice this and decide on an appropriate strategy. Task 1: What is the weight of the paper clip (hint: g=9.8 ms -2 ). Compare this with the mass of the paper clip that you measure on the digital balance. Charging the sphere There is one 25kV power supply that is shared between the experiments. It is located away from your apparatus to minimise any induced charge that might complicate your results. The diagram below shows the high voltage supply. On no account touch the exposed metal point (a). Initially, set the power supply to the maximum voltage (25 kv). This will result in the strongest forces. To charge a sphere, touch it on the exposed metal point, (a) in the diagram. It will help to charge the sphere if you hold the earthed metal rod while doing this. It is very important that you do not touch the exposed contact yourself. You will receive an electric shock similar to (or worse than) that given by an electric fence.

4 The unit is regulated to limit the current flowing, but the experience will still be extremely unpleasant. Fig 2. The high voltage supply used to charge the metal spheres used in this experiment. On no account touch the exposed metal point (a). Measuring the charge on the sphere You will use the electrostatic amplifier to measure the charge that has been induced on the sphere. Before you start, touch the metal cup with the earthed rod. The output voltage should now read zero. Hold the metal rod and carefully hold the charged sphere inside the metal cup. It is vital that you do not touch the metal cup if this happens you will have to recharge the sphere. The electrometer works by using the charge on the sphere to induce a charge on a capacitor. The voltage on the capacitor can then be measured using the multimeter. The electronics inside the box prevent any of the induced charge leaking away through the multimeter. Recall that the charge on a capacitor is given by, Q = CV, where C is the capacitance (in this case 1 nf) and V is the voltage across it. Experiment with measuring the induced charge. Do you always get the same charge on the sphere? How quickly does the sphere discharge? If the sphere discharges

5 significantly in 5min, charge must be leaking away down the insulating rod, and this will need to be cleaned. You should practice this procedure so that you can charge the sphere, quickly measure its charge each time you start to take force measurements. Fig 3. The electrometer used to measure the charge on the metal sphere. It is vital that the sphere does not touch the sides of the metal cup, or else it will discharge. Preparing to make the force measurements Once you have measured the charge on the sphere, locate it in the force meter as shown in the diagram. Now zero the force meter, and then place the metal plate underneath the sphere. Set the adjustable stand to the top of its range (the zero position), and adjust the height of the force meter so that the sphere almost touches the plate. Alternatively, you may find it easier to set the sphere in approximately the right position, and then adjust the stand to measure the height of the plate at which they just touch (if you use this method, you will then have to subtract this value from your subsequent measurements). Now lower the stand so that you slide the plate out of the way without it touching the sphere.

6 Full Experiments Fig 4. The force meter and the earthed metal plate have been set up ready to take force measurements. It will be best to practice some measurements of force and perfect the alignment of the equipment before carrying out final measurements. This will also help you decide how to space your measurements, and how many individual readings you should average over. Measuring the Electrostatic force Start at a separation of a few mm. Take several reading of the force so that you can compute an average value. You can use the spread in your readings to estimate the error. Adjust the height of the plate and take further readings, so that you can plot the force as a function of the separation. Note that you should space your readings sensibly. You need several readings while the separation is small, and fewer readings at large separations. Note that the stand has a vernier scale, so that you can measure the height to.1mm. The scale is explained in the appendix. When you have finished taking force readings, check that the force meter reads zero when you remove the plate (if it does not, this indicates the level of systematic error

7 in your force measurement) and re-measure the charge on the sphere so that you can quantify the amount of charge that has been lost during the experiment. Repeat this part of the experiment so that you can assess the systematic errors in your measurements. Task 2: tabulate your force measurements as a function of distance. And Finally Task 3: Measure the diameter of the sphere using the vernier callipers (d ). Coulomb s Law and the permittivity of free space Begin by computing the average force at each plate separation (and its associated error). Coulombs law suggests that the force between the sphere and the plate should be: 1 F = 4 πε Q 2 2 (2s + d ) (where s is the separation between the edge of the sphere and the plate, d is the diameter of the sphere and ε = C 2 /(N m 2 ) is the permittivity of free space). Rather than applying this directly, it is more convenient to express the relation as: Plotting 2 4πε 1/ F = (2s + d Q 1/ 2 F as a function of ( 2 d ) ) s + should give a straight line graph with 4πε gradient. The line should pass through the origin. Note that the relative error Q in 2s + d ) will be much smaller than in your measurement of the force. ( Task 4 Confirm that your data are reasonably well fitted by a straight line (i.e., that Coulomb s Law is correct). You may find that the force at small separations is larger than you would expect: this is due to the distribution of charge on the sphere becoming non-uniform. Task 5 Determine the value of ε and its uncertainty and compare it with the standard value.

8 Further Investigation In order to confirm Coulomb s law, we still need to confirm that the force is proportional to the charge on the sphere. Set the power-supply to a lower voltage (e.g., 1kV) and repeat the measurement of force. Is the force reduced by the factor that you would expect? Appendix: Reading the scale on the vertically adjustable stand. Bibliography H. D. Young and R. A. Freedman, University Physics, 13 th Ed., Pearson Addison- Wesley, San Francisco (212).