Work on your kitchen table with the forces that hold matter together
The air in Denver is quite dry, making it a good location for electrostatics experiments. Indeed, Nikola Tesla set up shop near here at Colorado Springs for his high-voltage work. Lots of electricity can be picked up from carpets and cats in this area. In view of this, I have decided to try some experiments with static electricity, and report on them here, giving explanations for readers not up to date on electricity. The basic properties you need to engage seriously in electrostatics are shown in the photograph at the right. The Wimshurst machine, which you surely recoginize, is a source of static electricity, and this one easily gives more than 60,000 V, making a two-centimetre spark with only a few turns. To its right is a gold-leaf electroscope, a sensitive detector of static electricity. A very fragile gold leaf stands away from the post on which it is suspended when there is a voltage difference between the central pillar and the grounded case. The case has two glass sides to view the state of the gold leaf, whose position can be calibrated relative to a plastic quadrant. On top of the electroscope is a metallic cup, to facilitate charge transfer to it. In front of these two items is a metal ball on an insulating handle, used to transfer small amounts of charge. The Wimshurst machine and the electroscope can be purchased from the source in the References. They were made in China, which deserves our deep gratitude for making these things available to the impoverished experimenter at a price that pleases everyone. The metal ball on the insulating handle you can make yourself. Cut the head off of a machine screw that threads into a suitable ball (one is furnished with the electroscope) and cement or thread it into a plastic (lucite) cylinder of the desired size.
Experimental electrostatics is the only activity that can give the student a concrete feel for the forces that hold matter together. In no other place will the student actually see the direct effects of electrical forces, and how they can be analyzed and used. It takes so little time and effort, but is abundantly profitable in insights. Only something this valuable could be allowed to vanish completely from education, as geometry has already done. Experimental electrostatics is a great deal of fun, besides. All the experiments are clean and safe, with the additional thrill of slight shocks now and then when you forget Leyden jars are charged.
I'll avoid mathematics here as much as possible, but some is still necessary for an understanding of Coulomb's Law and Gauss's Law, and for the definition of the electric field and potential. A great deal of mathematical analysis can be done with electrostatics, in the theory of potential and other branches, but it is much more easily found in the literature than information on experimental electrostatics. More observations may be added to this article from time to time.
The word "electricity" is from Greek hlektron, a word for "amber," but also for the natural alloy of gold and silver now called "electrum." When rubbed with cloth, amber attracts small, light bits of material, and often repels them again. Other insulating materials also behave this way, and the curious phenomenon was known from ancient times. However, it was far less important than magnetism, whose attractions are much stronger and more permanent.
William Gilbert (1540-1603) is best known for his study of terrrestrial magnetism, but he also described electric attractions and distinguished them from magnetic attractions. Otto von Guericke of Magdeburg (1602-1686) made a frictional electric machine with a revolving ball of sulphur, with which he performed a number of demonstrations. From this time on, electricity was produced mainly by similar machines depending on frictional electricity. Francis Hauksbee (1666-1713) of the Royal Society replaced the sulphur globe with a hollow glass cylinder, which was more rugged. When the glass cylinder was evacuated, he noticed sparks and glows in it caused by electrical discharges (movement of charge through gases).
Stephen Gray (1666-1736) distinguished insulators ("electrics") and conductors ("nonelectrics") in 1720, and showed how electricity could be conducted along wires. C. F. de Cisternay duFay (1698-1739) distinguished positive ("vitreous") and negative ("resinous") electricity, produced by rubbing glass or hard resin, respectively, and that electricities of the same kind repelled each other, while electricities of the different kinds attracted.
Andreas Cunaeus (1712-1788) and Pieter van Musschenbroek (1692-1791) of Leyden discovered the Leyden Jar, or condenser, while trying to bottle electricity around 1745. Cuneus held a bottle containing water in his hand (one electrode), while charging the water (the other electrode) by touching a rod immersed in it and coming through the cork by an electrical machine. A considerable charge could be stored with this device, which greatly facilitated electrical experiments. The hand and water were later replaced by foil coatings on the inside and outside of the bottle. A small Leyden jar, 150 mm high and 50 mm in diameter, with a bottle 3 mm thick, has a capacitance of around 200 pF.
Benjamin Franklin (1706-1790) showed that atmospheric electricity was the same as common electricity, and invented the lightning rod to discharge buildings and their vicinities, making a lightning stroke less probable. He argued that electricity was of one kind, and that exesses and deficiencies were responsible for the effects noted by duFay. Joseph Priestley (1733-1804) met Franklin in London in 1766 and was encouraged by him to study electricity. Priestley found that if an electrified body was touched to the inside of a conducting cup, all its charge would be communicated to the cup and appear on its outside. This is a result of the inverse-square nature of the electrical force, like gravity, and suggested that this was the key to electricity.
Alessandro Volta (1745-1827) discovered the electrophorus in 1775. A resinous plate, electrified by rubbing, induced charges of opposite signs on a conducting disc laid on it. One sign of charge could be removed by touching it with a grounded body (a finger), and the disc would then retain a charge opposite to that of the resin when it was lifted. An unlimited amount of electricity could be obtained in this way without friction. The Wimshurst machine mechanizes this procedure, called induction.
Henry Cavendish (1731-1810) perfected the suggestions of Franklin and Priestley with his ingenious concentric sphere experiment in 1772. The inner of two concentric conducting spheres was first charged, then connected to the inner surface of the outer sphere. On then testing the inner sphere for charge, it was found to be uncharged. This showed that the exponent in the law of force differed by less than 0.02 from 2. Cavendish's work was nearly forgotten until resurrected by James Clark-Maxwell nearly a hundred years later. The experiment was redone and brought the limit even closer to zero. Modern experiments have refined the measurement even further. In modern terms, this is equivalent to measuring the "mass of the photon," which on theoretical grounds should be zero.
Charles Augustin de Coulomb (1736-1806), a very ingenious and prolific French engineer, used the torsion balance (invented earlier in England by Michell, where it was used by Cavendish to determine the gravitational constant) in 1777 to show that the force between electrical charges had the same inverse-square dependence as gravitational forces. He quantified the concept of electrical charge and the unit of charge is named in his honor. Coulomb's measurement of the inverse-square law is much less sensitive than Cavendish's, but is easier to understand without mathematics.
Volta, who invented the electrophorus, made the momentous discovery of the chemical battery for producing current electricity in 1800. It was soon shown that this electricity was identical to that produced by electrical machines, but at a much lower pressure and in much greater quantity. It was now necessary to distinguish between static electricity and current electricity, because the experimental arrangements were almost completely different. The connection of electricity and magnetism was soon established using current electricity, and this connection has been of primary importance in the technical applications of electricity, while it plays no role in static electricity.
Examine the Wimshurst machine, noting the method of driving the wheels at the same speed in opposite directions. Although it may appear old and antiquated (one was shown in the PBS program on Benjamin Franklin), it is not. It was invented in 1883 by a British engineer, James Wimshurst, and uses induction, not friction, to generate electricity. The one in the picture has 9" discs, and so is quite small, but can still create a potential of 100 kV. The sparking potential in air is about 30 kV/cm, and I regularly test the machine with 2 cm sparks, corresponding to 60 kV. A large Wimshurst machine may have 36" discs and generate corresponding larger potentials. This machine is the most convenient way to obtain the high voltages for the usual classroom electrostatic demonstrations and experiments, and is safe to use. In modern high-voltage testing, alternating currents are used instead, which are easily transformed to high voltages by transformer action (Tesla coil, Ruhmkoff coil). Since the golden age of electrostatic experimentation was the 17th century, these experimenters and the Wimshurst machine have become confused in the minds of the public. There are other induction generators, such as Kelvin's water dropper, as well as Armstrong's steam electrostatic generator, but these are not as handy as the Wimshurst machine. For much higher voltages, the van de Graaff and Cockroft-Walton generators also produce DC static electricity. Small van de Graaff generators can be purchased (see References), but they are not cheap.
Each wheel has 20 conducting sectors made from a metallic paint, 45 mm long, 10 mm wide at the centre, separated by 9 mm on the outer faces of the discs. At different places on the two discs, brushes at the ends of a shorting rod connect diametrically opposite sectors momentarily. If we consider one disc, then the shorted sector is oppposite, say, a negatively charged sector on the other disc. A (+) charge is induced on the inner side, while a (-) charge appears on the outer side, under the brush. At the diametrically opposite sector, the polarities are reversed, so the (+) charge appears on the outer side under the other brush. The charges on the outer sides are neutralized as a brief current flows, leaving the upper sector with a greater (+) charge, and the lower with a greater (-) charge. The induction of the charges does not depend on how much charge is already there, and equal signs of charge will not neutralize. The sectors charge to higher and higher potentials each time they pass beneath a brush, and their higher potentials serve to induce even greater charges.
Two Leyden jars are provided to store the charge. These are cylinders with an outer grounded metallic coating and an inner metallic coating connected with the electrodes that are supported at the centres. They are, of course, just capacitors that can stand a very high voltage. Each jar is connected with a rod that places sharp points near the moving sectors of the wheels. The sectors that approach from opposite directions on the two discs are always charged similarly but oppositely on the two sides of the disc. They have been separated from the opposite charges by the distance travelled since they last passed under the shorting bars, and so work has been done on them, and they have been raised to a high potential. An invisible corona discharge caused by the high potential gradient at the sharp points sprays charge to and from the sectors, connecting them to the Leyden jars, in effect.
The charged Leyden jars, one charged positively, the other negatively, can give a powerful shock if they are touched. However, since there is very little charge stored, a dangerous current cannot be produced. The Leyden jars hold charge very well here in Denver, and can still give a good shock hours after they have been charged. If you work with them, this will become obvious. If you want to see if you can make your hair stand on end, be certain that you are NOT grounded, even slightly. The girls with their hands on a van de Graaff generator electrode are well isolated. If grounded, they would receive a dangerous shock. A Tesla coil uses high frequencies that, due to the skin effect, pass over the surface of the skin and do not cause an internal shock.
Take a small transistor radio and set it to the AM band. When you operate the Wimshurst machine, you will hear a rather loud roar or hiss, that stops when you stop turning, and starts again when you start cranking again. This comes from the corona discharge at the pickup electrodes. Now switch to the FM band. You won't hear the corona, but every time a spark flashes, a loud crack will be heard in the receiver. The spark is an oscillatory discharge, with the current rapidly reversing at a high frequency determined by the capacity of the Leyden jars and the inductance of the connecting wire, and the RF produced is picked up by the radio. After a few sparks, you will begin to smell a whiff of ozone. The radio term "static" refers to static electricity, that can produce noises like we have just made. Most radio static comes from lightning, often from distant storms.
When the Leyden jars are charged (before they spark; widen the distance between the electrodes), touch one with the ball on an insulating handle, and then bring the ball inside the cup of the electroscope, but do not touch the inside surface. Watch the gold foil as you insert and remove the ball. Now put the ball in the cup and touch the inside surface of it. You can remove the ball, and the electroscope will remain charged. Discharge the electroscope with your hand until the foil hangs limply. Now see if the ball you previously used to charge it is still charged (of course, it should not have touched anything). If the ball still has any charge, it will be a small one. Most of the charge originally on the ball was transferred to the electroscope.
Coulomb's Law for the force between point electrical charges is f = qq'/r2, where f is the force in dynes, r the distance between the charges in cm, and q,q' are the amounts of charge in esu (electrostatic units) or statcoulombs. These units are very convenient for electrostatic experiments, and are simple to use, so they will be used here. In SI units, a factor 4πεo = 1.11 x 10-10 F/m appears in the denominator, f is in newton, r is in metres, and q,q' are in coulomb. From these figures, 3 x 109 esu = 1 C. The direction of the force is along the line joining the charges, attractive if qq' < 0, repulsive otherwise. It should be appreciated that this law was unknown to early experimenters in the 18th century and before. It can be applied to distributions of charge by using integration. It is the basis of all electrostatics, which is fundamentally about force.
The integration with respect to distance of the component of force in the direction of the path is work, and represents a change in energy. Therefore, W = -∫(qq'/r2) dr = qq'/r + const. If we consider q' as fixed and of unit magnitude, then W = q/r + const. The minus sign was chosen in the integral for work so that W would correspond to the work done on the charge q, which would be positive if r increased from a smaller to a larger value. If the constant is further taken as zero, then W = q/r. This means that W is positive if q is brought from ∞ to r, against a repulsive force, so that this amount of work must be done on the charge. W is seen to be proportional to the charge q. The work per unit charge is called the electrostatic potential V, which is 1/r for a unit charge. The potential of a charge q' will then be V = q'/r. The dimensions of V are seen to be either erg/esu or esu/cm, which are equivalent. From Coulomb's law, dyne = esu2/cm, so we can see that these units are indeed equivalent.
The erg/esu is called a statvolt. The equivalent SI unit, measured in J/C, was named a volt in honor of Volta. Since 1 J = 107 erg (a joule is a newton-metre, while a newton is 105 dynes and a metre is 100 cm), and 1 esu = 3.33 x 10-8 C or 1 C = 3 x 10 esu, we find that 300 V = 1 statvolt. Field strength in SI is in V/m, so 300 V/m = 0.01 esu/cm, or 1 esu/cm = 30,000 V/m = 30 kV/m.
The electrostatic potential function V(x,y,z) can be differentiated with respect to distance to give the electric field. The three vector components of E are found by the derivatives in the x,y and z directions. It is much easier to visualize and represent the function V(x,y,z) than the three components of the electric field. Much of the theory of electrostatics is in terms of the potential V. The surfaces V = constant are called equipotential surfaces, and the electric field is perpendicular to them, pointing in the direction of less positive values. Electrostatic fields can be visualized by imagining the equipotential surfaces, or even better by imagining the lines of force perpendicular to them. Lines of force will be discussed at more length below. The force on any charge can be derived from the potential due to all other charges, taking into account all influences of the charge on which the force is desired.
To demonstrate the reality of electrostatic forces, we can experiment with light spheres suspended by threads. Hobby stores sell 1/2" cork balls that will serve very well, and thin metal leaves used for gilding. This metal is not gold, but looks like it, and is conducting and very light, so it will serve our purposes. We will also need some thin adhesive, and the adhesive sold with the metal leaf will do. The cork balls can be pierced with a normal sewing needle. Press the needle halfway through before threading it and pulling the needle the rest of the way with pliers. Tie a knot at the end of the thread, and pull it up to the cork. Now coat the cork lightly with adhesive, and wrap it with a piece of metal leaf. Excess leaf is easily removed. A mess can be made of this, but there is enough metal leaf in the package sold to gild a hundred cork balls, so if you don't succeed at first, try again. It is necessary for the cork balls to be gilded, since they are nonconducting and can hold various charges at various places that will give exceedingly annoying behavior. I used a 2-m piece of thread between two balls that can be suspended from a rod notched to hold the thread in place. This will be found to be a very annoying device, with the strings winding about each other and so forth. Nevertheless, with care it can be brought to a neat state with the balls just touching. At times, we may want to work with a single ball, which is easily done.
Consider a conducting ball of radius a. When it is given a charge q, the charge distributes itself as a result of the repulsive force uniformly over the surface of the sphere (if the spherical surface is uniform, of course). The effect at a distance is the same as if the charge were concentrated at the centre of the sphere, a result taken over from the theory of gravity and proved by Newton in the Principia. Therefore, the potential at the surface of the sphere must be V = q/a, where q is the charge on the sphere. The ratio of charge to potential difference is called the capacitance, so the capacitance of a sphere is C = q/V = a, the radius in cm. Since esu/statvolt = esu2/erg, and farad = coul/J, we find 1 cm = 1.11 x 10-12 F, or 1.11 pF. A 1/2" ball has a radius of about 0.88 cm, which is its capacitance in esu. When charged to 10,000 V = 3.3 statvolt, its charge will be 2.9 esu. The earth is a conducting sphere of radius about 6.37 x 10 cm, so its capacitance is about 0.0007 F. A charge of 1000 C would raise the potential of the earth by 0.7 V.
The force on a charged sphere can be measured by suspending it as a pendulum, with a thread of length s. In the diagram, the force f deflects the sphere to the right, and the tension T in the thread is balanced by the sum of the force f and the weight of the sphere, mg. The cork balls weighed 167 mg each; let's say 200 mg with the metal leaf coating. Their weight is then 0.2 x 981 = 196 dynes. If s = 100 cm, then a deflection of 1 cm is a tangent of 0.01, so with this deflection the force would be about 2 dynes. Quite small forces can be measured in this way. We are making the approximation that the sine, x/s, is nearly equal to the tangent. For x/s = 0.1, the tangent is 1.005, so the error is only 1/2%.
Arrange two metallized balls so that they hang at the same height and in contact. Charge the transfer ball by means of the electric machine. Grasp the threads carefully a little ways above the balls, so that they are held in contact, and move the transfer ball so it contacts both balls and shares its charge with them. Then release the balls; with luck, they will not move around too much, and will soon settle down with a certain distance between them. I can get them to separate as much as 6 cm with care, but several cm is easy. With s = 100 cm, a deflection of 3 cm of 200 mg balls means (0.2)(981)(3)/(100) = 5.9 dyne. Assuming the charges on the balls equal (they are the same radius and will share charge equally) we have q2 = (5.9)(6.0)2, or q = 14.6 esu. The balls are charged to 14.6 / 0.63 = 23 statvolt or 6900 V.
To determine the sign of the charge, we can note the direction of deflection in an electric field of known strength. We define the field strength as the force per unit charge, or dyne/esu. Multiplying by 1 = cm/cm, the field strength becomes dyne-cm/esu-cm = erg/esu-cm = statvolt/cm. The force on a charge q esu in a field E statvolt/cm is then f = qE. I have a DC power supply that I can crank up to 450 V, or 1.5 statvolt. If I put two electrodes 5 cm apart, the potential gradient will be 1.5 statvolt per 5 cm, or 0.5 statvolt/cm, which is the average electric field strength between the electrodes. A ball with a charge of 15 esu will experience a force of 7.5 dynes, which will cause a deflection not only easily seen, but even capable of measurement. The sign of the charge on the ball can be deduced from the direction it deflects. If toward the + electrode, the charge is negative; otherwise, it is positive. There is an easier way to find the sign of the charge, as we shall see in the next section.
The electroscope is an essential tool in electrostatics experiments. It can be used not only to detect charge, but to give a rough indication of potential and to determine the sign of a charge. A metal-leaf electrometer is shown at the left, and one can easily be constructed. These are called gold-leaf electrometers because the original leaves were gold foil, still an excellent choice if you have it. Much less expensive metal leaf is available at hobby stores, and will do just as well for our purposes. It comes in gold color, so it even looks the same even though it may tarnish. The electroscope is made in a 250 ml Erlenmeyer flask. Pierce a hole through the No. 13 cork so it fits the 3/32" or #12 AWG uninsulated copper wire snugly. A No. 6 rubber stopper would do as well, since I could not measure any leakage across one, but it is harder to make the hole. After pushing the wire through the stopper, bend the wire to make a horizontal section at the lower end, and a horizontal circle at the upper end. Any sort of small metal can can be used as the receiver. I have my eye on a small wasabi-ko can, but I shall have to eat a lot of sushi before it becomes available. Clean the bottom of the can to bare metal, and tack it to the wire with solder. Meanwhile, cover the outside bottom of the flask with metal leaf, following the instructions that come with the metal leaf, using an adhesive. This step can actually be omitted, since it will have little effect on the operation of the electroscope if held in the hand or placed on a grounded conducting mat. Do not use a fixative, since the leaf must remain conducting. Then, carefully hang a strip of metal leaf, perhaps 8 mm wide by 60 mm long, over the place for it at the bottom of the wire. This is much easier said than done, but can be accomplished eventually, and you will have lots of metal leaf. Then carefully put the cork in the flask, and the electroscope is complete.
To use the electroscope, rest it on a grounded conducting mat (such as is used with integrated circuits) or a metal plate, or just hold it in your hand. I used the electroscopes I purchased from ScienceLab.com for my experiments. I repaired the one that arrived damaged, so now I have two. The first thing you can do with an electroscope is to investigate frictional electrification. I found a silk tie a good cloth to rub with. A black hard-rubber (ebonite) comb was easy to electrify. A piece of glass tubing proved impossible to electrify to the slightest degree, even using different fabrics, including wool and cotton flannel. However, a small glass tumbler electrified readily. The comb should provide me with resinous electricity, the glass tumbler with vitreous. Surely enough, if the electroscope was electrified with the comb, approaching it with the rubbed tumbler discharged it. The same thing happened when the tumbler first charged the electroscope, and the electrified comb was brought close. The leaf collapsed, then expanded again as the comb charge predominated. Quite clearly, there are two kinds of electrification, as duFay asserted.
When a body is electrified by friction, it is not the friction so much as intimate contact of two different materials that allows the exchange of charge. Electrons will stay on the object with the greater affinity for them, which will become negative, and the other object will become positive. The charges are equal and opposite. It is hard to prove equality without special apparatus, but the opposite nature is easily proved. A list of materials can be made, in order of electron affinity, to predict what will happen when one is rubbed on another.
When electrified combs and tumblers are used to influence the electroscope, it is clear that only a small part of the charge at most stays on the electroscope when the comb or tumbler is taken away. If you use a charged metal ball instead, and touch it inside the receiver, the electroscope will become highly charged, while the ball will be found to be uncharged. This demonstrates the difference between "electrics" where the charges cannot move, and the "nonelectrics" where they can readily move, as Stephen Gray found.
The electroscope is a capacitor, with the central wire and metal leaf as one electrode, and the metallic coating as the other one. It should be most sensitive when the metallic coating is grounded. The voltage on the electroscope is V = Q/C, and may be something like 50 pF, though of course this varies greatly with construction. I have so far not been able to detect any difference in the operation of grounded and ungrounded electroscopes.
My electrometer is deflected to the first graduation when connected across a 450 VDC power supply. In this way, the electrometer can be given a charge of known sign. It is then easy to confirm that resinous electricity is negative, and vitreous is positive, according to the modern convention. When I face the Wimshurst machine on the crank side, the Leyden jar on the left is positive, and the one on the right is negative.
To find the sign of a charge, then, transfer some of the charge to an electroscope; it should give about 50% deflection. Now rub glass with a silk cloth (or similar fabric). This creates a vitreous, or positive, charge on the glass rod. Bring the glass rod to the electroscope. If the deflection decreases, then the vitreous electricity has neutralized some of the charge, which must then be negative. On the other hand, if the deflection increases, then the charge is the same as vitreous electricity, or positive. The results of this test can be compared with the results of the electric field test to see if they agree on the sign of the charge. The same thing can be done with the negative resinous electricity of the comb. Use the cup electrode for these tests.
A proof plane is a small conducting disc on an insulating handle. I made one in a few minutes by epoxying a one-cent piece to the end of a 5 mm x 150 mm plastic rod. If you place the proof plane against a charged conductor, the proof plane will pick up exactly the charge that occupied the area that it covers, and this charge will go with it when the proof plane is removed. Now the proof plane can be touched to the inside of an electrometer cup to receive an indication of how much charge was there. I charged a conducting cup from the Wimshurst machine. Then I put the proof plane against the outside of the cup, and brought it to the electrometer. A small amount of charge registered. I discharged the electroscope and proof plane by touching them, and then put the proof plane against the inside of the charged cup. When brought to the electrometer, no charge was evident. As expected, this shows that all the charge on the cup is on the outside. More careful work can be done, but this illustrates the principle of using a proof plane.
Another experiment with an electroscope is shown at the left. I made a capacitor from two 5" x 7" galvanized steel plates that are available at low cost at builder's supply stores. Aluminium could also be used, but at greater cost. One plate was secured to the post of my electroscope (I used the ScienceLab electroscope, not one like that shown in the figure) by drilling a 6 mm hole in the centre and using an M5 x 0.8 nut and lock washer. An aperture about 15 mm square was nibbled in the other plate so it could lie flat on the other. A piece of paper to serve as dielectric was glued to the upper plate, and two corners were turned up to allow it to be grasped, and contact to be made with the lower plate.
The upper plate and electroscope body were grounded. A DC power supply was used to charge the electroscope. I used 100 V, but perhaps even less would do. The electroscope will deflect little if at all at this point. I then disconnected the power supply, and carefully lifted the upper plate off the lower. As I did so, the electroscope was strongly deflected. Incidentally, if the paper was not glued to the upper plate, and was allowed to lie where it was, very little deflection would be noted. Most of the charge on the upper plate was actually on the paper.
Assuming the dielectric constant of paper is 3, the capacitance calculated from C = κA/4πd, with A = 226 cm2 and d = 0.015 cm is 3600 cm. Measurement with a capacitance meter showed this to be a good estimate. The charge stored is Q = CV, or 1200 esu, with V = 1/3 esu. This is a considerable charge. When the upper plate is lifted off, the charge on the lower plate must remain constant, while the capacitance of the electroscope is greatly reduced, say by a factor of 100 to 36 cm. However, Q = CV = C'V' = constant, so V' = 100 V or 33.3 esu, 100,000V. A potential of this amount will cause a large deflection of the electroscope. We see, incidentally, that it is potential that the electroscope measures, not the amount of charge.
A capacitance meter measured the capacity of the electroscope with upper plate on at 3500 pF, and 18 pF with the plate off. The capacity depends very sensitively on the plate separation. It is not easy to keep this uniform. With the upper plate resting only loosely on the lower, the capacitance may be no higher than 500 pf. The effect, however, is still very evident. When using a capacitance meter, subtract the reading with nothing attached from the observed reading.
Dielectrics and capacitors have not been discussed in this article, but the reader is probably well enough acquainted with them to make sense of this experiment.
We have noticed that when a charged conducting ball is touched to the inside of the receiver of the electroscope, all of its charge is transferred to the electroscope and the ball becomes uncharged. Before the ball has been touched to the receiver, moving it around has no effect on the electroscope charge. These and other observations are a consequence of the fundamental inverse-square nature of the electric force, as we shall now explain. The subject is called Faraday's Ice Pail Experiment because Faraday used a metal ice bucket as the receiver for the electroscope. Done with care, it is Cavendish's Experiment, that confirms the inverse-square dependence with precision.
The electric field in esu, statvolt/cm, around a point charge q is E = q/r2, where q is the charge in statcoulombs and r the distance in cm. directed outwards from a positive charge and inwards to a negative charge. This means that the force on a charge q' is f = q'E at any point, provided that the charge q is not disturbed. The surface area of a sphere of radius r is 4πr2, so the product of this area and the electric field normal to it (which in this case is the same everywhere, except for direction), is 4πr2(q/r2) = 4πq. Since r has dropped out, this is true for any distance r. The product of the area and the field normal to it is called the electric flux by analogy with fluid flow, where the product of area and normal velocity is the volume rate of flow.
Faraday envisioned the field as described by lines of force, like the streamlines of fluid flow. There are no physical lines, of course, but they give a graphic picture of the field and can take the place of mathematical analysis. We use as many lines as we need to give a good representation, with the number of lines leaving a positive charge or arriving at a negative charge proportional to the charge. We could say that a unit charge gives one line of force in each steradian of solid angle, or 4π lines in all. This makes it easy to conceive that through any closed surface surrounding a charge q there is a flux of 4π lines. We saw that above for the special case of a spherical surface, where it is easy to calculate the flux. We can, of course, prove the general case mathematically. This is the result known as Gauss's Law: through any closed surface surrounding a charge q the flux of the electric field is 4πq.
Lines of force begin on positive charge, and end on negative charge of the same absolute value. The density of lines, which is the number that cross any normal area, is proportional to the electric field. More crowded lines mean a higher field. Lines of force tend to repel each other, and exert a tension in their direction. Remember that lines can always be subdivided to any fineness necessary, but the tensions and the associated charges subdivide the same way. Lines of force is simply a way to visualize the electric field, but Maxwell gave it a precise mathematical meaning.
Gauss's Law is sometimes called Gauss's Theorem, but this term is best reserved for the mathematical result that the surface integral of the outward normal derivative of a function over a closed surface is equal to the volume integral of the divergence of the function over the volume enclosed by the surface. This result is also called the divergence theorem, and can be used in a mathematical proof of Gauss's Law. Gauss's Law is equivalent to Coulomb's Law but more general, and is one of Maxwell's equations of electrodynamics. In its magnetic form, that the net flux of the magnetic field across a closed surface is zero, it is a second of Maxwell's equations. Electrostatics is based on Gauss's Law, and on the law that the line integral of the electrostatic field around any closed curve is zero, which is a form of conservation of energy, and which we tacitly assume in this article. Experimental electrostatics gives concrete examples of the meaning of Gauss's Law.
Now we can explain how to charge a conducting cup. In the diagram, we have a closed surface with an aperture for access to the inside. Ideally, the surface should be completely closed, but as a practical matter an access port makes very little difference to the result. A small transfer ball is charged with a charge +q, which spreads uniformy over it due to the mutual repulsion of the elements of charge. When we put the charged ball inside the the cup, and approach the inner surface, the positive charge on the ball becomes distorted as it attracts negative charge from the cup. The positive charge liberated then spreads itself uniformly over the outer surface of the cup. Actually, of course, only the mobile negative charge of electrons moves, uncovering positive charge as it does so. The result is exactly the same as if the positive charge could actually move, although it does not. A Gaussian surface around the charged ball will still show a charge +q, a Gaussian surface enclosing the cup inside and outside (two spherical surfaces) will still show charge 0 as it did before the ball was introduced, and a Gaussian surface outside everything will show a charge +q. The charge +q on the outer surface arranges itself uniformly, while the equal and opposite charge -q comes as close as it can to the charge of the ball.
When the ball touches the inner surface of the cup, the positive charge +q is neutralized with a negative charge -q from the inside of the cup, leaving a uniformly distributed charge +q on the outer surface. The transfer ball is now uncharged, and can be removed from the cup. The charge +q on the cup has been uniform since the ball was introduced, and is absolutely independent of the position of the ball within the cup, so long as an equal and opposite charge is induced on the inner surface. The charge +q acts as if it were concentrated at the point P, the centre of the spherical cup. This operation can obviously be repeated as often as necessary, adding a charge +q to the charge already there. This is how the receiver of a van de Graaff generator becomes charged to a high voltage as the result of the addition of many small charges carried into the receiver on the belt.
The inside of a closed conducting surface becomes an equipotential volume. If the surface is grounded, the potential is zero by convention. The surface does not have to be solid, but can be a conducting mesh or screen, and it will work just as well. The lines of force from any charge within the surface will terminate on the inner walls of the surface, and the lines of force from any external charge will terminate on the outer walls. To prove this, consider a Gaussian surface entirely within the closed conductor. The flux across it is zero, and so the net charge inside it must be zero. Therefore, the electrostatic fields inside and outside will be completely independent. Such a closed surface is called a Faraday Cage. Delicate electrical experiments are usually done inside a Faraday Cage. A Faraday Cage will not protect against magnetic fields, however.
The electrophorus was discovered by Alessandro Volta in 1775, as mentioned above. It is a very simple device, but the basis for induction machines that produce static electricity much more copiously than can be done by friction. I'll improvise an electrophorus from a comb and the transfer ball, since I lack a full-scale one at present. The comb is first electrified by rubbing with silk, and is laid on the desk. I take the transfer ball and lay it on the comb, and touch the upper part of the ball briefly with a finger. Then I bring the ball to the cup of the electroscope, and transfer its charge. It will not be much, but it will definitely be something. Then I bring the ball back to the comb, and repeat the operation. The electroscope receives additional charge. In a short time, I can fully charge the electroscope by carrying the ball back and forth only a few times. The charge, of course, will be positive, opposite to that of the comb, as you can easily demonstrate.
A proper electrophorus consists of a resin plate, and a metallic disk with an insulating handle, as shown in the diagram at the left. The resin is first electrified by friction, as the comb was in the improvised electrophorus. Then the plate is laid on the charged resin. It does not discharge the resin, since the charge is quite fixed to the resin and does not move. If this were any problem, a thin insulating film could be laid on the resin, but this is unnecessary. The upper surface of the plate is touched with a finger or a grounded wire, draining off charge of the same sign as the charge on the resin. Then the plate is lifted, and will be found to be charged with the opposite sign of charge (positive in most cases). This can be repeated as often as desired, until the charge on the resin is depleted, when a few rubs will bring it up to snuff. The electrostatic potential of the charge on the plate is raised by separating it from the resin; this requires work to be done against electrical forces (the attraction of the charges), as in all cases when we gain electrical energy. We also have a graphic demonstration of the different behaviors of electricity on insulators and conductors.
Note that the induced charges of opposite sign on Faraday's ice pail or the electrophorus plate are separated by metal, in which there is no electric field. Any electric field in a metal is instantly neutralized by electric currents that put charge where it is needed to bring the field to zero. These charges do not interact directly, but tend to move as far apart as possible to reduce the field as much as possible. That is the reason that the positive charges on the top of the electrophorus plate run down the wire to ground so readily, instead of being attracted to the negative charges on the other side of the plate. Those charges are fully in love with the charges on the resin. What both charges are trying to do as best they can is to make the electric field inside the metal of the plate plus wire equal to zero. This is another consequence of Gauss's Law, that there can be no charges inside a conductor because there can be no fields there, and so the flux through any closed surface will vanish.
I speak of positive and negative charges as moving this way or that, although I am thoroughly aware that only the negative charges move in (most) metals. In semiconductors, both positive and negative charges move, as they do in ionic solutions. Whatever the actual situation, motion of positive and negative charges in opposite directions are perfectly equivalent so far as electric currents and effects are concerned. When the electrophorus is grounded, electrons actually rush up from ground to neutralize the positive charges, but this is no different from the postive charges moving to ground. The essential thing is that electric charge is an algebraic quantity, unlike mass.
For this experiment, you need a candle in a candlestick, and a short piece of plain wire. I used #22 AWG wire, one end of which was wrapped between the small and large ball on the wand connected to the negative Leyden jar of the Wimshurst machine. The other end projected horizontally. The lighted candle was placed with its flame close to the wire. When the crank was turned, the flame appeared to be blown away from the wire, as if by a breeze. Stopping and starting turning verified that the effect was connected with the electrification, not a random breeze. Also, the Leyden jar was quickly discharged, unlike the usual case where it remains charged for a long while if not disturbed. The effect was much less, if observed at all, at the positive Leyden jar. This is another way to determine the polarity of the Wimshurst machine.
In the dark, a small bluish brush can be seen at the end of the wire. This is the visible part of a corona discharge, a high-pressure glow discharge often found around highly charged electrodes or wires in air. In particular, this is a negative DC point corona discharge. There are positive discharges as well, but they are somewhat different. The high field near the point accelerates random electrons until they can form positive ions by collision with atmospheric gases. The positive ions are strongly attracted to the negative electrode, and on striking it knock out electrons. These electrons are accelerated in turns through the sharp potential rise near the point, create more positive ions, which then make more electrons cooperatively. In this way, a glow discharge cathode spot is formed, which emits electrons copiously. These electrons move in the electric field away from the cathode, and form negative ions by collision with oxygen molecules. The negative oxygen ions are then also accelerated by the field, creating the wind as they transfer momentum to the predominant neutral molecules. This is a likely description of what goes on, but the processes are very complicated in detail.
Similar discharges may be seen at the end of lightning rods (though normally they are not luminous enough to be seen), or at the ends of the masts or yardarms of sailing ships, where they were called St. Elmo's Fire. This happens when the atmospheric electric field becomes very large due to the passage overhead of a cumulonimbus. The major charge in the lower part of a thunderhead is negative, so the discharge is usually a positive corona. However, sometimes there is a positive charge at the very base of the cloud, which may produce a negative corona. In Graeco-Roman times, St. Elmo's Fire was attributed to Castor and Pollux. St. Elmo was apparently St. Erasmus of the 4th century, patron saint of sailors, and many forms of his name have been used. It had superstitious significance as well, and was sometimes called Corposanto, which English sailors called Corbie's Aunt. It was generally a favorable omen, but a man whose face was illuminated by it was doomed to die within a day, some believed. Since relatively dry air is required, it usually signified the end of rainfall and stormy weather. It can also occur on aircraft flying in the strong electric fields around thunderstorms. Corona discharges usually produce radio noise, as we found in the First Experiments above. Corona is effective in transferring charge without actual conductive contact, of which the Wimshurst machine and the van de Graaff generator furnish examples.
The economy Wimshurst machine and the gold-leaf electroscope are available online from ScienceLab.com, 4338 Haven Glen Dr., Houston, TX 77339. The cost is $187.17 plus around $16 shipping. This is a reputable company that has given me good and honest service. Allow several weeks for delivery. I hope the packing of the electrometer has improved. ScienceLab promptly replaced one that arrived damaged beyond repair.
Wimshurst machines ($119.95), 30" van de Graaff generators ($424.95), and 50 kV Tesla coils ($259.95) are available online from Edmund. The picture of the Wimshurst machine shows the one I have, but the ad says "black wheel model" and I do not know what this means. A proof plane, which is a metal disc on an insulating handle, is also available. Edmund, of course, is also reputable.
R. C. Brown, A Textbook of Physics (London: Longmans, 1961). Chapters LVI and LVII. This is an encyclopedic sixth-form text in classical physics (1424 pages) that covers everything, including a good account of experimental electrostatics. You will find nothing on this in any general physics text, with or without calculus. It is little wonder students develop no understanding of fields and forces and charges, and treat physics as a collection of formulas. One university physics text, at least, presented a little account of forces resulting from particle exchange that the authors themselves probably didn't understand too well, and which is not a very good introduction to classical fields anyway.
Composed by J. B. Calvert
Created 12 April 2003
Last revised 10 August 2003