This is a version of the apparatus that Charles Coulomb used to perform his classic experiment to determine the law of electrostatic repulsion (or attraction), which he did in 1785. It is a torsion balance, with one of the charged spheres on the arm of the balance, and one held by a sliding wooden stick set over a scale. An arm with a vane suspended in oil damps oscillations, and with a counterweight hung on the arm, the vane counterbalances the arm with the sphere. The shaft that carries these arms is suspended between two identical torsion fibers. At the top of the shaft is a mirror, which reflects the beam from a laser onto a meter stick that is mounted horizontally. When the spheres carry a charge of the same sign, they repel each other, causing the arm of the torsion balance to deflect, which you can observe by the position of the laser spot on the meter stick. The mount for the top end of the upper torsion fiber has a dial with a scale marked in degrees, which allows you to cancel the deflection and measure the angle through which you had to turn the end of the fiber to do so. In addition to the sphere on the arm of the torsion balance and the one on the sliding stick, there is a third sphere, which you can use to remove charge from the sphere on the sliding stick, and also to charge the two spheres initially.
The deflection angle gives a measure of the repulsive force between the spheres, since the torque, τ, equals kθ, where k is the torsional constant of the torsion fiber and θ is the angle. To return the sphere to its original position, since it is suspended between identical fibers, you must turn the angle dial through twice the angle of deflection. (Turning the dial through any given angle turns the center of the torsion pendulum through half that angle.) The scale on the sliding stick allows you to find the zero position and then subtract this from subsequent scale settings to obtain the distance between spheres. (You must, of course, include the diameter of the spheres to account for each sphere’s radius at zero separation.)
To perform this experiment so as to obtain a set of data from which you could find Coulomb’s law (the electrostatic force is proportional to q1q2/r2; Coulomb’s initial results showed merely that the force was proportional to 1/r2) would take too much time for a lecture demonstration, but you can certainly show in a few minutes the basic workings of this apparatus, and explain how one could use it to determine Coulomb’s law.
To start, note the position of the laser spot when the torsion balance is in equilibrium (zero torque). Carefully ground both spheres by touching them. Now you can charge the spheres in either of two ways. First, rub the plastic rod vigorously with the cloth. Then you can either stroke it on the stationary sphere, removing as much charge as you can, and then move the sphere towards the one on the torsion balance until they just touch, or you can carefully set the two spheres so that they touch and use the third sphere to charge them, by stroking the plastic stick on the third sphere, then touching it to the stationary sphere. In the first case, each sphere will carry half the initial charge that you put onto the first sphere. In the second case, each sphere will carry one-third the charge you put on the sphere that you used to charge them. The first method has the advantage that you start with greater charge on the spheres, but as you bring the spheres close, the uncharged sphere is attracted to the charged one by polarization, so it deflects towards it before it is repelled. You can get around this by having the uncharged spheres touch and noting the scale reading for the fixed sphere. Then if you slide the sphere away, charge it and then slowly touch the other sphere with it, you can set it at the original scale reading (though you will probably wish to start with some distance greater than the sphere diameter, i.e., with the spheres not touching). You can also charge each sphere separately, which will give greater charge on both spheres and make the experiment a bit easier to do, but you will not be able to be sure that each sphere carries the same charge. This should not matter, though.
However you choose to charge the spheres, the one suspended by the torsion fiber will deflect. Then, first grounding it by touching it, you can take the third sphere and carefully touch it to the sphere on the sliding stick. This removes half the charge on that sphere, reducing the electrostatic force and, thus, the deflection. The laser spot moves back towards its initial position. If you charge the spheres well enough at the beginning, you can repeat this operation once or twice, the motion of the laser spot showing the change in deflection. For at least the first two deflections, the change in position of the laser spot is large enough to be seen by the class. In doing this, you will not have kept the distance between the spheres constant. If you wish to do this, you should perform one set of measurements according to the procedure below.
To obtain Coulomb’s law without the proportionality constant, ground the two spheres by touching them, and let the suspended sphere relax to its equilibrium position (zero deflection), noting the position of the laser spot on the meter stick. Move the sphere on the sliding stick until the two spheres just touch, and note the scale reading. Now charge the spheres and move the stick back some distance from the initial setting. (Alternatively, you can set the sphere on the stick at the desired distance and then charge the spheres separately.) Turn the angle dial to return the suspended sphere to its zero position, noting the angle of deflection (which equals half the reading (vide supra)). You can do this for several distances and thus obtain the force vs. distance relation. Then, with the third sphere, you can remove half the charge on the sphere on the sliding stick and repeat the process. This gives you data for q2, q2/2, q2/4, etc., which yields the relationship between the force and the product of the charges. (If you’ve charged the spheres separately, the data will be for q1q2, q1q2/2, q1q2/4, etc.)
A second way to perform the experiment would be to begin the same way, but then instead of cancelling the electrostatic force by twisting the fiber to return the sphere to its zero position, note the scale reading for the deflection, then remove half the charge from the sphere on the sliding stick. Then, with the spheres discharged, you can turn the angle dial to reproduce the scale reading for each deflection, slide the stick until the balls touch, and then calculate (from the stick scale reading) the original distance. Repeating this for various distances will give essentially the same data as the first method.
To obtain the constant in Coulomb’s law, you would need to measure the moment of inertia of the pendulum, and then obtain the torsional constant by measuring the frequency of oscillation of the pendulum. In addition, you would need to know the amount of charge on the spheres. This would be difficult, if not impossible, to do if you charge the spheres via the charged rod as described above. To do this, you could use a high-voltage D.C. power supply to charge the spheres, or an electrostatic generator that achieves a known potential, then measure the charge on one of the charged spheres by means of a Faraday pail and electrometer.
One can deduce Coulomb’s law from Gauss’ law and symmetry considerations. In fact, the most sensitive tests for deviation of Coulomb’s law from inverse square (the deviation of the exponent on r from 2) have used apparatus based on Gauss’ law, such as two concentric conducting spheres, with an electrometer inside the inner sphere, a removable connection between them, a means of charging or grounding the spheres, and a conductive window through which to view the electrometer. (See, for example, the work of Plimpton and Lawton, and of Williams, Faller and Hill.)
Coulomb’s law stated in full is: F = (1/4πε0)q1q2/r2, where ε0 is the permittivity constant, which equals 8.854187818 × 10-12 C2/N·m2.
Because of the difficulty of making accurate electrostatic measurements, the SI unit of charge, the coulomb (C), is defined in relation to the unit of electric current, the ampere (A). If a steady current of one ampere is flowing through a wire, then one coulomb is the amount of charge passing through any cross section of that wire in one second, or 1 A = 1 C/s. The charge on a single electron is 1.602 × 10-19 C.
References:
1) Halliday, David and Resnick, Robert. Physics, Part Two, Third Edition (New York: John Wiley & Sons, 1977), pp. 568-9, 606-10.
2) Clarion, Geoffrey R. Coulomb’s law (instruction manual to PASCO Coulomb’s law apparatus).