Coulomb's Law The torsion balance was invented by the Englishman John Mitchell in the mid 1700's. A torsion balance is a precision instrument which allows for the measurement of a very small force. A wire is used to hang a lever arm. The wire is uniform in diameter and composition. An object at the end of the lever arm is what is experiencing the force. The force experienced by the object causes the twisting of the wire. As the wire twists it is possible to measure the angle through which it twists. The twisting force on a wire is related to the angle through which it twists by a linear relationship, namely, twist = H6 where H is a constant of proportionality. By measuring the angle through which the wire twists, you can measure the force that is being experienced by the object. f^m ilnmh 1 In some ways the relationship between force and angle for the torsion balance is similar to the relationship between force and displacement found in Hookes' Law F = kx, where x is the displacement. In the case of the torsion balance, the angle 6 is the comparable to displacement in Hookes' Law. In 1785 Charles Coulomb borrowed the idea of the torsion balance from the english to do an experiment to determine the relationship between the electrostatic force on charged spheres, and the distance between the spheres. In his experiment Coulomb took two similarly charged spheres, one on a fixed rod and a second on the torsion balance-lever arm. By touching the two spheres, Coulomb was able to ensure the two spheres were equally charged. Pnnlnmh 9 Since like charges repel there is a force pushing the two spheres apart from each other. It is this repulsive force between similarly charged spheres which Coulomb measured using his torsion balance. MM a q tt flftMIg *mm By turning the knob at the top of the torsion arm he was able to change the distance between the fixed charged spheres. How the measurements were taken: > An angular scale near the knob allowed Coulomb to measure the angle through which the wire twisted. > A scale near the spheres allowed Coulomb to measure the distance between the spheres. rnnlnmh ^ Coulomb did two experiments: The first was to determine the relationship between the force and the distance, and the second was to determine how the charges on the two spheres affects the force. Experiment 1 : The Relationship between Force and Distance Consider a sample experiment with the following results (distance and angle are in arbitrary measurements): Distance Angle (9) 1.6 0.39 1.4 0.51 1.2 0.69 1.0 1.0 0.8 1.6 0.6 2.8 0.4 6.3 Analysis: 1 ) Graph the data with the distance as the independent variable and the angle as the dependent variable. 2) Use logarithms to find the functional dependence between distance and angle. 3) If force is proportional to the angle, what is the relationship between the force and the distance? Pm ilrxmh A Experiment 2: the Relationship between Charge and the Force Coulomb was not interested in an absolute measure of the charge on the spheres. By touching the two spheres to begin with, Coulomb was able to ensure the two spheres were equally charged. By grounding the sphere on the torsion arm from time to time he was able to divide the charge several times over so that if he started with a charge Q, on the first run he was comparing the force between two spheres with charge Vz Q, and then two spheres with charge % Q, and soon .... Consider a sample experiment with the following results (charge and angle are in arbitrary measurements): Charge Angle (9) 2.0 2.0 1.0 1.0 1/2 0.50 1/4 0.25 Analysis: 1 ) Graph the data with the charge as the independent variable and the angle as the dependent variable. Oni ilnmh R Coulomb's Law In the his first experiment Coulomb determined that the electrostatic force and the distance between the two charges are related by Foe 1 /d 2 The electrostatic force relationship is an INVERSE SQUARE LAW . In his second experiment Coulomb determined that electrostatic force and the charge is related by F <k Q! Q 2 where Qi is the charge on the first sphere and Q 2 is the charge on the second sphere. Combining these two relationships and using a constant of proportionality Coulomb arrived at his Law of Electrostatic Force F = K Q 1 Q 2 /d 2 In MKS units K = 8.998 x 10 9 Nm 2 /C 2 , where the unit of charge (C) is known as the Coulomb Cm ilnmh ft If two charges each of 1 C are one metre apart they would experience a repulsive force of 8.998 x 10 9 N. A force of 9 billion Newtons is an enormous force, equal to about 1 million tonnes. Charges we typically experience are around a microcoulomb (1 uC = 10~ 6 C) and so we experience electrostatic forces on the order of 1 Newton or less. The charge on an individual electron (e) is - 1 .602 x 10~ 19 C. The charge of a proton is equal in magnitude but opposite in sign. It is a yet to be explained fact that the charge of the electron and proton are equal yet opposite. 1 C of charge is equal to 1 19 electrons. In one mole of matter there are therefore on the order of 10 5 Coulomb of charge! If the charge in matter were not neutral the matter would fly apart with a considerable force. For instance, when a uranium atom splits during nuclear fission (the nucleus is no longer able to stick together) most of the energy that is released from the nuclei is a result of the electrostatic repulsion of the 92 protons of positive charge so close together within the nuclei. ' PSCB/physics 12/2004 nr\i ilnmh 7