Coulomb s Law Objectives To demonstrate the veracity of Coulomb's Law. To do this you will show that the Electrostatic Force between two charged bodies is directly proportional to the product of their charges. Apparatus Required Coloumb s law Appratus Assembly Power Supply Introduction Two charged objects always exert force on each other. The direction of such force depends on the nature of the charges whereas the magnitude of the force depends only on the magnitude of the product of the charges and separation between them. According to coulomb s law the magnitude of electrostatic force between two point charges q 1 and q 2 separated by distance r is Where, = 9 x 10 9 Nm 2 /C 2 is electrostatic force constant (or coulomb s constant) and ε 0 (=8.85 x 10-12 C 2 /(Nm 2 ) is permittivity of vacuum. Each isolated metallic sphere can be treated as a point charge. If two metallic spheres carry equal charge q and are separated by distance r the force between them is (1) (2) But we know for an isolated charged metallic sphere, the charge q on the sphere is directly proportional to the potential V of the sphere and the relationship between them is given by (3)
Where, a is the radius of the sphere. Torsion Pendulum: A torsion pendulum is a pendulum that rotates about the vertical axis of a wire. Instead of swinging like an ordinary pendulum it twists the wire and rotates about its axis. The twisting force reverses the direction of rotation, so the torsion pendulum oscillates slowly, clockwise and counter-clockwise. When a wire of torsion pendulum is twisted by an angle (θ) it generates a torque (τ) on the pendulum, where If "F" is the force on the torsion pendulum and "d" is the perpendicular distance of the force from the axis of rotation Since "d" is constant, After combining equations (2), (3) and (4) we obtain (4) (5) Exercise 1: Relationship between force (F) and charge (q) In this experiment you will keep the distance constant between two charged metallic spheres. The metallic spheres will be charged to different potentials (V) and you will measures the corresponding torsion angles ( ) to establish the relationship between (equivalent to force) and V (equivalent to charge) According to coulomb s law the force between two identical charged spheres placed at constant distance relative to each other is Imagine you don t know the Coulomb s law and you want to find the relationship between F and q when charges are placed at constant separation. Let s say the force varies to the n th power of the charge, then (6)
Taking natural log, ( ) ( ) (7) where, C is the proportionality constant. However, using equation (3) and (4) above equation can be written as ( ) ( ) (8) In this section of the experiment you will find the value of n by varying voltage (which is equivalent to charge) and measuring the corresponding torsion angle θ (which is equivalent to force) and compare it with the standard value from the Coulomb's law. (a) Figure 1 (b) Experimental Setup Procedure Note: The experimental set up will be ready for you, as shown in Fig 1. By reading the following points make sure everything is fine with your set up. If there is some problem please inform your instructor or lab manager. 1. The experimental set up is shown in figure (1 a). It has two segments one is torsion pendulum and another one is sliding arm. 2. Two metallic spheres each of radius 1.9 cm are provided. The sphere, one in the torsion pendulum must be at same height to the sphere at the sliding arm. 3. If they are not at same height talk to your instructor. 4. Make sure dial of the torsion pendulum at the top reads zero angle.
5. Make sure the torsion pendulum is well aligned with the index alignment as shown in figure (1 b).please look carefully in figure 1(b) when the torsion pendulum in at rest two lines, one from the pendulum and another from the index arm must coincide. 6. Make sure one of the copper rings is positioned well inside the magnet while the torsion pendulum is well aligned with the index alignment. 7. Position the slide arm so that the centimeter scale reads 3.8 cm (this distance is equal to the diameter of the spheres). 8. Position the spheres by loosening the thumbscrew on top of the rod that supports the sliding sphere and sliding the horizontal support rod through the hole in the vertical support rod until the two spheres just touch. 9. Tighten the thumbscrew You are now ready to perform the experiment. Let s make sure again the degree scale at the top should read zero, the torsion balance should be zeroed (the index lines should be aligned), the spheres should be at same height and just touching, and the centimeter scale on the slide assembly should read 3.8 cm. (This means that the reading of the centimeter scale accurately reflects the distance between the centers of the two spheres.) Experimental Procedure: 1. Bring the sliding arm slowly at 4.0 cm. Make sure the torsion pendulum is still aligned well. 2. Connect the high voltage power supply to the electric source if it is not already connected. 3. Connect the banana probes to the high voltage power supply. Make sure red probe is connected to high voltage and black (blue) probe is connected to the ground. Note: Be careful don t touch the high voltage probe! Always switch off the power supply if you are not taking data. 4. Switch on the power supply and discharge yourself by touching the black (ground) probe. Make sure you are touching the ground probe not the high voltage one! 5. Discharge both the spheres by touching them to ground probe. 6. Adjust voltage at 6.0 kv. 7. Charge each sphere gently by touching them to high voltage probe (You can touch each sphere separately but if it is hard to do so you can touch both the spheres together) 8. Switch off the power supply after charging the spheres. 9. Due to identical charges in each sphere the spheres will repulse each other. You can see the sphere on the torsion pendulum moves away by misaligning the index. 10. Rotate the dial at the top of the torsion pendulum slowly to bring the index alignment back. This task need to be done very gently. 11. Note the angle in the table (1) as θ 1. 12. Repeat steps 4 to 9 and record the angle as θ 2. 13. Find the average angle at this distance 14. Repeat steps 4 to 11 for different voltages as in the table (1) below. 15. Complete table (1) 16. Plot ln(θ) versus ln(v) and find the value of n from the graph and equation (8).
17. Compare the value of n with the standard value 2 by calculating the percent difference. r(cm) V (volt) θ 1 (dig) θ 2 (dig) θ av (dig) Table 1 4.0 3.5 4.0 4.0 4.0 4.5 4.0 5.0 4.0 5.5 4.0 6.0 Similarly you can verify the relationship between electrostatic force and the separation between the charges i.e. electrostatic force between two charges is inversely proportional to square of the distance between them.