= [1] Coulomb s Law. 1. Objective

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1 PHYS-102 LAB-01 Coulomb s Law 1. Objective The objective of this experiment is to demonstrate that the force between two stationary charges is directly proportional to the product of the charges and inversely to the square of the distance between them. 2. Theory According to Coulomb s Law, the magnitude of the electrostatic force between two charged particles with charges Q 1 and Q 2 and separated by a distance r is given by: Q Q = [1] 1 2 Fe k r 2 Where k = 8.99x10 9 N.m 2 /C 2 is the Coulomb constant. The unit of charge is taken as a Coulomb (C). The constant k is also written as k = 1/4πε o where the constant ε o is called the permittivity of free space ε o = 8.854x10-12 C 2 / N.m 2. Since force is a vector quantity, in the vector form Coulomb s law is expressed as: Figure 1. The free body force diagram of two charged particles.

2 Where r 12is a unit vector directed from Q 1 to Q Experimental Procedure The basis of this experiment is very simple. Two graphite-coated spheres are charged and the force between them is measured. This is done to study the dependence of the electrostatic force on [1] the separation between the charges and [2] on the product of the two charges. We will study only the proportionality of electrostatic force on the separation distance and the product of the charges. The separation distance dependence is studied by charging the two spheres and measuring the force between them by varying the distance between the charged spheres. The charge product dependence is similarly studied by varying the charge on the two spheres kept at a fixed distance and measuring the resulting force. Note: You will measure the force between the two spheres by measuring torque resulting in a torsion (angular twist) in the wire holding the one of the charged spheres on a light insulating lever. One end of the wire is fixed were as another one can be rotated. The amount of rotation is proportional to the force between the spheres. The torsion balance can be calibrated to precisely determine this constant of proportionality between deflection θ (the twist) and the force. However, in this experiment we will not do so. 3.1 Apparatus 1. A Coulomb s Law apparatus with associated accessories. Two views of the apparatus in schematic form are shown below. 2. A high-voltage power supply. 3. Faraday Ice Pail, charge producers, electrometer and associated accessories

3 Figures 1a and 1b. Schematic views of the Coulomb s Law Apparatus.

4 3.2 Experimental Procedure Important precautions 1. The wire in the torsion pendulum is very fragile and should be handled with utmost caution. Do not adjust screws holding the wire. You should never tap the sphere attached to the torsion wire from above. Do not touch the spheres with your bare hands as it can contaminate and damage the spheres graphite coating. 2. When using the high-voltage power supply, you should never touch anything except the spheres the high-voltage probe. Do not point the probe toward any other person or your own body. Use the probe with one hand, keeping the other hand free. Do not handle the probe if you are wet or are standing on a wet floor. 3. Discharge and re-charge both spheres before every measurement. Always charge spheres at maximal distance from each other, afterwards bringing the sphere on the adjustable support to shorter distances. Initial setting. 1. Move the sliding base to the 4-cm mark and set the sphere support rod so that the sliding sphere is very close to but not actually touching the suspended sphere. Do not readjust the rod for the remainder of the experiment. 4 cm is the minimal distance between centers of the spheres. 2. Make sure the spheres are fully discharged (touch them with the green grounded probe) and move the sliding sphere as far as possible from the suspended sphere. Rotate the torsion dial to align black lines at the counter weight vane and on the fixed plate (this is null position). Write down the angle on the torsion dial θ 0. This angle is corresponding to zero force between spheres. 3.3 Distance Dependence 1. With the spheres at maximum distance, charge both the spheres to a potential of 6 kv, using the red charging probe connected to the High Voltage Power Supply.

5 2. Position the sliding support at the 14 cm mark, Adjust the torsion dial as necessary to balance the forces and bring the pendulum marker to null position. Record the distance R from the base scale and the angle θ from the torsion dial. Each member of the lab group should perform this measurement separately. 3. Repeat steps 1 and 2 for 10, 9, 8, 7, 6, 5, and 4 cm distance. 4. Plot the distance R as a function of the displacement angle θ - θ 0. Add a power trendline to fit the data using Excel and include the equation of the line, as well the R 2 value, on the chart. 3.4 Charge Dependence 1. Charge both spheres to 6kV. Set the sliding sphere to 4 cm and record the voltage and the displacement angle θ. (Hint: this was the last reading you took from the previous section) 2. Discharge both spheres and bring the sliding sphere to the maximal distance. Recharge both spheres to 5.5 kv bring the sliding sphere to the same distance of 4 cm as in step 1. Record the voltage and displacement angle θ. 3. Repeat step 2 for 5, 4.5, 4, 3.5, and 3 kv. 4. Plot the Voltage as a function of the displacement angle θ. Add a trendline to fit the data using excel and include the equation of the line, as well the R 2 value, on the chart. 3.5 Correction to the data. The Coulomb s law (eq.[1]) applies to point charges. An isolated charged conductive sphere will act as a point charge for measurements made outside the sphere. In the presence of another similar sphere, a redistribution of charge will take place to minimize the potential energy of the charges. This redistribution of charge resulting in a deviation from the point charge approximation can be significant for separation distances comparable to the radii of the spheres. A correction factor will be used to correct for this deviation. This correction factor is B = 1-4[a/R] 3, where a is the radius of the sphere and R is the separation between two

6 spheres. To apply the correction to your data simply divide the measured value of θ (or θ average ) by B. The spheres used in the present Coulomb torsion balance apparatus have an average radius a = 1.75 cm.

7 LAB-01 Coulomb s Law Name: Sec./Group Date: 4. Prelab 1. Consider a charged metallic spherical shell of radius a = 1.75 cm. The potential at a distance of r = 6.0 cm from the center of the shell is measured to be 1.5kV. What is the charge on the shell? Calculate the potential on the surface of the shell. (Note: V = kq/r. You may also like to read the chapter of your textbook that discusses the Electrostatic Potential.) 2. Using an Excel spreadsheet make a plot of force F vs r, where F and r are as defined in equation [1]. Use Q 1 = Q 2 = 5.0µC and vary r from from 0.5 cm to 10 cm in steps of 0.5cm. Plot F along the vertical axis.

8 3. Using an Excel spreadsheet make a plot of F vs. Q 1 Q 2 in eq.[1]. Take r = 10.0 cm and Q 1 = Q 2 vary from 1.0 µc to 10.0 µc in steps of 1.0 µc. Plot F along the vertical axis.

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10 LAB-01 Coulomb s Law Name: Sec./Group Date: 5. Data 5.1 Distance Dependence V = (v) R (in cm.) θ (in deg.) θ avg (in deg.) B=1-4[a/R] 3 θ corrected (in deg.)

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12 LAB-01 Coulomb s Law Name: Sec./Group Date: 5.2 Charge Dependence R = (cm) B =1-4[a/R] 3 V (in kv) θ (in deg.) θ avg (in deg.) B=1-4[a/R] 3 θ corrected (in deg.)

13 6. Analysis Use the computer to create the two graphs as specified in Procedures 3.3 and 3.4. Submit these to your lab instructor. 7. JUST FOR THE FUN OF IT Fun with the Faraday s Cage Often one has to construct large shielded volumes such as rooms in laboratories where sensitive electronic measurements are to be shielded from external electrical interference. Such rooms are also crucial when the electronic communication has to be secured against unauthorized information sharing - also known as spying. Such electrically shielded regions are called Faraday cages. Michael Faraday used a metal ice pail to study how the charges distributed themselves when a charged object was introduced inside the conducting pail (Note: great scientific experiments are often carried out not by using the fanciest of gadgets, but by imaginatively using whatever is easily available). You will be repeating Faraday s experiment with the Faraday cage a cylinder made out of wire mesh isolated from and surrounded by an outer metallic cylinder made from the same wire mesh. Once, say, positively charged object placed in the inner cage positive charges from inner cage will have higher energy than before and therefore will try to escape to the outer cage. Thus difference in potential energy (voltage) will be created. It can be measured by

14 precision voltmeter (electrometer). The voltage (V) will be proportional to the charge Observe carefully and have fun. V Experimental Procedure 1. Rub the two charge producers together to create a charge on them. 2. Insert the blue wand into the lower half of the ice pail (the inner cylinder). Make sure the wand does not touch the surface of the pail. Note the magnitude of the deflection and its direction (to the right or to the left, or positive or negative). 3. Withdraw the wand from the cage and note the electrometer reading again. 4. Insert the wand again, and let the wand touch the inner surface of the inner cylinder. Note the electrometer reading. Remove the wand from the inner cylinder. 5. Ground the ice pail with green wire and then touch the pail again with the wand from step 4 above. Note the electrometer reading. 6. Repeat step 2 with the white wand (or steps 1 and 2 if you observe very small or no deflection). Conclusions:

15 1. What can you conclude from the direction of electrometer deflection in steps 2 and 7 about the polarity of charges on the white and blue wands? 2. What can you conclude from the direction of electrometer deflection in steps 2 and 4 about the polarity of charges on the blue wand and the charge induced on the outer surface of the ice pail? 3. What can you conclude from step 5 about the charge on the wand? What happened to the charge? Make a hypothesis (an educated guess). Suppose you rub the two wands together as in step 1 of Experimental procedure, but this time inside the ice pail. What do you think (and why) the reading on the electrometer would be? Choose from one of the three possible outcomes: [1] deflection to the right [2] deflection to the left [3] no deflection. Now do the experiment and see if you were right.

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