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Coulomb’s Law The goal of this lab is to experimentally establish the distance and charge dependence of the electrostatic force using the apparatus shown in Figure 1. The apparatus includes three graphene-coated polystyrene spheres—one hanging on a thin string inside the enclosure and two attached to guide blocks on the left and right sides. Let’s refer to these spheres, viewed from left to right, as spheres A, B, and C. Materials – Coulomb’s Law apparatus, polystyrene balls, electronic balance, triboelectric charging materials (vinyl strips, felt, wool, etc.), ruler, computer with Logger Pro or Microsoft Excel

Part A: Finding the distance dependence for a fixed charge 1) Before any measurements are made, make sure sphere B hangs in a straight line with spheres A and C. Then measure the vertical distance 𝐿 (shown in Figure 2) from the top of the apparatus to the center of sphere B. (Note that 𝐿 is not the length of the string, but the length from the center of the sphere to the point at the top-center of the enclosure.) 2) Remove the block with sphere C and set it aside for now. Take the block with sphere A and charge it by induction. To do this, rub the vinyl strip to create a static charge. Place the strip as close to one side of sphere A as possible to polarize it and ground the opposite side of the sphere by gently tapping it with your finger. Sphere A now has a charge 𝑄#.

Figure 1 - The apparatus for today's experiment, shown with the three spheres aligned in a row. A mirror located behind the spheres helps eliminate parallax errors when measuring the positions of the spheres.

Figure 2 – A diagram showing the key variables in the experimental setup as sphere B is pushed away from sphere A.

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3) Touch sphere B with your finger to ground it and insert sphere A through a hole near the bottom of the chamber. Describe what happens when sphere A approaches and contacts sphere B. What is the physics behind the behavior you see? 4) Afterwards sphere A and B will both have charge 𝑄#/2 and will repel each other. Continue to push sphere A towards sphere B in small steps and for each step measure and record the distance 𝐷 between the centers of the spheres and the distance 𝑑 that sphere B has moved from the vertical. To control for the parallax effect of measuring a sphere’s position by eye, make sure that you cannot see the sphere’s own image in the mirror when you make your measurement. 5) Discharge both spheres and repeat this charging and measuring process at least twice more. It may take multiple attempts to achieve a good initial charge on sphere A. The spheres will gradually discharge into the air, especially if it’s humid, so making timely measurements is important for the assumption of a constant charge to hold. Part B: Exploring the case of varying charge 1) Repeat the procedure above to charge sphere B and make a measurement of 𝐷 and 𝑑. Now leave sphere A in place for the rest of the run. Discharge sphere C and insert it in the hole opposite to sphere A. Allow sphere C to touch sphere B and siphon off half of its charge. The charge on the system of spheres A and B is now different and the values of 𝐷 and 𝑑 have changed to compensate for this. 2) Repeat the previous step by measuring the new values of 𝐷 and 𝑑, and then proceed to siphon more charge off of sphere B using sphere C. (Make sure sphere C is discharged each time after siphoning charge!) Do this as many times as you can without allowing spheres A and B to touch. Realistically you will only be able to get a few data points. 3) Just as in part A, repeat this part of the experiment to acquire the data for two more runs. Be prepared for some trial and error involved in getting clean data. Part C: Graphing and analysis 1) Consider sphere B to be deflected from vertical as shown in Figure 2. Draw a free-body diagram of the forces on sphere B. 2) Using the symbols in Figure 2, apply Newton’s second law to sphere B and show that the following relationship holds between the electrostatic force 𝐹) and the gravitational force 𝐹*:

𝐹)(cos𝜙 cos 2𝜃 + sin𝜙 sin 2𝜃) = 𝐹* sin 2𝜃. By a trig identity, this is also equivalent to 𝐹) cos 2𝜃 − 𝜙 = 𝐹* sin 2𝜃. Since sphere B does not deviate too far from the vertical, both 𝜙 and 𝜃 remain small. In the equation above, apply the small-angle approximation to the trig functions and argue that it simplifies to

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𝐹) = 𝐹*𝑑𝐿 = 𝑚𝑔

𝑑𝐿.

3) On a single graph plot 𝑚𝑔 ;

< vs 1/𝐷> for each of your runs from part A. (You will need to know

the mass of sphere B for this. Find the average mass of some extra polystyrene spheres using the electronic balance and use that value as an estimate of sphere B’s mass.) What shape does your data take on this graph? Fit a simple function to each run. What do you think accounts for the variation in the coefficients from run to run? Charles Augustin de Coulomb published a law in 1784 that, when applied to the conditions in our experiment, claims the following should be a good model of the relationship between spheres A and B: 𝐹) = 𝑘𝑄#𝑄@/𝐷>. If we take 𝑘 =8.99×10E𝑁𝑚>/𝐶>, calculate how many electrons are on each sphere. 4) On another graph plot 𝑚𝑔𝐷> ;

< vs 𝑄#𝑄@ for each of your runs from part B. Perform fits to each

run on this graph as well. What do you notice about the relationship between the different runs? Can you explain why you see this result? 5) Assume you have a method for charging sphere A with a known charge 𝑄#. With sphere B initially uncharged, describe an experiment you could perform to increase its charge to a value greater than 𝑄#/2. 6) Is there a maximum amount of charge you can provide to sphere B using your technique? If so explain what it is and show why that value is a maximum. What are the physics underlying your conclusion? 7) Your TA may choose to hold a class discussion at the end of the lab period. If so, be prepared to present your results using a whiteboard. Be sure to include in your lab report both of your graphs, any calculations needed for your analysis, your conclusions about the relationship between electrostatic force, charge, and distance, and make sure you address the questions posed throughout this lab manual.