Coulomb’s Law
Coulomb’s Law: Design Customization
TEMPLE UNIVERSITY PHYSICS
It is commonly understood that like charges repel, and opposites attract, but the strength of this force (be it attractive or repulsive) also depends on the distance between the two charges. Through careful observation of the magnitude of this force, Charles-Augustin de Coulomb discovered an inverse square law relating the force between two charges to the distance between them
(1) |
where is the Coulomb constant (8.988 109 N·m2/C2), and are the charges on point charges 1 and 2, respectively and is the distance between the two charges. The direction of the force in this special case is always along a straight line drawn between the two charges.
Figure 1. Experimental setup showing the electroscope chamber with a suspended sphere, two guide blocks with spheres, and materials for charging the spheres.
It is important to remember that charge is conserved: we can physically move it around, but we can neither create nor destroy it. In this lab, we will move charges around to set up a test of Coulomb’s law on a simple electroscope. First, we’ll measure the force as a function of distance, and see how well it fits the inverse square relationship. Second, we will look at how the magnitude and sign of the charges affects the force. Finally, we will use Equation 1 to determine the amount of charge (in Coulombs) that we can generate with static electricity. Along the way, we will see a few interesting effects that arise from the accumulation of static charge.
Learning Goals for this Laboratory:
- Become familiar with static charge and how it is transferred.
- Become familiar with induced charge and its effects.
- Practice using Coulomb’s Law in a real system with multiple forces acting.
- Practice graphing 1/r2 function.
Apparatus
electroscope chamber with suspended sphere and top cover, 2 guide blocks with spheres, cotton & wool squares, plastic rods, white vinyl strip
Part I.
Charging a Sphere
In this experiment, we’ll charge a plastic rod using friction, and then transfer the charge to metal-coated Styrofoam spheres. Unfortunately, this static charge won’t stick around forever, it will dissipate into the air (this could take seconds to minutes depending on the humidity) so be sure to read through and follow the charging procedure carefully.
- a) Inductively charge the sphere attached to the left guide block by doing the following. Check out Figure 2 to help you visualize the steps. The order of the steps is critical to success!
- Stand up the guide block so the sphere is at the top.
- Rub the wool square on the plastic rod to transfer charge to the rod.
iii. Bring the charged rod to about ½ cm away from the sphere but do not touch them together. While the rod is close to the sphere, touch the sphere with your finger and then remove your finger. (When you touch it, you are acting as a ground, a path for electrons to move along.)
- After you have removed your finger from the sphere pull the sphere away from the charged rod. The sphere on the guide block should now be charged.
- Note: If you hear a crack or pop sound while the rod is close to the sphere, this means that they were too close to each other and charge jumped across the gap. The sphere is now uncharged so you will need to start over, recharging the rod and repeating the process. Also, work carefully from this point: if you touch the charged sphere to anything it will immediately discharge and you will have to charge it again.
- b) Slowly slide the guide block with the charged sphere into the left side of the electroscope chamber and carefully observe the two spheres as they approach each other and just before they touch. Record your observations. If nothing happens you might not have enough charge on your sphere, recharge it and try again.
Question 1. When charging the guide block sphere, why do we have to remove our finger before removing the charged rod? In other words, would the sphere be charged if we removed the rod before removing our finger? Why or why not?
Question 2. Figure 2 illustrates the charges in the conductive spheres with electrons that are free to move around the fixed positive molecules. Use the concept of opposites attract and likes repel to explain how the procedure we used charged the guide block sphere.
ground
Figure 2. Charging by induction. Note how the sphere becomes polarized when the charged rod comes close, but its net charge is still zero.
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Question 3. According to your observation, did the suspended sphere experience a force when we moved the guide block sphere near?
Question 4. Can you explain why the suspended sphere would experience a force even though it is neutral having no net charge? Doesn’t this contradict Coulomb’s law which says that you need two charged objects to have a force? Figure 3 may help you to understand what is going on. Notice in the left panel of Figure 3 that the size of the spheres is important because the negative charges in the neutral sphere (+0) are much closer to the charged sphere than the positive charges. How does the difference in this distance affect the net force acting on the neutral sphere? Use Coulomb’s Law to explain why the unequal separation distances result in a net force
Figure 3. Force acting between a charged and uncharged sphere (left) and two equally charged spheres (right).
Question 5. You may have noticed that the spheres are Styrofoam with a conductive metal coating. Why must the spheres be conducting spheres for the arrangement of charges to occur as illustrated in Figure 3?
Part II.
Dependence of Force on Distance
Now we’ll charge both spheres and look at how the force between them depends on their separation distance. A video is provided showing how pushing the sliding block closer to the hanging sphere repels the hanging sphere away from its equilibrium position. The greater the force, the more the hanging sphere can be pushed out of position.
- a) Watch the video now. In the video, the sliding block sphere was charged as described in Part I. Then it was brought into contact with the hanging sphere so both spheres shared the charge (and had the same potential). This was repeated a few more times until both spheres were well-charged and repelled each other strongly.
- b) Now we will measure the separation distance between the two spheres for different amounts of displacement of the hanging sphere from its equilibrium . These values are shown in the figure below.
- c) We’ll use video analysis to first measure the absolute position of the two spheres, then use the position values to calculate and in Excel. To measure the positions using video analysis follow the steps below.
- Make sure you have Pasco Capstone installed (the install link is on Canvas). Open Capstone click on Video Analysis and open the video file. Navigate to the zoomed in view at the end of the video.
- Usually Video Analysis mode is already active but if not, click the “Enter video analysis mode” button on the toolbar at the top of the video (see figure below). This toolbar has several other useful tools that you can explore.
Video Analysis mode
iii. To define the coordinates and origin location for your experiment, click and drag the axes of the yellow x-y coordinates tool, which appears overlaid on your movie by default. Any choice of origin is fine but align the x-axis so that it is along the horizontal direction as the video is not perfectly horizontal.
- Set the distance scale in your movie using the yellow caliper-shaped tool that is by default overlaid on your movie. Adjust the ends of the caliper to line up to the diameter of the sphere. The sphere is 1.2 cm in diameter. Enter this value in meters into the caliper’s text box.
- Next click the gear button in the top right of the video and click Overlay and set the Frame Increment to 10 so the video will auto-advance by 10 frames each time you mark a position. See figure below.
Settings Menu
- In the video navigation tools use the slider and the Next Frame button (see figure below) to advance the video to the point where the spheres are the farthest the right. We will mark the position of the two spheres from this point on.
vii. To make the spheres easier to see, make the video larger and use the magnifying glass tool. Choose one of the spheres and mark the location of the center of the sphere with your mouse. A marker should appear at the location you click, and the video will automatically advance 10 frames. Repeat clicking on the position of the center of the sphere to track its position as the video advances. Stop once the hanging sphere is no longer affected by the sliding block sphere. You should have at least 10 data points. If you make a mistake or want to start over, either create a new analysis object or delete the marks you made.
viii. Use the previous frame button to back the video up to the point you started tracking the first sphere and in the toolbar at the top, click to add a new tracked object for tracking the other sphere. Mark the second sphere’s location the same way as you did for the first; the video should auto advance each time you click.
- In the Displays menu at the far right of Capstone, select table, and click Select Measurement at the top of each column and display the x-position data of each of the spheres. Copy and paste this data into Excel for further calculation.
- d) Use Excel to calculate the separation distance between the two spheres. Do this for all the values you collected. Also calculate for the suspended sphere, which is it’s displacement from its equilibrium position. The equilibrium position is the position of the suspended sphere when the sliding block is far away.
- e) We can now analyze the dependence of the electrical force between the two spheres on the distance between the spheres. Because we know the suspended sphere is not accelerating we can use Newton’s 2nd Law to write down Equations 2 for the suspended sphere
(2) |
where is the electrostatic repulsive force between the spheres, and is the weight of the suspended sphere. Combine these equations using the definition of and the parameters shown in Figure 5 to derive the following expression for the force between the spheres
(3) |
where is the displacement of the suspended sphere from its equilibrium position and is the length of the wire from which the sphere is suspended.[footnoteRef:1] [1: We assume that the angle is small enough that the change in elevation of the suspended sphere can be neglected. This allows us to take the electrical force between the two spheres to be completely horizontal, and the displacement from equilibrium to be simply related to the length and the angle as shown in Figure 5.]
Figure 5.
Forces acting on the hanging sphere with the distances and shown.
The mass of the suspended sphere and the length have been determined to be = 0.08 g and = 23.5 cm.
- f) Use your measurement of to determine the repulsive electrostatic force using Equation 3 for each separation distance.
- g) Plot a graph of the force as a function of the separation of the two spheres.
Question 6. Comment on the general trend of the data: does it look as you would expect? Why or why not?
- h) Plot a graph of the force as a function of. Fit the graph to a line and find the R2 value.
Question 7. Is the graph linear? What would a linear dependence between and tell you? What does the slope of this plot represent physically (see Coulomb’s law – in y=mx+b form)? From this slope determine the charge on a sphere (remember the spheres are equally charged).
- i) Plotting vs is one way of determining that you have a dependence. Another way is to plot the log of the data and look at the slope. You can either do this by literally taking the log of your x and y data and then making a new plot, or for a simpler way, you can just make a copy the plot of vs you already have and then change the axes to a log scale instead of the default linear scale. Either way, make a linear fit to the resulting plot and you can determine if you have an inverse square dependence of force on distance. (you may want to convince yourself what the slope would be by taking the log of both sides of Equation 1).
For your lab report:
Refer to the grading rubric and lab syllabus for what to include in your lab report. Be sure to include all graphs with clearly labeled axes and fitted equations, and answers to all questions.
Coulomb’s Law: Physics Lab Report
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