2022W 136-2 01 Coulomb

Lab 1: A Law of Electrostatic Force In this lab, you will use an online simulation to investigate how the charges of two objects and the distance between them effect the electrostatic force between them. After this lab, you should be able to describe the functional dependence of the force on charge and radius and draw simple conclusions from this understanding. You should also be able to linearize data and understand how to investigate functional dependence of a physical quantity by systematically changing related physical quantities. Introduction You are probably aware that like charges repel, while unlike charges attract. You may also have learned the quantitative formula for the attractive or repulsive force. These facts, like all physical laws, were established through rigorous experiments. In this lab we will use a simulation to experimentally derive the equations for electrostatic force. Electric charge is measured in units of Coulombs , abbreviated as C. In practice, a single Coulomb is a lot of charge, so in this lab we will mostly be dealing with µC (microcoulomb, = 1 × 10 − 6 C). Familiarization and Setup First, open the simulation 1 and click on the simulation. Play with the simulation for a little while to familiarize yourself with its controls. We will use the “Macro Scale” option. a) The vertical dotted lines through the centers of the two charges, q 1 and q 2 indicate the locations of the center of the charges. The ruler itself is also movable. b) The legends above the two charges give the force of interaction. The “force values” checkbox at the bottom right enables this option. c) The controls at the bottom allow you to change the value of the two charges. 1 https://phet.colorado.edu/sims/html/coulombs-law/latest/coulombs-law_en.html Page | 1

Force versus Charge First, determine how the electrostatic force depends on the electric charge. This simulation allows us to vary the charge on the two objects ( q 1 and q 2 ) as well as the x-positions of those charges ( x 1 and x 2 ). To do this, we will set x 1 , x 2 , and q 2 to arbitrary values and leave them alone while varying only q 1 . 1.1 (5pts) Record your values for x 1 , x 2 , r and q 2 below. ( r = x 2 − x 1 , the distance between the two charges.) A positive q 2 and x 1 = 0 will be most convenient, but any values will work. x 1 (cm) x 2 (cm) r = x 2 − x 1 (cm) q 2 (  C) 4.00 7.00 3.00 8  C 1.2 (5 pts) Now, vary q 1 as shown in Table 1 and complete the table. If the force is attractive (arrow on q 2 points toward q 1 ), enter a negative number. If it is repulsive, enter a positive number. Table 1: Electrostatic force versus electric charge data from PhET simulation. Charge 1 (  C) Force on Charge 2 (N) -9 -639.115 -6 -479.336 -3 -239.668 0 0.000 3 239.668 6 479.336 9 639.115 1.3 (20 pts) We will guess that force and q 1 have a linear relationship : that is, the force obeys an equation of the form F = k 1 q 1 (1) for some as-yet-unknown value of the constant k 1 . Using the LineFit.xslx file included in the Canvas assignment, plot the force (vertical axis) vs. the charge (horizontal axis) and fit the data to this straight-line equation (set the intercept to zero). Save the graph as an image file, and insert the image here. Do not just copy and paste the graph into this document – if you do that and submit your work as a .docx file, your graph may not show up in the submitted file! Record the slope k 1 (determined by the straight line through the data) here. Don’t forget to state the units of Page | 2

[Side note: of course, it is possible that the F vs. q 1 graph is not a perfectly straight line after all, and that the relation between the force and charges is entirely different. This is particularly likely if our range of charges covers only a small part of what is possible in nature, so that our linear fit is only an approximation. For example, we didn’t test values larger than 9 µC or closer to zero than 3 µC, so it’s possible that something strange happens there. Ideally, we would verify the linear dependence with additional data, but you do not need to do this now.] Page | 4

Force versus Distance 2.1 (5pts) Next, determine whether (and how) the electrostatic force depends upon the distance r between the two charges. Move the first charge all the way to the left, x 1 = 0 . Select arbitrary, non-zero values for q 1 and q 2 and record them here. q 1 (  C) q 2 (  C) -4 8 2.2 (10pts) Vary the position of Charge 2 and complete Table 2 with 7 data points. Use all of the ruler and try to spread out your points uniformly. Also, for each position, calculate the value of the constant k 12 from the previous section using the known q 1 and q 1 and measured F . Table 2: Electrostatic force versus distance between electric charges for simulated data. r = x 2 − x 1 (cm) Force (N) k 12 = F / q 1 q 2 (N/  C 2 ) 2 -719.004 22.469 3 -319.557 9.986 4 -179.751 5.617 5 -115.041 3.595 6 -79.889 2.497 8 -44.938 1.404 10 -28.760 0.899 2.3 (20 pts) In the previous section, we thought that k 12 was a constant because it didn’t depend on q 1 or q 2 . However, to explain the data in Table 2, k 12 clearly must depend on r . We will assume that the dependence is a power law: k 12 = Ar B (3) for unknown constants A and B . B is an exponent so it will not have units, but A will have whatever units are necessary to balance Equation (3). To determine values for A and B , we would normally use a computer program to fit Equation (3) to our data. However, not all data analysis software can fit arbitrary non-linear equations such as Equation (3). Also, when we start talking about fitting uncertainty later in the course, it is more complicated to think about that for non-linear fittings. So, we’d really like a trick to let us fit Page | 5

Results The electrostatic force law is known to be F = 1 4 π ϵ 0 q 1 q 2 r 2 (5) ϵ 0 is a fundamental physical constant, called the “vacuum permittivity” or the “permittivity of free space.” Other scientists have carefully measured ϵ 0 to have a value of 8.8542 × 10 − 12 C 2 / N m 2 . We have now collected enough data to determine this natural constant for ourselves. 3.1 (10 pts) In the first part of this lab, you determined k 1 , defined by F = k 1 q 1 . Thus k 1 = 1 4 π ϵ 0 q 2 r 2 (6) with q 2 and r having the fixed values you selected in 1.1 above. Use the value of k 1 you wrote down in 1.3 above to calculate ϵ 0 and enter it here. Please express ϵ 0 in units of C 2 / N m 2 , which may require converting from cm or µC. 3.2 (10 pts) Comparing Equation (5) with our expressions F = k 12 q 1 q 2 and k 12 = A r B from Equations (2) and (3), we get A = 1 4 π ϵ 0 (7) Use the value of A you entered in 2.3 to calculate ϵ 0 and enter it here. Again, please express ϵ 0 in units of C 2 / N m 2 , which may require converting from cm or µC. Page | 7 ϵ 0 = 9.54 * 10 -12 C 2 /Nm 2 ϵ 0 = 8.85 * 10 -12 C 2 /Nm 2

3.3 (5 pts) Are the two estimates of ϵ 0 close to each other? Are they close to the established value for ϵ 0 ? We have not yet discussed a formal way to decide whether two values are close— that is the topic of a future lab—so for now, investigate the percent difference. The percent difference between two values v 1 and v 2 is their difference over their average: v 1 − v 2 ( v ¿¿ 1 + v 2 )/ 2 ¿ For this lab, we will say that if the two values are within a few percent, they are in decent agreement. If you were writing a report on this exercise, you would want to report your measured value for the physical constant, ϵ 0 , in your Conclusions. Additionally, you would state whether you have verified that your data supports Equation (5) as the electrostatic force of interaction between two electric charges. History and Summary These results were first published (with a few variations) by Charles-Augustin de Coulomb in 1785. Unlike the simulation, Coulomb could move his charges around in three dimensions, and noted that the force (a vector) is always directed along the line connecting the centers of the two charges. Equation (5) relates only the magnitude (not the direction) of this force. Feel free to play with the “Atomic Scale” simulation for a more realistic set of achievable charge distributions. Page | 8 Using value from 3.1: 0.0746 or 7.46% - not very accurate Using value from 3.2: 4.74 * 10 -4 or 0.474% - more accurate