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Lab 03 - Discharge Rate Designers at your local science museum are perfecting an exhibit on electrostatic discharge

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Lab 03 - Discharge Rate

Designers at your local science museum are perfecting an exhibit on electrostatic discharge. The exhibit involves two long “bifilar” pendula. Each pendulum has a conducting ping-pong ball of known mass hanging from a pair of long non-conducting threads. They want visitors to “charge” the balls using a Teflon rod stroked with fur and see what happens. The designers plan to have a local sign made while the set up is being perfected. As a student intern at the science museum you have been asked to help one of the designers estimate how long it will take for the charges to drain off the balls before they touch. In addition, the sign makers have been asked to report on approximately how much charge is on each ball after a visitor charges the balls and touches them together. Your job is to figure out how to use the laws of mechanics and Coulomb’s Law to find the information the sign maker needs.

q L L q R   L = 1.683 m m = 2.9 g k = 8.99 x 109 N•m2/C2   Figure 1: Diagram of the exhibit apparatus

You have to measure the distance between the centers of the balls as a function of time in order to do the needed calculations. But, you can’t make these measurements directly without disturbing the system. Also, it seems to take about a half hour for the balls to discharge enough so they touch each other.

Since the discharge process is quite slow, your supervisor makes a digital video of the discharge by setting the camera to record only 1 frame every 20 seconds. The title of her edited movie is <Discharge.mov>. Next she inserted her movie into a Logger Pro file entitled <Discharge.cmbl> and used video analysis to determine the positions of the ball centers in each frame for just under nine minutes of discharge. She then asks you to analyze her Logger Pro data and: (1) derive the equations needed to determine how much charge is on each ball as a function of their separation; (2) use the Logger Pro New Column features to calculate the charge, q, on each ball as a function of time to determine the discharge rate, and (3) estimate how long it will take for the balls to discharge fully enough to touch each other.

Before starting, you should view the movie entitled <Discharge_TimeLapse.mov>. This is a time-lapse version that plays back 120 times faster than the original movie, so the first nine minutes of a much longer discharge plays back in about five seconds

 

a)  Initially each ball is charged separately with a Teflon rod so one ball has charge q₁ and the other charge q₂. Is the excess charge on each ball positive or negative in this case? Explain the reason for your answer. Hint: Teflon and rubber have the same electrostatic properties as each other.   10 pts

 

 

 

 

 

 

b)  According to Coulomb’s Law, how does the electrostatic force exerted on one ball by the other depend on the charge, q₁ and q₂, on the balls and on the separation R of their centers? Note: Assume that the balls’ centers are far enough apart to avoid inducing a non-uniform distribution on each other. Note: The centers of the balls must be at least twice a ball diameter apart.   10 pts

 

 

 

 

 

 

c)  If the balls are “identical” what should happen to the charges on them if they are forced to touch. Explain.   10 pts

 

 

 

 

 

 

a)  On the diagram on the right to draw a free body (or vector) diagram with arrows showing the direction of each of the forces on the mass, m, including the gravitational force, , the force the string exerts, , and a horizontal Coulomb force due to the other ball, .   Use “Insert>Shapes” to draw the arrows. Note: The diagram is not to scale. Actually the string length L>>R/2 and  is small enough so that we can make the approximation that        where R/2 is half the distance between the centers of the two balls.

L R/2

b)  Show that when q  is small and the balls are in equilibrium, the net force on each of them is zero and the magnitude of the Coulomb force on each ball is given by.  Show your derivation here.   10 pts

 

 

 

c)  Write the equation needed to calculate the distance between the centers of the balls, R, as a function of the locations of the center of each ball where X is the left ball’s location and X₂ is the right ball’s location.   Remember the distance is a function of bot the x and y coordinates of the objects.   10 pts

 

 

 

d)  Open the Logger Pro experiment file <Discharge.cmbl>, which has the movie inserted. Click on the “R” column header and add the appropriate equation for R using the Calculated Column option. You should find that R seems to decrease linearly in time between about 100 s and 520 s. Does that seem to be the case? If the answer is no, check your work.   10 pts

 

 

 

e)  If the balls are far enough apart to behave more or less like equal point charges, then F^coul should be given by F^coul = kq²/R². Write an equation that can be used to determine the amount of charge, q, on each ball as a function of m, R, and L as well and the gravitational and Coulomb constants, g and k.  Show your derivation here.   10 pts

 

 

 

f)  Your next task is to use the same Logger Pro file and the new column feature once again to determine the magnitude of the charge on each ball for times between 100 s and 520 s. Pay attention to the notes in the box in the <Discharge.cmbl> file, and note that L = 1.68 m, m₁ ≈ m₂ ≈ 0.0029 kg and k = 8.99 x 10⁹ N•m²/C². If you enter the equation needed to calculate q properly, you should find that q also seems to decrease linearly in time between about 100 s and 520 s. Use the Examine tool to select the times of interest. Then perform a linear fit to determine rate of charge decrease and write it in the space below with appropriate units. Warning: The charges in Coulombs are quite small. If your linear fit shows 0.000 for slope, you should double click on the box that displays the fit parameters and choose four significant figures rather than four decimal places for the Display Precision. Also explain what fit parameter gives you the value of .   10 pts

 

= ______________ C/s

 

 

g)  Copy and paste your graph, including the fit, here.   20 pts

 

 

 

 

h)  Recall that your final challenge is to estimate how many minutes it should take for the balls’ centers to be one diameter apart (i.e., when R = 0.0377 m) so the balls touch ? ending the discharge process. Your supervisor has suggested that you find the slope and intercept for the linear portion of the R vs. Time graph (from t = 100 s to t = 520 s) using the Examine tool and then the Linear Fit tool. Then you can use the linear discharge equation to find the time in seconds at which R = 0.0377 m. You also realize that you need to calculate that time in minutes instead of seconds. Show your work in the space that follows and report your estimated number of minutes to two significant figures.   10 pts

 

 

 

a)  Can you think of reasons why the charge on each ball decreases over time and where the charges might go?   10 pts

 

 

 

b)  The linear model you used for times between 100 s and 520 s matched your data very well. During the first 100 s the rate of loss of charge differs from your linear model. This suggest that some other path for charge loss may be important when each ball carries even more excess charge. Can you think of a physical reason why this might happen?   10 pts