Coulomb's Law(last edited June 25, 2013) 

Dr. Larry Bortner 

Learning ObjectivesDescribe the concepts of a torsion balance. State how the charge deposited on a metallic sphere depends on the charging voltage. State how the magnitude of the force between two charges at a given separation depends on the values of the charges. State how the magnitude of the force between two constant charges depends on the distance between them. Purpose 

To investigate and verify Coulomb’s Law that quantitatively describes the force between two point charges. 

Background 

What Affects the Force Between Two Charges? 

At this time in your academic endeavors, you are probably quite aware that there are two types of electric charges, positive and negative. (No, we’re not going to count the electric bill as a charge here.)
Like charges repel each other and opposites attract.
To be true scientists, we should discover this experimentally, but this is one of those things that is sort of universally known, so we’ll accept it as fact.
Aside from the polarity of the charges, what other single quantity might possibly affect the force between these two things? Let’s make a list: 1. How much charge one of the charges has. 2. How much charge the other has. 3. The mass of one of the charges. 4. The mass of the other. 5. The shape of one of the charges. 6. The shape of the other. 7. The distance between the charges. You may have thought of one or two other things but let’s concentrate on these items.
To simplify matters, let’s assume point charges. This eliminates items 5 and 6. Next we’ll assume that any gravitational force is too small to worry about, so that gets rid of items 3 and 4. This leaves us with the charge amounts and the distance between them. Your task is to experimentally determine how the force between two charges is affected by these quantities. Since this experiment has already been done before, initially by Charles Augustin de Coulomb back in 1785, you are being asked to verify Coulomb’s Law: the force F between two point charges q_{1} and q_{2} separated by a distance R is given by 



(1) 



where Coulomb’s constant k = 8.988 x 10^{9} Nm^{2}/C^{2}. Do not assume that this “law” is correct. Take data and fashion an experimental mathematical model of the force (called an empirical relation) and see how that compares to the theoretical model. 



Measuring Small Forces 

Because keeping many, many charges of the same polarity right next to each other for any reasonable time is difficult, we work with small amounts of charge. This means the force between two charges is weightofamosquito tiny. How exactly do you measure this force? With a tiny scale, of course! Well, not really, but sort of. A torsion balance is used, where the force is proportional to how far a wire twists instead of how far a spring is stretched. In an extension of Hooke’s Law, a measure of the angle twisted from equilibrium gives a measure of the force: 



(2) 



where K is a constant specific to the wire. There are two different kay constants in this experiment, this constant and Coulomb’s constant (a lower case k). Be sure to keep them separate in your mind. 



The torsion balance that we use in the experiment is the PASCO Coulomb Balance (Figure 1). A conductive sphere S_{T} is mounted on a rod, counterbalanced, and suspended from a thin torsion wire. An identical sphere S_{M} is mounted on a slide assembly so it can be positioned at various distances from S_{T}. (A charged sphere acts like a point charge most of the time. Exceptions are dealt with below.) 



Figure 1. Diagram of the full Coulomb Balance assembly. 

To find the force, both spheres are charged, and the sphere on the slide assembly is placed at a fixed distance from the equilibrium position of the suspended sphere. The electrostatic repulsion between the spheres causes the torsion wire to twist. The experimenter then twists the torsion wire to bring the balance back to its equilibrium position. From Eq. (2), the angle through which the torsion wire must be twisted to reestablish equilibrium is directly proportional to the electrostatic force between the spheres. 

Charge Production 

We could set up a patch of carpet for you to shuffle along to generate a charge that you could transfer to each sphere. (You did wear your sneaks today, right?). Alas, quantifying such a charge is problematic. A better, repeatable way of providing a charge is to use a voltage source. When a charging probe connected to this source touches an electrically isolated metallic sphere, a charge q is deposited. This charge is proportional to the voltage V: 



(3) 



where C is a yet another constant (at least it’s not K). The SI unit for voltage is a volt (V) and the unit for charge is a coulomb (C). The unit of voltage of the source used in the experiment is a kilovolt (kV). 

Angle Correction 

The spherical shape is used for the charge holder because it acts like a point charge when the charge is evenly distributed across the surface. In other words, the force of this charge on other charges is as if all the charge were concentrated at the center of the sphere. Unfortunately, the charge does not always spread evenly around the sphere. When two spheres of the same charge are brought close to each other, some of the charges on the near side slide over to the opposite side due to the repellant nature of the electric force. There is a higher density of charge on each sphere’s opposite side compared to the near side. You can account for this redistribution by dividing the measured angle θ by a geometric factor (Look! Another constant!): 



(4) 



where a equals the radius of the spheres and R is the separation between sphere centers. So the angle required to calculate the force correctly is 



(5) 

Experiment Outline 

We have identified three independent variables that we need to see how the dependent variable (the force) changes. This means three investigations are necessary, in each case changing one variable while keeping the other two constant (the control variables) to see how the force changes. We can cut that down to two investigations if we assume that the two charges are symmetric. It doesn’t matter which of the two charges are varied, you expect the same force between them, no matter which charge is on which ball.
The experiments to conduct are these: one to probe the charge dependence of the force, the other to look at the distance dependence. The following section describes various tips and tweaks that you can do to optimize data collection for these two cases. Unfortunately, they may not be enough to collect viable data on a humid day, so we will go back to three investigations by looking at equal charges dependence of the force. 



Static Electricity Tips and Considerations 

Experiments with the Coulomb Balance are straightforward and can be accurate as well as frustrating. A charged shirt sleeve, a draft or breeze, somebody walking by a little too quickly, an excessively humid day any of these and more can affect your experiment. Following the tips listed below gives you a good start toward a successful experiment. 

When performing experiments, stand directly behind the balance and at a maximum comfortable distance from it. This minimizes the effects of static charges that may collect on clothing. Of course, please be aware and considerate of people experimenting behind you. 

When charging the spheres: 

Charges move very quickly when given an easy path. A momentary, solid connection is all you need to charge (or discharge). 

Turn the power supply on, charge the spheres, then immediately turn the supply off. The high voltage at the terminals of the supply can charge a little bit of the surrounding air, which affects the torsion balance. The voltage supply has a convenient toggle switch. 

Hold the charging probe at the opposite end of the handle, so your hand is as far from the sphere as possible. If your hand is too close to the sphere, it can significantly change the constant C in Eq. (3), increasing the charge on the sphere for a given voltage. You want to minimize this effect so that the charge on the spheres can be accurately reproduced. 

Charge leakage always occurs to some extent remember that you are dealing with a powerful force. Perform measurements as quickly as possible after charging in order to minimize leakage effects. 

When charging or discharging the sphere on the torsion wire, always touch the probe on the opposite side of the attached rod so that you do not add excessive torque to the wire. 

The voltage is set with a sliding control. Be aware that the source “warms up” after you turn it on with the toggle switch and that the voltage as read from the dial increases slightly the longer the source is on. If you are charging a ball at, say, 4.0 kV, be sure that the needle is as close to the 4.0 KV mark as you can make it. (It’s OK to tweak the slider control.) 

The “ground” is essentially a vast reservoir that sucks up any excess charge. The process of grounding is tantamount to discharging excess charge, or making sure that an object is electrically neutral. Before each measurement, always ground yourself and each sphere. 

Always charge spheres when they are at their maximum separation, R 38 cm. 

Because the torsion constant is so small, the period of the torsion pendulum is very large. When a change is made, the sphere attached to the wire may oscillate, but very slowly. Be aware of this when measuring an angle or zeroing the torsion balance. 

Procedure 

You need the following items: 

PASCO Model ES9070 Coulomb Balance; use the following values in your calculations unless a different one is posted on the apparatus: 

K = 1.51 ± 0.02 μN/deg 

C = 1.92 ± 0.03 nC/kV 

Teltron Limited 813 kV power unit (voltage source) 

MHC digital calipers 

Charging probe (red) 

Grounding probe (black) 

1. Record the given values of K and C, along with explanations of what they are. 

2. Measure and record the diameter of the sphere on the sliding assembly. Include your estimated uncertainty. (Assume that the spheres are the same size.) Take at least two readings at different spots. The calipers give you a reading to the nearest hundredth of a millimeter; the metal balls may not be spherical to this precision. a. Measure and record the diameter of the business end of the charging probe. (Should be ~ 2 mm) 

Force Dependence on Equal Charges 

3. Move the sliding sphere to R = 4.0 cm. 

4. Set the torsion dial to 0° (Fig. 2). The spheres should be touching or close to it. a. Zero the torsion balance by appropriately rotating the bottom torsion wire retainer until the pendulum assembly is at its zero displacement position as indicated by the index marks (Figs. 2 and 3). See note above about the torsion pendulum and large periods of oscillation. 

b. Ground yourself (touch the metal end of the grounded probe) and the spheres (contact S_{T} with the grounded probe so that S_{T} touches S_{M} {See Figure 1}). 



Figure 3. Side view of Coulomb Balance. 

Figure 2. Photo showing torsion dial. 

5. Set the torsion dial to 170° (Fig. 2). The spheres should be touching. 6. Turn on the Teltron voltage source and adjust the output to 5.00 kV (or the maximum it will go). 

a. Charge both spheres by touching S_{T} in the same manner that you grounded the two. Assume that both charges are charged as if they were charged with the same voltage separately. 

b. Holding the charging probe so that it is straight up and down, place it so that the metal portion is at the point of contact of the two spheres. Apply slight pressure against S_{M} so that electrical contact is assured. What is happening physically is that the force from the twisted wire exceeds the repulsive electric force, and S_{T} remains in place. 

c. Reduce the torsion knob angle until S_{T} just begins to separate from the probe and S_{M}. At this angle the twisty force and the repulsive force are the same magnitude. Record the angle θ. 7. Ground yourself and the spheres as in Step 4.b. Repeat Step 6 using a voltage of 4.00 kV. 8. Repeat Step 7 three more times using charging voltages of 3.00, 2.00, and 1.00 kV. √ Checkpoint! Have the TA check your data before you proceed further. 

Force Dependence on Charge 

9. Move the sliding sphere to 4.0 cm and ground yourself and the spheres as before. 10. Move the sliding sphere as far as possible from the suspended sphere, R 38 cm. 

11. Set the torsion dial to 0° (Fig. 2) and double check the torsion balance equilibrium (zero displacement) position. 

12. Use a voltage of V_{T} = 2.50 kV to charge S_{T}. 

a. Charge S_{M} with a voltage of V_{M} = 0.50 kV. 

b. Move S_{M} to a position of R = 8.0 cm. 

c. Adjust the torsion knob as necessary to bring the pendulum back to the zero position. Record the angle θ required to do this. 

13. Repeat Steps 912 for the following V_{M} in kV, in this order: 1.00, 1.50, 2.00, 2.50, 3.00, 3.50, 4.00, 4.50, 5.00, 4.50, 4.00, 3.50, 3.00, 2.50, 2.00, 1.50, 1.00, 0.50. 

a. Include V_{M} = 0.00 and θ = 0 as data points. 

b. These are the minimum number of points. More points are better. If an angle doesn’t seem right, retake the data. √ Checkpoint! Have the TA check your data. 

Force Dependence on Distance 

14. Repeat steps Steps 912 for V_{T} = V_{M} = 5.00 kV and for the following R in cm: 20.0, 17.0, 14.0, 12.0, 10.0, 9.0, 8.0, 7.0, 6.0, 5.0. √ Checkpoint! Have the TA check your data. 

Additional Data 

15. Repeat steps Steps 912 for V_{T} = V_{M} = 4.50 kV and for R = 7.0, 6.0, and 5.0 cm. 

16. Repeat steps Steps 912 for V_{T} = V_{M} = 4.00 kV and for R = 5.0 cm. 

Analysis 

1. Start a Coulomb’s Law spreadsheet from the Second Semester selection of Templates in the Start Up menu. 

2. Enter known values. 

3. Calculate the radius a of a sphere. 

Force Dependence on Equal Charges 

4. In labeled columns enter your values of V_{M} in kV (V_{T} is echoed), R in cm, and θ in degrees for your first set of data, a constant sphere separation distance with the same varying charge applied to both spheres. 

5. For each of your data points, calculate the labeled quantities in the appropriate column. 

6. The plot of F vs. q with error bars is automatic, with linear and quadratic trendlines. a. In least squares fits, the R^{2} value is a measure of how good the fit is. The closer to one, the better the fit. On this basis, which of the two exhibited functions gives the better functional fit? That is, based on your data, does the force F between two equal charges vary as q or as q^{2}? 

Force Dependence on Charge 

7. In labeled columns enter your values of V_{M} and V_{T} in kV, R in cm, and θ in degrees for your second set of data, a constant sphere separation distance with one varying charge. 

8. For each of your data points, calculate the labeled quantities in the appropriate column. 

9. The plot of F vs. q_{M} with error bars is automatic, with a linear trendline. Does the data fall on or close to this line? 

Force Dependence on Distance 

10. Repeat Steps 7−8 for your second set of data, where the charges were held constant and the distance varied. 11. Again the plot with error bars, this time of F vs. R, is automatic. But this graph has three trendlines, a line, a polynomial of order four, and a power law. a. Based on the R^{2} value, which of the three functions is the best fit? What an Excel Trendline (a least squares fit) gives is the best function of that type that fits the given data. In Physics Lab, we need to go a step further and use our intuition. In this case, what do we expect the force to be for very large or very small values of R? As R gets smaller and smaller (approaches zero), do we expect the force to get larger, smaller, or stay the same? As R gets larger and larger (approaches infinity), do we expect the force to get larger, smaller, or stay the same? b. Using physical intuition, which is the better fit? √ Checkpoint! Have the TA check your calculations and best fit conclusion. 

Challenge: Finding Coulomb’s Constant 

12. Repeat Steps 7−8 for the second and third data sets combined. 13. Again the plot with error bars is automatic. This is a graph of F vs. q_{M }q_{T }/R^{2}. We have assumed that the results of your two experiments were that the force depended linearly on the product of the charges and on the inverse square of the distance. 

14. Do a linear least squares fit to find the experimental value of k and its uncertainty in μN·cm/nC. 

15. Convert the accepted value of to and compare with your experimental value. 

Questions 

1. Show a Sample Calculation of one of your values of u{q_{M}}. 

2. Propagate the uncertainty for . 