Coulomb’s Law for Two Charged Spheres

In 1767 British natural philosopher Joseph Priestley became the first scientist to propose that electrical force followed an inverse-square law, similar to Newton's law of universal gravitation. However, he did not elaborate on this.[1] The general law was enunciated by French physicist Charles Augustin de Coulomb. In 1784 Coulomb used two charged spheres arranged in a torsion balance to verify that the interaction forces varied as the inverse square of the distance between their centers.
Your goal in this assignment is to use very simple apparatus along with video analysis to explore the inverse square law proposed by Coulomb. The equipment consists of two charged spherical conductors – one that hangs from a long pair of threads and another one attached to the end of an insulating rod. Before starting this activity you should watch the movie entitled <coulomb.mov>. / Figure 1: Simple Coulomb apparatus with two charged conducting spheres.

1. Preliminary Questions

Note: You will receive full credit for each prediction made in this preliminary section whether or not it matches conclusions you reach in the next section. As part of the learning process it is important to compare your predictions with your results. Do not change your predictions!

(a)  Suppose the two spheres shown in Figure 2 are charged in the same manner so that they are both either positively charged or both negatively charged. Will the two charged spheres attract or repel each other? Explain the reason(s) for your answer.

(b)  Use the diagram below to draw and label a vector showing the electrical force that the sphere with charge q1 exerts on the sphere with charge q2. Also draw and label a vector showing the electrical force that the sphere with charge q2 exerts on the sphere with charge q1. Do the vectors have the same length? Point in the same direction? Explain. Hint: What does Newton’s Third Law suggest?


2. Activity-Based Questions

Theory: Briefly assume that the prod in Figure 2 is far away from the ball. If you were to use your finger to push on the hanging ball with a very small but steady horizontal force in the positive xdirection, the ball would rise through an angle until the x-component of the tension along the string that holds the ball would just counter the constant pushing force you are exerting. At the same angle, the upward ycomponent of the tension force would just counterbalance the downward gravitational force acting on the mass, m. By resolving these forces, you could then find an equation that could be used to determine that horizontal push you exerted as a function of m, g, and x where x is the horizontal distance that you displaced the ball.

If the action-at-a-distance force exerted by the charged sphere on the charged hanging ball replaces your push, then the same equation can be used to find the electrostatic or Coulomb force on the hanging ball. Since each of the <coulomb.mov> frames represents a “photo” showing the prod sphere being held steady so that the suspended sphere shown in the movie is stationary, you can then determine how the Coulomb force in each frame varies as a function of the distance, r, between the centers of the two charged spheres.

You can begin your exploration of Coulomb’s Law by deriving an equation that allows you to calculate the magnitude of the Coulomb force on the hanging ball as a function of its displacement, x, from the vertical.

(a)  Use Figure 3 to draw a free body diagram for the forces acting on the mass m. Include vectors with labels showing the approximate relative strengths of the gravitational force, , the tension in the string, and the horizontal electrostatic force due to the charge on the prod’s sphere, .

(b)  If the hanging ball in each frame shown in the <coulomb.mov> is stationary, explain why

[Eq. 1] and also

[Eq. 2]

(c)  Next show that

(d)  Refer to Figure 4 below, and find the equation for as a function of x and L. Hint: Start by using the approximation for very small angles that so then the lengths y and L can be assumed to be almost equal, provided that xball < L.

(e)  Combine the equations you derived in parts (c) and (d) to find the equation needed to calculate the experimentally determined magnitude of the Coulomb force as a function of L, xball, m, and the local gravitational constant g.

Experimental Verification of the Inverse Square Force: You now have an equation you can use with the Logger Pro software to analyze the movie <coulomb.mov> and determine the relationship between the magnitude of Coulomb force on the hanging ball, Fcoul and the distance, r, between the ball on the prod and the hanging ball. In order to proceed, open the Logger Pro experiment file Coulomb.cmbl>. The movie has been calibrated using the meter marker of the first frame. Also, the origin has been moved to the center of the hanging ball in the first frame when no horizontal forces act on it. It is hidden. Columns have been established for video analysis of the locations of the prod and the ball.

(f)  Use the information on the title screen of the movie to determine the effective length, L, of the string suspending the ball in meters (that is, the vertical distance given for the suspension line from the ceiling to the hanging ball). Also write down the mass of the ball in kg.

(g)  Use the Set Active Point tool () to proceed with video analysis for the positions of the center of the prod’s ball, denoted x_prod and y_prod for each frame starting with frame 1, the first frame after the title frame. Then repeat the process to determine the positions of the center of the hanging ball for the same frames, denoted x_ball and y_ball. There are more detailed instructions in the text box included in <Coulomb.cmbl> Beware! In Step (h) you’ll need to calculate the distance, r, between the ball centers in each frame by taking the difference between the locations of the two spheres. Be consistent in marking the same location on each ball because when doing the r calculations location errors will be magnified by small differences in the exact location of each ball.

(h)  Next use the New Calculated Column feature to calculate, r, the distance between the balls from the data. Since the column has been established for you, choose Column Options from the Data menu and select r . The Calculated Column Options dialog will open so you can enter the equation you need.

(i)  Use the equation you derived in part (e) together with the x_ball data to have Logger Pro calculate the experimental value of the electrostatic force in each frame from your data. Repeat the process in the F_coul column to enter the equation needed for the calculation.

(j)  Sketch the data points that appear in the Fcoul vs. r graph frame that has been established. Note: If your data points don’t fit in the graph frame that has been established, check your calculations.

(k)  Analyze your data using the Logger Pro Curve Fit tool in the Analyze menu to fit your plot of Fcoul vs. r with an equation of the form Fcoul = A/r2. Write down the value of A and its uncertainty with units to 3 significant figures. Note: If you know how to do modeling (or manual fitting) you can select values.

A = ±

(l)  Draw conclusions. Does the inverse square law proposed by Coulomb seem to hold so that the relationship Fcoul = A/r2 holds within the limits of experimental uncertainty? What is the percent uncertainty?

3. Reflections on Your Findings

(a)  Describe the most plausible sources of experimental uncertainty in your data.

(b)  If the charges on the spheres were increased, what do you think would happen to the value of A?

Physics with Video Analysis 25 - XXX

[1] Schofield, Robert E., The Enlightened Joseph Priestley: A Study of His Life and Work from 1773 to 1804. University Park: Pennsylvania State University Press, 2004, pp 144-156.