The Ball Bearing High Voltage Generator – A Demonstration Apparatus Illustrating Electrostatic Concepts at Multiple Levels

The BBHV generator, shown schematically in Fig. 1, consists of two rectangular, parallel metal plates, each carrying a charge +Q and –Q, respectively, to form the homogeneous electric field of the classic parallel plate capacitor. The two plates can simply be charged with a piece of rabbit fur and a small PVC tube.¹ Guided by two slightly inclined plexiglass tracks, two ball bearings roll side by side² through this electric field, which causes induced charges +q_B and –q_B on these two ball bearings. If the plexiglass tracks were to continue in a parallel fashion, the +q_B and –q_B charges on the two ball bearings would simply neutralize as they exit the electric field; however, the two tracks separate from each other (this is literally charge separation!) while still within the field, preventing the charge neutralization. Once outside the electric field, the two oppositely charged ball bearings are guided to two conducting spheres where they deposit their charge. This process is then repeated with many more pairs of ball bearings. An electro-meter connected to the conducting spheres shows that with each additional, separated pair of charged ball bearings dropping into the open conducting spheres, an additional quantum of charge +q_B and –q_B is being deposited in each sphere, respectively. For a brief YouTube clip of the operating BBHV generator, click on The BBHV Generator.MOV and Depositing BB’s Charge.mov at Ref. 3.

The basic operation of the BBHV apparatus is so simple that it can be explained to middle school students who have just been introduced to the existence of positive and negative charges for the first time. At this level, students know, or have been told, that equal charges repel, and opposite charges attract, and that charges (electrons) can move through a conductor. Let’s assume that the plate closer to us carries a negative charge and that the plate farther away from us carries a positive charge, as shown in Fig. 1. The two ball bearings, rolling side by side through the space between the metal plates, form a figure-eight shaped conductor, consisting of two small spheres that are in contact at one point. Therefore, some of the negative charges get pulled toward the positive plate and pushed away from the negative plate. Similarly, positive charges end up on the ball bearing closer to the negative plate.

If the teacher has revealed to their students that in a conductor it is only some of the negatively charged electrons that can move and that the rest of the atoms that make up the conductor stay in place, then the students can argue a bit more precisely that electrons move from the ball bearing that is closer to us toward the positive plate, i.e., making the ball bearing that is farther from us negatively charged and leaving the ball bearing closer to us positively charged, i.e., with a lack of electrons.

The students then observe that as the tracks separate, it is impossible for the charges to return to their original place, keeping both ball bearings charged. As they drop into the open conducting spheres at the end of the track, they deposit these excess charges on each conducting sphere. A digital electroscope, or a traditional gold leaf electroscope, clearly shows the students that each set of ball bearings adds a bit more charge to the conducting spheres.

This is also a good time to emphasize again that the direction of a vector gets reversed when it is multiplied by a negative number, i.e., that the direction of the force on a negative charge is opposite to the direction of the electric field. The electric field is pointing from the positive plate (farther from us in Fig. 1) to the negative plate (closer to us). Therefore, the force on the electrons is directed away from us and they end up on the ball bearing farther away from us, leaving the ball bearing closer to us positively charged.

The question remains, how big is the electric field that we need in the previous segment to calculate the force on the charges? For students in an introductory college course this presents a good opportunity to review the application of Gauss’s law for a case with planar geometry. At this level we allow them to make the simplifying assumption that the plates are very large compared to the ball bearings; in fact, they may assume that the plates are infinite in extent and that they contain a uniform surface charge density σ.

In reality, the two metal plates are not infinitely large. Therefore, in a numerical physics course, the BBHV generator motivates the need to calculate the electric field or the electric potential between two finite-sized plates carrying, for simplicity, a uniform charge distribution σ, or the actual charge distribution σ(x,y).

For introductory physics students (at the HS or college level) the question might arise how the positive and negative induced charges are distributed on the two ball bearings. For example, (i) are the negative excess charges distributed evenly ‘throughout’ the ball bearing? (ii) Are they spread throughout the ball bearing but mostly in the half that is closer to the positive plate due to the attraction of all those positive charges on the positive plate? Or (iii) are they in the half of the ball bearing that is touching the positively charged ball bearing because of their attraction to those positive charges? Or (iv) are the charges exclusively on the surface of the ball bearing? (v) Are they evenly distributed over the surface, or (vi) are they non-uniformly distributed over the surface? And finally, if they are non-uniformly distributed over the surface of the ball bearing, (vii) what is their charge distribution?

After having shown that there cannot be an electric field inside a conductor in an electrostatic situation, this presents a good opportunity for our students to use Gauss’s law to prove that the interior of either one of the two ball bearings must indeed be free of any excess charges.⁴ 

In contrast, when Gauss’s law is used on a charge configuration to calculate its electric field, it is only “useful” when that charge distribution has planar, cylindrical, or spherical symmetry and we can place the Gaussian surface such that all angles between E and da are either 0˚, 90˚, 180˚, or 270˚, E is constant over the Gaussian surface, such that the integral $∯E⋅da$ becomes trivial.

One of the learning goals at the HS and introductory college level E&M courses is to have students acquire practice in drawing electric field lines. In particular, we would like our students to understand that (i) a region with a stronger E-field is represented by field lines that are closer together, (ii) an area with sparser field lines is indicative of a weaker E-field, (iii) field lines start and end perpendicularly on conductors, (iv) a tangential E-field component would lead to electrons moving in the conductor surface until the field lines finally end up meeting the conductor surface perpendicularly.

While it quickly becomes boring for students to draw the homogeneous electric field inside a parallel plate capacitor, the electric field between the charged parallel plates of the BBHV generator, containing two ball bearings with their respective induced charges as shown in Fig. 3, is a configuration that is not trivial and also not found in a textbook.⁶ It is perfectly suited for students to show that they have a firm grasp of the shape of an electric field in the space that contains several conducting bounding surfaces.

Requesting students to draw the field lines in this situation can also be a quick diagnostic tool for the instructor. Students can demonstrate here that the field lines start and end perpendicularly on the conducting surfaces, i.e., the plates as well as the ball bearings. Their field line pictures also reveal if they have an understanding that the charge density on the plates near the pair of ball bearings is larger than on other parts of the plates. Furthermore, their field line pictures show if they understand that a finite length capacitor has a fringing field⁷ and that there is a higher charge density at corners, such as at the ends of the parallel plates.⁸ 

Finally, the fact that the parallel plates with the pair of ball bearings is a configuration that is not found in a textbook makes this a suitable problem for the conceptual part of a midterm or final exam.

At the advanced undergraduate level, the BBHV generator serves as an ideal example, a paragon, to motivate the use of the “Method of Image Charges.”⁹ We suggest that before we investigate the configuration of two ball bearings between two metal plates, we look at the much simpler situation of one ball bearing located in front of one plate. In fact, we shall first look at just a point charge in front of an infinite, conducting, and grounded plate.

where the V_i represent the value of the equipotential contour.

The simple extension renders this as an ideal homework problem, solving the real problem of an apparatus the students have seen and played with during their class. It should be fairly easy for students to use again, e.g., GeoGebra to plot a series of equipotential contours as shown in Fig. 5.

The extension from the situations described in the previous two sections to one where one (or two) charged ball bearing(s) is (are) located between two metal plates is just a bit harder, as this involves images of images ad infinitum.¹² The problem can be solved by students of an advanced undergraduate course on electrodynamics or graduate students in an introductory graduate electrodynamics course.^13,14

Of interest to the students might be the non-uniform charge distribution on the metal plate as the charged ball bearing moves past it. The answer to this question comes immediately from the method of image charge approach. A derivative of the potential expressed in either Eq. (3) [or Eq. (5) for the case of two ball bearings] yields the electric field in the area surrounding the ball bearings. Evaluating the field at the surface of the conductor plate then yields the surface field, and σ(x, y) = ε₀ E(x, y) is the desired surface charge density (cf. Fig. 6).¹⁵ 

Clearly, there are many more questions that arise from the BBHV generator. However, due to space limitations, I cannot address all of them here. Thus, in this last segment, I will simply pose a few more questions that may serve as an inspiration to the reader. View the “Construction Manual for the BBHV Generator” online.³ 

• What are the charges +q_B and –q_B that will accumulate on each of the ball bearings as they move side by side through the electric field?

• What are the surface charge distributions +σ_B (θ, ϕ) and –σ_B (θ, ϕ) on the two ball bearings?³ 

• What would be the shape of the electric field if just one ball bearing is located in the previously homogeneous electric field?¹⁶ 

• Approximately, how much charge is required on the two copper spheres such that there will be an arc from one sphere to the other?

• Who does the work in separating the charges?

• What is the minimum inclination of the plexiglass track inside the parallel plate capacitor such that the two ball bearings will not be held back when they leave the electric field?

• Could we use the electricity generated by charge separation to power the motor that recycles the ball bearings (i.e., the ball bearing elevator motor, see the video Dual Ball Bearing Elevator at Ref. 3).

• Why is the BBHV generator not a perpetual motion machine?

• In the first version of the BBHV generator, the conducting spheres at the end of the track did not have openings at the top; they were perfect spheres. Thus, the ball bearings did not drop inside the conducting spheres; they simply bounced off of their outsides. In this situation, the ball bearing bouncing off the conducting sphere takes some amount of charge with it. How much charge would each ball bearing take with it as it bounces off the conducting sphere?

• What if we connected the right conducting sphere to the left capacitor plate and the left conducting sphere to the right capacitor plate? How much quicker would we accumulate charge on the conducting spheres?

I would like to express my gratitude to former Physics & Astronomy Department technician Rick Lindsey, who put together the first version of the BBHV generator. The dual ball bearing elevator was built by our expert technician Jonathan Barrick. My thanks also to Sophie Kirkman and Catrina Hamilton-Drager for proofreading the manuscript.