Comment on "Walking a charged pith ball perpendicular to an electric field".1

Paul J. Dolan, Jr., Dept. of Physics & Astronomy, University of Maine, Orono, Maine 04469-5709a
 
 

Abstract: Two geometric arguments are presented that assist in visualizing the fact that a pith ball, between two almost parallel plates, will acquire a net motion in a direction perpendicular to the electric field, as demonstrated in a recent issue by Ouseph & Davis.
 
  I would like to point out two, primarily geometric, arguments, that conceptually suggest and clarify that a pith ball between two (nearly) parallel plates will acquire a net motion perpendicular to the electric field, as demonstrated mathematically by Ouseph and Davis recently in "Apparatus and Demonstration Notes".

For both arguments, one needs to keep in mind that the distance the pith ball travels is only a few diameters (the spacing previously quoted is between 1.0 and 2.5 cm, with a 0.3 cm diameter ball), and more importantly that the angle (phi) between the plates is small (0.06 rads is used by the authors). The pith ball starts its journey 'from rest' from the lower (horizontal) plate.

Consider first the purely geometric situation, at very small angle. The initial direction of the ball is normal to the plate (as it must be, since the electric field is normal to the surface of the conductor). As a first approximation, if the angle is very small, we may ignore the arcing shape of the electric field, and the ball will arrive at the upper plate at the small angle phi, with respect to the normal of the upper plate. After a (somewhat elastic) collision, the ball leaves the upper plate at this small angle phi, but to the other side of the normal. As a result of this collision, the ball has acquired a velocity perpendicular to the electric field! Geometrically, one can easily show that the ball will now encounter the horizontal plate at an angle of 2*phi, and leave at this same angle. As long as our assumptions remain valid, the ball will gain an additional 'phi' with each collision with either plate, and thus an additional component of the velocity perpendicular to the electric field, in qualitative agreement with the authors' Figure 3.

However, the electric field is not straight, which leads to the second conceptual argument. When the ball initially leaves the lower (horizontal) plate, from rest, the direction of the electric field, and thus of the acceleration, will tend to bring the direction of the ball's path back toward the normal of the upper plate. Even if the ball follows the arc completely, and hits the upper plate at its normal, when it leaves the plate it will NOT leave 'from rest', but from the recoil will have a component of the velocity away from the vertex -- it has again acquired a velocity perpendicular to the electric field! This component of the velocity also propels the ball into a region of lesser field strength. At the very least, the ball will acquire an additional component of the velocity in this direction at every collision with the upper plate. If the field strength is insufficient to return the ball to the normal, prior to any collision, then there will also be a component of the velocity simply due to the reflection, as discussed above. Additionally, as the ball moves further from the vertex, the electric field strength decreases and so the increment to the ball’s velocity will tend to decrease, in qualitative agreement with the authors' Figure 3.

Thus, this quite interesting demonstration is useful for both qualitative and quantitative discussions, making it useful for students at many different levels.
 
 

(1) P. J. Ouseph and C. L. Davis, "Walking a charged pith ball perpendicular to an electric field", Am. J. Phys., 69(1), 88-90 (2001).

  1. Permanent address: Physics Dept., Northeastern Illinois University, Chicago, Ill. 60625, P-Dolan@neiu.edu.