San José State University
Department of Economics

San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 The Force due to a Distributed Charge Compared to the Force from the Center of the Distributed Charge

This is an investigation of the relative error involved in replacing the force due to distributed objects with the force with the force those objects would have concentrated at the center of their distribution. The analysis has relevance for phenomena at the nuclear and atomic level all the way up to the galactic cluster level.

There is a wonderful theorem in mathematical physics to the effect that a uniform charge on a spherical surface has the same effect for an inverse distance squared force as that charge concentrated at the center of the sphere. This theorem extends to a charge distributed uniformly throughout a spherical ball. This theorem however does not apply if the charged object has a shape other than a sphere, such as a disk. It also does not apply is the charge on the spherical shape is not uniform. This in particular means that if the shape is spherical but the charge (or mass) is located only at specific points on the spherical surface. And, the theorem does not apply if the force is not an inverse distance squared force.

What the analysis deals with is the circumstances in which the theorem holds approximately and asymptotically holds exactly.

## The Simplest Case

Consider first two equal objects located at ±σ distances from the origin. For an inverse distance squared (IDS) force the force experienced per unit charge at a distance x from the origin is

#### F(x) = 1/(x+σ)² + 1/(x+σ)²

The force that would prevail if both objects located at the origin is