San José State University

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Thayer Watkins
Silicon Valley
USA

 Forces, Charges and Potential Energies in Systems of Particles

Suppose there are two type of particles, A and B, and that like particles repel and unlike particles attract. It is very convenient to explain such a situation in terms of charges, say qA and qB, where qA and qB are of opposite signs. The force between two particles, i and j, would be of the form

#### F = Hqiqjf(si,j)

where H is a constant and si,j is their separation distance. If i and j are of the same type then qiqj is positive for the force is a repelling one. On the other hand, if i and j are of different types then qiqj is negative and the force is an attraction.

The potential energy involved for the two particles is then

#### Vi,j(s) = Hqiqj∫s∞f(z)dz

The values of potential energy for the three possible interactions at the same distance; i.e., VA,A(s0), VA,B(s0) and VB,B(s0). What conditions do these three quantities have to satisfy in order that their values can be explained by a pair of charges qA and qB?

If such a pair of value do exist then

#### VA,A/VA,B = qA/qB = VA,B/VB,Bwhich reduces to VA,AVB,B = V²A,Bor, equivalently VA,AVB,B − V²A,B = 0

This is the relation that has to be satisfied in order that the values of VA,A, VA,B and VB,B can be explained by two charges qA and qB. The same relation would have to prevail for the forces FA,A, FA,B and FB,B.

If the effects are arrayed as a matrix; i.e.,

#### | VA,AVA,B | |VB,AVB,B |

then the condition to be satisfied is that the determinant of this matrix be zero.