applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley 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 q_{A} and q_{B},
where q_{A} and q_{B} are of opposite signs. The force between two particles, i and j, would be of the
form

F = Hq_{i}q_{j}f(s_{i,j})

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

The potential energy involved for the two particles is then

V_{i,j}(s) = Hq_{i}q_{j}∫_{s}^{∞}f(z)dz

The values of potential energy for the three possible interactions at the same distance; i.e., V_{A,A}(s_{0}),
V_{A,B}(s_{0}) and V_{B,B}(s_{0}). What conditions do these three quantities have to
satisfy in order that their values can be explained by a pair of charges q_{A} and q_{B}?

If such a pair of value do exist then

V_{A,A}/V_{A,B} = q_{A}/q_{B} = V_{A,B}/V_{B,B} which reduces to
V_{A,A}V_{B,B} = V²_{A,B} or, equivalently
V_{A,A}V_{B,B} − V²_{A,B} = 0

This is the relation that has to be satisfied in order that the values of V_{A,A}, V_{A,B} and
V_{B,B} can be explained by two charges q_{A} and q_{B}.
The same relation would have to prevail for the
forces F_{A,A}, F_{A,B} and
F_{B,B}.

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

| V_{A,A}

V_{A,B} |

|V_{B,A}

V_{B,B} |

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