Physics

Non-static interpretation of the Coulomb’s Law

Coulomb’s Law is the basis of electrostatics. While deriving the law we consider the observer to be sharing the reference frame of one of the two point charges. Therefore, for the observer, one of the charges is at rest while the other charge moves in the electric field of the static charge. Many books explain this geometrical setup by calling one charge ‘fixed’ at a position, while the other moving towards or away from it. This explanation implicitly  assumes the observer to be sharing positions with one of the charges, hence the ‘fixed at position’  assumption. Do note that in such cases  the zeroth value of our distance-measurement instrument positioned at one of the charges becomes the observer.

Though this gives the right value of the electrical force due to one charge on the other, it does not feel intuitive.  In experiments, the observer is never sharing positions with any of the charges. He instead keeps one charge at rest with respect to the reference axes (which may be the walls of his laboratory in the current case) and varies the distance of the other in order to measure the resultant electric force.

In order to make our approach as general as possible, in the forthcoming interpretation I position the observer at rest with respect to some object other than the two charges such that for the observer both charges appear to be moving towards or away from each other. The presented interpretation is a qualitative explanation of the electrostatic force experienced between a pair of moving charges as the product of the potentials developed in bringing both charges in the electric field of the other simultaneously.

This interpretation was derived by reverse-engineering the Coulomb’s Law. It was noticed that the equation of the electrostatic force acting between two point charges q1 and q2 seperated by a finite distance r can be manipulated to match the product of potentials developed at the endpoints of the hypothetical line joining the two charges.

Equation

F=k*(q1)*(q2)/(r^2),

where k=1/(4*pi*E0) for vaccum, E0=permittivity of vaccum,pi=3.14 approximately,  q1 and q2 are the two point charges and r is the distance between the two charges; can be split up to give

F={k*(q1)/r12}*{k*(q2)/r21}/k ,

where r12 is the distance of q1 measured from q2 and r21 is the distance of q2 measured from q1. Do note that r12=r21 , always. This can also be written as

F=V(q1)*V(q2)/k ,

where V(q1) and V(q2) are the potentials developed on q1 and q2 repectively when they traverse the electric field  of the other charge. Here k becomes a constant of proportionality. The equation can also be written as,

F=(4*pi*Eo)*V(q1)*V(q2) , for vaccum.

It is important to note the consequences of this interpretation. Firstly, the choice of the position of observation has become arbitrary. The point charges are not given preference for force measurement. The constituents of the sample space of the set of points of observation are {q1, q2, X} where X is any other point at rest with respect to an object or coordinate axes in which both charges q1 and q2 appear to be moving.

For the observer, the electrostatic force of the binary system is the product of the two instataneous potentials of q1 and q2 with respect to each other.

The individual potentials are found to be mutually dependent.

V(q1) is proportional to V(q2).

The only dangerous asumption that remains is that either the two ‘k’s that appear in the equation collapse to one for the observer, or that we have (1/k) as the proportionality constant for the equation of the resultant force and the product of the two potentials. The latter asumption suffices the mathematical rigour required for the equation while the former looks more like a grave mistake. The collapse hypothesis could be correct. It needs to be critically re-examined and established by supporting physics or exchanged, if possible, with a better explanation.

For the time-being we can assume that the Coulomb’s electrostatic force equation for a binary charge system is actually the superposition of two simultaneous relative potential equations, considered from a third reference frame in relative motion with both the point charges.

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