When our ancestors realized the existence of electricity by observing lightening or by rubbing a rod against amber, the physicist pushed ahead their research on it and since discovered the force of electric, and that this force obey the same law as Newtonian force.

Therefore, the delta of kinetic energy is also equal to the integral of electric force multiplied by the minor distance x.

It also means if the energy is unchanged or conservative, then it should be agnostic of the paths taken, or if it’s a closed loop of path, no matter what circumference it is, the total is zero.

To deduce the equation on two dimensions, gradient is introduced as a useful symbol. The following

Moving forward to three dimensional situation on gravitational force:

However, the real focus is to verify if the electric force is also, like gravitational force, a conservative force? How to do it?

The pictorial display of a unit charge is like this

Now if we focus on a unit charge, to find is electric field E, and prove it’s integral agnostic of path, the q*E, general charge is also integral-able agnostic of path. So we pick from above graph, from the field radiated by q0 in the center, a point 1 to point 2, the radius is r1 and r2 respectively.

In essence, after this verification, one can gain that the below equation stands, the kinetic part (1/2mv2) plus the electric potential (charge times unit charge potential at position r do not vary.

The great advantage to use this formula is that it exclude the complexity of calculating E (electric filed), which will have to deal with vector calculation. things are reduced to simply the sum up of unit charge at different position to reach V(r) first, then do the further integral.

The last part on dipole calculation is quite puzzling, showing the screenshot for further study…