# At infinite separation do charges get kinetic energy?

When charges are infinitely separated, the potential energy is reduced to zero. As we know that loss in potential energy is nothing but the gain in kinetic energy . So do charges start flowing at infinite distance of separation ? If not then their must be a drop in total mechanical energy.
Why energy is not conserved here?

• What kind of charges?
– nasu
Commented Apr 22, 2019 at 2:26
• @nasu The kind of charges aren't important.. potential energy of any two charge system is 0 when at infinite distance of separation Commented Apr 22, 2019 at 2:29
• "So do charges start flowing at infinite distance of separation" - Welcome New contributor Sreetama ghosh hazra! What does "infinite distance of separation" mean? Imagine two charges that are separated by a (finite) distance $x$. Can these two charges ever have "infinite distance of separation"? Commented Apr 22, 2019 at 2:32
• Potential energy of a system of charges can be anything you want it to be. It is most common to choose the value at infinite to be zero. It is important for the analysis to know if you mean same sign charges or opposite sign charges.
– nasu
Commented Apr 22, 2019 at 2:35
• @AlfredCentauri No, sir it can't never be infinite .. actually there's nothing called infinite.. I mean to a large distance such that it's potential becomes zero.. practically it must be impossible! Commented Apr 22, 2019 at 2:38

## 2 Answers

The answer depends on many factors.Firstly,in order to separate charges work must be done on the system of charges by an external force. You see, the conversion of potential energy into kinetic energy is a special case of the more general work energy theorem which states that the change in kinetic energy of a system is equal to the work done by all forces on the body.

Wnet = ΔKE

Since charges on their own will not separate to infinity(because of their attraction),an external force will have to be applied to move them to infinity.

Writing the work done by external force as Wext and work done against the electrostatic force (which is the same as change in it's potential energy) as - Welectro,we have

Wext - Welectro = ∆K.E

If the work done by external force is equal to that of electro static force, ∆K.E will be zero so no kinetic energy will be gained and if it is greater than work done by electro static force ∆K.E is positive meaning there is a gain in kinetic energy.

Hope this helps!Please ask if you have any doubts.

If you have two charges of opposite sign, their potential energy increases as the separation increases. So at infinite separation their potential energy is maximum and the kinetic energy is at the minimum value. In order to reach this "infinite separation" without the intervention of other forces their initial KE needs to be high enough so that at infinity is at least zero. So they move slower and slower as the separation increases and their KE decreases asymptotically to a finite value. The total value (PE + KE) is the same everywhere along the path.

If the charges have the same sign thy don't need an initial KE to reach the "infinite separation". They can start at rest and as they repel each other the separation increases. As the separation increases their PE decreases and the KE increases so that again the sum is constant. At infinite separation the KE is maximum and PE is zero.