Placing two similarly charged particles in space Now, I will make a hypothetical situation. Assume that we place two similarly charged particles (lets take electrons) in space. Imagine that there is no other force acting on the particles except the repulsive force and the gravitational force of the particles. In other words, only these two electrons are present in the universe. So they are free from any outside interference. Now by nature, these electrons will start moving away from each other due to the repulsive force. Since there is nothing to stop them (gravitational force will only slow them down and not stop them as it is of a lesser magnitude than the repulsive force) they will keep moving and never stop. Over here we exclude expansion of space also for no complications. Now since the particles will keep moving as there is a constant repulsive force acting on them, they will do infinite work because $Work = Force * displacement$ and the displacement over here will keep increasing. Please tell me what is the problem in my thought experiment because it violates conservation of energy.
 A: Ok, so we all agree that $W = \vec F · \vec x$. If the force varies, then the total work on each electron is calculated using an integral:

$W = \int_{x_0}^\infty\vec F · d \vec x$

Here, $W$ is work, $\vec F$ is the electrical force, $\vec x$ is the distance of the charge from the center of your universe, and $x_0$ is the starting point from where you begin the experiment.
And the electrical force between the electrons is :

$F = q {e_1 e_2 \over (2x)^2}$

$q$ is the Coulomb constant, $e_1$ and $e_2$ are the charges (they are equal), and $2x$ is the distance between them. You know, one $x$ to the left and one $x$ to the right. And since the force and the displacement are colinear we can omit the arrows.
Now we can solve the integral:

$W = \int_{x_0}^\infty q {e_1 e_2 \over (2x)^2} dx$
$W = {q e_1 e_2 \over 4 }\int_{x_0}^\infty {1 \over x^2} dx$
$W = {q e_1 e_2 \over 4 } [{-1 \over x}]_{x_0}^\infty $
$W = {q e_1 e_2 \over 4 } {1 \over x_0}$

And that is basically the formula for the eletric potential energy!
A: as they move farther apart,, the magnitude of potential energy of the system reduces(which is 0 at infinity and infinity at 0 seperation in terms of magnitude),,, and the kinetic energy increases(which in this case tends to a specific value at infinity given by (2q1q2/4.pi.e)rm)^(1/2) where r is initial seperation
),, so the sum is conserved,, which means energy is conserved,,, hence , no it does not violate the principle.
A: The problem is not so simple. In fact I think the final kinetic energy of the electrons is not equal to the initial potential energy. But this not means the energy conservation is violated. A moving charge that accelerate emits electromagnetic waves. This waves transport energy so the electron slows down while moving to infinity. The final total energy is:
$E_{final} =  E_{waves} = \frac{e^2}{4 \pi \epsilon_0 x_0} = E_{initial}$
So there is no kinetic energy at infinity since, due to the self force (http://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force), the electron stops after some time. 
