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When discussing electric potential energy, the conservation of energy is a fundamental concept that plays a crucial role. It states that the total energy in an isolated system remains constant over time. This means that energy can neither be created nor destroyed; it can only be transformed from one form to another or transferred from one object to another.

In the context of point charges, as seen in the exercise, conservation of energy implies that the electric potential energy combined with any work done by or against electric forces results in a new form of energy or a redistribution of energy within the system. If we have a charge in an electric field and move it from one point to another, the work done on the charge changes its electric potential energy.

For our point charges example, this can be observed when the charge is moved from point a to point b. The electric potential energy at point a and the work done to move the charge to point b together must equal the electric potential energy at point b, assuming no other forces are doing work. This concept ensures that the initial and the final state of the system can be connected through energy transformations without loss or gain, adhering to the conservation of energy principle.

The work-energy principle is another pivotal concept to understand electric potential energy in a physical context. This principle tells us that the work done on an object is equal to the change in its kinetic energy. However, when dealing with point charges in an electric field, the principle extends to relate work done to the change in electric potential energy, especially when no kinetic energy is involved, as charges can be held stationary.

Applying this principle to our exercise, the work done by the electric force when the charge moves from point a to b is negative because the force opposes the displacement. This negative work indicates that energy is being taken away from the charge, thereby reducing its electric potential energy. To calculate the final electric potential energy at point b, we can deduct the work done from the initial electric potential energy at point a.

This aligns with our intuition that if you have to 'work against' a force to move an object, you're decreasing its potential in the field created by that force. Since the scenario provided does not include kinetic energy, the focus lies solely on this transfer between work and electric potential energy.

The term point charges refers to hypothetical charges that are concentrated at a single point in space with no volume. Despite this being an idealization, it serves as an extremely useful concept in understanding electric forces and fields.

In practice, when considering point charges like charges q_{1} and q_{2} in our exercise, the electric potential energy of the system depends on the relative positions of these charges as well as their magnitudes. The closer the charges are, the stronger the interaction and the higher the potential energy if they are of opposite signs, whereas charges of same sign would have higher potential energy when separated.

Calculating Electric Potential Energy

To calculate the electric potential energy between point charges, you can use Coulomb's law, which relates the potential energy to the product of the charges divided by the distance between them and multiplied by the Coulomb constant. However, in our exercise, we already have the initial electric potential energy and the work done, so we do not need to calculate this from scratch, but rather apply conservation of energy to find the new electric potential energy after charge q_{2} moves to a new location.