Question

Show that work done in an electric field is independent of path.

Answer 1

Incidentally, we can be sure that the field returns the energy to the particle without loss because if there were any loss then this would imply that non-zero work is done in taking a charged particle around a closed loop in an electric field generated by fixed charges. We call a force-field which stores energy without loss a conservative field. Thus, an electric field, or rather an electrostatic field (i.e., an electric field generated by stationary charges), is conservative. It should be clear, from the above discussion, that the concept of potential energy is only meaningful if the field which generates the force in question is conservative.
A gravitational field is another example of a conservative field. It turns out that when we lift a body through a certain height the increase in gravitational potential energy of the body is actually stored in the surrounding gravitational field (i.e., in the distortions of space-time around the body). It is possible to determine the increase in energy of the gravitational field directly, but it is a very difficult calculation involving General Relativity. On the other hand, it is very easy to calculate the work done in lifting the body. Thus, it is convenient to calculate the increase in the energy of the field from the work done, and then to ascribe this energy increase to the body, via the concept of gravitational potential energy.
In conclusion, we can evaluate the increase in electric potential energy of a charge when it is taken between two different points in an electrostatic field from the work done in moving the charge between these two points. The energy is actually stored in the electric field surrounding the charge, but we can safely ascribe this energy to the charge, because we know that the field stores the energy without loss, and will return the energy to the charge whenever it is required to do so by the laws of Physics.