Question:

A proton a absolutely charged elementary particle is released from rest at a point in space where the electric potential is 15 volts. Following to its release the particle is subject to no force other that due to the electric field characterized by the electric potential. Afterwards when the particle is at a point in space where the electric potential is  25 volts. Are the conditions discussed in the above quote possible?

Answer:

For the positively charged particle the electric potential energy has the similar algebraic sign as the electric potential of the point in space the particle occupies. Consequently in going from a point in space where the electric potential is 15 volts to a point where it is 25 volts the proton is going from a point where it has one positive value of potential energy to a point where it has a greater positive value of potential energy. Now once we obtain to talking about potential energy rather than potential I like to think concerning the gravitational potential energy of an object near the surface of the earth. It isn't exactly relevant here but it makes for a useful analogy. Regard as a rock near the surface of the earth. I know that while it goes upward its potential energy is increased. Now it is probable for a free rock to go upward but to do so it must have some initial upward velocity. A liberated rock at rest near the surface of the earth will never go upward. It will certainly not go to a region where its gravitational potential energy is greater than that at its current location. The similar goes for any kind of potential energy. A proton beginning at rest in the absence of all forces but the electric force characterized by the electric potential will never go to a location with a greater potential energy than that at its initial location. To do thus would be like releasing a rock near the surface of the earth and seeing it fly upward. It would be a abuse of the conservation of energy.

The idiom at rest is absolutely key here. Given sufficient initial velocity the proton could definitely make it from a point where the potential is 15 volts to a point where it is 25 volts. In doing consequently it would be going against the force of the electric field just like a rock thrown upward, on the way up is going against the force of gravity. However the proton would be slowing down the whole way just as the rock would be slowing down the whole way up. The potential energy of the proton would be rising while its kinetic energy would be decreasing, such that the total energy would remain the same.