# Why don't electrons gain potential energy as they near the positive terminal of a battery?

If electrons move through the wire because of the force applied by the electric field that attracts them from negative terminal to positive terminal and this force is setting them to motion in a direction not opposing to the direction of the force, why don't they gain potential energy as there is work being done on them?

And like this even if they lose the potential energy, they gained from the chemical reactions inside the battery, on a lamp or a resistor they should gain some energy as they move.

You are partially correct, as Potential due to a positive charge is given by V= $$\frac {kq}{r}$$. As an electron moves towards a proton it goes from a region of lower potential to higher potential. But I assume that you know $$\Delta U$$=$$\Delta$$Vq. Hence Change in Potential energy is equal to Change in potential multiplied by Charge as Change in Potential is positive, also as q is negative Change in Potential Energy is negative and electron loses Potential Energy and gains Kinetic Velocity. Hope you understand. Cheers