When moving a charge against an electric field, we need a force or acceleration to overcome the constant force of the electric field.if the force applied is equal to the force of the electric field there should be no motion or work done, so why does the charge move at all? And why is there any work done?
In a static electric field, work done on a charged particle is equal to the change in potential energy. If the charged particle is held still in the electric field by an external force, no work is done on the charged particle or on the field itself, because the charge does not change its electric potential energy or its position
If the charged particle is moving initially in the same direction as the force the field is exerting on it, and the force is exactly balanced by an external force, the charged particle will not accelerate -- but it will keep moving at a constant velocity. Because the charged particle is moving, its potential energy will change. Without the external force that change in potential energy would show up as an opposite change in the kinetic energy of the charged particle. Because the external force prevents the charged particle from accelerating, the kinetic energy does not change. In this case, the change in potential energy shows up as the external force times the distance the charged particle moves (that is, the dot product of the position change vector and the force vector). That means the external force does do work on the charged particle. That work will be negative or positive, depending on the relative directions of the two vectors.
The force on a charge $q$ in a uniform electric field, $E$, is $F=qE$ and is constant. The work required to move the charge a distance $d$ in the field is
If the charge is positive, and the charge moves in the direction of the electric field (+ to – by convention) solely under the influence of the field, the field does positive work on the charge. It accelerates the charge gaining kinetic energy equal to the work done by the field.
If an external force is applied to the charge against the electric field (- to +) and the force is equal to the force of the electric field so that it moves at constant velocity, the external force does positive work on the charge, but the electric field does an equal amount of negative work, so the net work done on the charge is zero. The end result is the work done by the external force on the charge is stored as electrical potential energy of the charge.
Hope this helps.