# Calculate the final velocity of the charge

### Newton’s second law

F=qE and F=ma. Equating these we obtain a = qE/m.

Now from the equations of motion we have v2 = 2as + u2. As the charge starts from rest u = 0 and substituting a = qE/m we see that the final velocity reached by the charge will be v = √(2qEs/m), where:

- the displacement(s) is the plate separation

- q and m are the charge and mass of the charge respectively

- E is the electric field strength.

### Conservation of energy

As the force on the charge is F =qE, the work done by the field on the charge when it accelerates the charge between the plates will be given by: work = Fs = (qE)s.

In a vacuum, the charge will have no collisions with gas molecules; all the work done on the charge will go into the kinetic energy of the charge.

We can therefore write: work done = energy gained or qEs = 1/2mv2.

Rearranging these quantities gives v = √(2qEs/m), the same expression for the final velocity obtained using Newton’s second law above.