An easy way to accelerate a charge is to allow it to move through a potential difference. For instance, take a charge and place it inside a parallel-plate capacitor. We'll assume the capacitor has a uniform field E, and a potential difference with a magnitude of:

|ΔV| = Ed, where d is the plate separation.

A positive charge released from beside the positive plate will accelerate towards the negative plate. Cutting a hole in the negative plate allows it to escape. Similarly, a negative charge released from near the negative plate will accelerate across the gap and leave the parallel plates at high speed.

Applying conservation of energy ideas to find the speed:

U_i + K_i + W_nc = U_f + K_f

There are no non-conservative forces acting, the initial kinetic energy is zero, and we can define the plate where the charge exits the capacitor as the zero of potential energy (i.e., U_f = 0).

This gives K_f = 1/2 mv² = q ΔV so:

v = [ 2qΔV /m ]^1/2

Can you think of any practical applications of such a system for accelerating charges?

There is a useful energy unit that's particularly applicable to accelerating electrons, protons, or ions. This unit is the electron-volt (eV).

1 eV is the amount of energy associated with moving one electron through a potential difference of 1 volt.

1 eV = 1.6 x 10^-19 J