The drop generators in inkjet printers can fire over 100k droplets per second. Some of these droplets are charged and can be steered to different points on the paper by electric fields. The droplet has mass m, charge -q reaching deflecting plates at a speed v.The plates are a  distance x apart, have a length and the p.d. between them is V. a)Show that the vertical acceleration of the droplet is qv/mx. Assume that gravitational forces are negligile. b)Describe the motion of this ink droplet (i) as it moves between the deflecting plates (ii) as it moves beyond them.

a)

As the second principle of physics states for a mass m having an acceleration a, the force acting on it is

`F =m*a`

The electric force acting on a particle having charge q in an electric field E is

`F_e = q*E`

From the two above relations we have

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a)

As the second principle of physics states for a mass m having an acceleration a, the force acting on it is

`F =m*a`

The electric force acting on a particle having charge q in an electric field E is

`F_e = q*E`

From the two above relations we have

`q*E = m*a` or equivalent `a = (q*E)/m`

Now by DEFINITION the relation between the electric field E and electric potential V (for a distance x) is

`E =U/x`

which gives a total value for acceleration

`a =(q/m)*(V/x)`

b) For the trajectory of the particle between the plates and outside the plates please see the figure below.

To describe the motion of the particle between plates we write down the equations of motion on the horizontal h and vertical v axes.

`d_h(t) = v_h*t =v*t`   and  `v_h(t) =v = "constant"`

`d_v(t) =a*t^2/2`   and   `v_v(t) =a*t `

which means that the projection of the particle trajectory on the vertical axis v will be a parabola (see second equation) , between the plates.

The equations of the particle motion (on v and h axes) outside the plates are

`d_h(t) = v_h*t =v*t`  and `V_h(t) = v = constant`

`d_v(t) = v_v*t`  and `v_v = constant`

which means the projection of the particle trajectory on the vertical axis will be a straight line, outside the plates.

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