Transverse Electric Field

Consider a charged particle (e.g. an electron) moving along X-axis with constant velocity v, i. At x = 0, it enters into a region where a uniform constant electric field E exists along Y-axis, i.e. for x > 0, E_y j, E_x = E_z = 0. This electric field applies a forceq E along the Y-axis, i.e. for x > 0, E = E_y j, E_x = E_z = 0. This electric field applies a force q E along the Y-axis, while particle continues to move along X-axis with constant initial velocity vx i. Such a case is realized in laboratory by passing a beam of electrons between the plates of a parallel plate capacitor, as in cathode ray tube (CRT).

The electric field applies a constant force, qEy, on the particle along Y-axis. Hence,

 

which gives v_y = ay t, y = ½ a_y r²                       (ii)

where ay = qEy/m ; y and vy at t = 0 are zero. In the same time t, particle moves a distance x along x along X-axis with constant initial velocity vx i, so that

x = vx t                                                              (iii)

Eliminating t, we find the trajectory of the particle:

 

which is equation of a parabola. This is analogous to parabolic motion of a projectile which moves horizontally with constant a velocity and vertically under constant acceleration g.

If the particle remains in the remains in the field only between x = 0 to x = l (e.g. l may be length of capacitor plates in CRT), then the deflection of particle along Y-axis, as it comes out at x = l, is

 

At x = l, particle velocity v is given by

 

which makes an angle θ with the X-axis, where

 

 

Since the displacement on screen is proportional to electric field, the position of the beam on screen tells about the value of (an unknown) electric field (or potential) which may be applied across the capacitor plates.