Acceleraton and Deflection of Electrons
In this experiment you will observe the acceleration and deflection of electrons by electric fields. To describe this motion we use a rectangular coordinate system where the z axis is along the tube axis (parallel to the undeflected electron beam), the x axis is horizontal, and the y axis is vertical.
An electron emitted from the cathode and passing through the
various apertures of the electron gun emerges from anode
with a velocity 
in the
+z direction; the magnitude of 
is determined
by the potential differences
between K and 
. In traveling from
K to 
the electron loses potential energy 
;
thus if it leaves the cathode with negligible initial kinetic
energy, its kinetic energy after emerging from
is given by the relation
The electron now passes through the deflection plates. If there is
no potential difference between the plates, it passes straight
through and strikes the center of the screen, making a small green
spot. If a potential difference 
is applied between
the vertical deflection
plates (the pair with the planes horizontal), then there will be a
transverse field 
between the plates and the resulting
force on the electron gives it a transverse velocity 
. The axial
component of velocity 
is unaffected. The electron emerges from
the plates traveling at an angle 
determined by
as shown in the figure below.
The deflection angle 
and the deflection distance D can be
calculated in terms of the applied voltages and the dimensions of
the electrodes and their spacing.
First, a potential difference 
between two plates separated
by a distance d, as in the figure,
produces a transverse electric field 
,
and a transverse force whose magnitude is
. During the time 
that the electron takes to pass between the plates, this force
gives the electron a transverse momentum 
equal to the impulse of this force.
This gives,
But the time interval 
is also the time the electron
takes to travel along the z-axis
a distance l equal to the length of the plates, at axial
velocity 
. Thus 
. This relation can be
solved for 
and the result substituted into Eq.(4).
The result is
Finally, the deflection angle 
is given by
Substituting the energy relation of Eq. (1), we obtain
This equation shows that the deflection increases with deflecting
voltage 
, as might be expected, and also increases with the
length l of the plates. With longer plates the deflecting electric
field acts for a longer time and causes greater deflection.
The deflection is
inversely proportional to d: The more closely spaced the plates,
the greater the deflecting field for a given potential difference.
Finally, reducing the accelerating potential
increases the deflection by reducing the axial velocities of the
electrons, permitting the deflecting field to act for a longer time.
After the elctron beam leaves the deflection region, it again
travels in a straight line, tangent to the path at the point where
it left the deflecting region. Thus the green spot on the screen is
deflected vertically a distance D given by the relation
, where L is
the distance from the plates to the screen. (We neglect
the slight curvature of the screen.) A more detailed analysis of the
motion between the plates shows that L should be measured from the
center of the plates to the screen.
Thus, we have
