Section 3.3 · Worked examples · 1 of 2
Worked example 3.3.1: Electron gun
The two worked examples below put numbers into the two applications. Each one uses only , and the three Laplace solutions of Problems 3.3.1–3.3.3, with no new physics. Numerical constants: , , , .
Electron gun
#cf. Application 3.3.1
Given. Electrons leave a heated cathode at rest, and the anode is away at , followed by a field-free drift space.
Find. (a) in the gap and the kinetic energy in J and eV, (b) the final speed, (c) the time in the gap and in the drift space, and (d) for , classically and relativistically.
The work done on a charge moved between two points is , whatever the path. An electron () going from the cathode () to the anode () therefore gains the kinetic energy , independent of the shape of the electrodes. Between plane electrodes the field is uniform (Problem 3.3.1).
The field is uniform and points from the anode to the cathode (the force on the electron points the other way):
From ,
In the gap the acceleration is constant, , so from
(equivalently , since the mean speed in the gap is ). In the drift space the speed is constant:
At the classical formula gives . Relativistically the kinetic energy is , with :
Units: . Both speeds are below , and agrees with the value from the acceleration.
One kilovolt already gives 6 % of the speed of light, and the whole flight takes a few nanoseconds. At 25 kV (a colour CRT) the classical speed is 3.6 % too high, at 100 kV about 14 %. The kinetic energy , however, is exact at any voltage, which is why particle energies are quoted in electron volts.