Velocity of Electrons in a Vacuum Tube

Author: J.B. Hoag

It is also possible with the apparatus of Fig. 1 B to measure the speed at which the electrons travel down the vacuum tube. It is found that they travel very fast; with velocities of the order of one million (10⁶) m/s or about ten million (10⁷) miles per hour. This is approximately one one-hundredth the speed of light. The velocity of electrons depends upon how many cells make up the battery which is used to speed them up. When the total accelerating voltage is the same as that of the electric light circuits in our homes, namely 110 volts, the electron velocity will be 6.55 · 10⁸ cms. per sec. When the voltage is doubled to 220 volts, the velocity will be increased to 10 · 10⁸ cms. per sec. It is to be noted that the velocity does not double when the voltage is doubled. In fact, the voltage must be made four times as great in order to double the velocity of the electrons, and it must be made nine times as great in order to treble the velocity of the electrons. In other words, the velocity of the electron is proportional to the square root of the voltage (provided the voltage is not too high). The relationship will be readily seen in Fig. 1 C.

Figure 1C: Velocity of electrons accelerated by various voltages.

The equation connecting the voltage E and the velocity v is as follows: Ee/300 = mv²/2, where e is the charge of the electron in electrostatic units and m is its mass in grams. This equation is accurate to within 1 per cent up to 7,000 volts. For higher voltages, a more complicated relativity equation must be used. The right-hand side of the equation is a well-known expression for the energy of moving bodies. This "kinetic" energy is expressed in units called "ergs". An energy of 1.60·10^-10 ergs will be acquired by an electron speeded up by means of a 100-volt battery. Thus 1 electron volt is equivalent to 1.60·10^-12 ergs. The electron volt is abbreviated "eV" and is often used today in referring to the velocity of a charged particle. It should be clear, however, that the volt is a measure of the energy of the particle rather than its velocity.