Thermionic Emission and Cathode Rays

Thermionic emission

Metals contain free electrons that move about inside the metal. At room temperature they stay in, because the metal surface holds them back: each electron needs a certain minimum energy to escape.

When a metal is heated strongly, its electrons gain kinetic energy. Some of them gain enough to leave the surface.

A metal filament (often tungsten) is heated by passing a current through it. The heated electrode that emits the electrons is called the hot cathode. The cathode is negative, so the emitted electrons are repelled away from it. If a positive anode is placed nearby, the electrons are pulled across to it.

Two conditions are essential:

• The filament must be hot. Heating gives the electrons enough energy to overcome the surface barrier. A cold filament emits essentially none.

• The tube must be evacuated. Air molecules would collide with and scatter the electrons, and the white-hot filament would also burn away in air. In a vacuum the electrons travel freely.

Saturation. Increase the filament temperature and more electrons are emitted each second, so the anode current rises. Increase the anode voltage and the current rises at first, but then levels off. At that point the anode is collecting every electron the filament emits, so a higher voltage cannot raise the current. Only a hotter filament can.

Cathode rays

The beam of electrons travelling from the cathode in an evacuated tube was first called cathode rays, because it seemed to come out of the cathode. Experiments in the 19th century showed that the rays behave like a stream of negatively charged particles:

Everyday devices that used cathode rays were cathode ray tube (CRT) televisions and computer monitors, and the oscilloscope. In a CRT the beam was steered across the screen by magnetic or electric fields to draw the picture line by line.

Accelerating electrons

An electron with charge $e = 1.6 \times 10^{-19}\ \text{C}$ released near the cathode is accelerated by the p.d. $V$ between cathode and anode. The work done on it by the field is

If it starts (almost) from rest and nothing is lost on the way, this work becomes kinetic energy:

This is energy conservation: electrical potential energy converted to kinetic energy. The mass of the electron is $m_e = 9.11 \times 10^{-31}\ \text{kg}$.

The energy gained by one electron through 1 V is a convenient small unit, the electronvolt (eV):

So an electron accelerated through 200 V gains 200 eV. Use joules in calculations unless told otherwise.

In a real tube a few electrons collide with stray gas molecules left in the imperfect vacuum, so the speed on arrival is slightly less than the ideal value.

Beam current

The current in a beam is the charge arriving per second, $I = Q/t$. Each electron carries $e$, so the number of electrons is $N = Q/e$.

When the electrons hit the screen or anode they are stopped, and their kinetic energy is transferred as heat (and, on a fluorescent screen, some light). The power delivered by a beam is the energy per electron times the number arriving per second, which simplifies to

where $I$ is the beam current and $V$ the accelerating voltage. This heating is why an anode needs a high melting point (tungsten, for example).

Deflection in an electric field

A beam entering a uniform electric field of strength $E$ between two parallel plates feels a force $F = eE$ on each electron. Because the electron is negative, the force is opposite to the field direction, so the beam bends towards the positive plate.

Why a parabola? The motion splits into two independent parts, exactly like a ball thrown horizontally:

• Horizontally: no force, so constant speed $v$. The time between plates of length $L$ is $t = L/v$.

• Vertically: constant force, so constant acceleration $a = \dfrac{eE}{m}$ (from $F = ma$). The sideways displacement is $s = \tfrac{1}{2}at^2$.

Eliminating $t$ gives $s \propto x^2$, which is a parabola. After leaving the plates, the electrons move in a straight line again.

For plates a distance $d$ apart with p.d. $V$, the field is $E = V/d$.

Magnetic fields and Thomson's experiment

A magnetic field acts on a moving charge with a force

for a charge $q$ moving at speed $v$ perpendicular to a field $B$ (for an electron, $q = e$). The direction is given by Fleming's left-hand rule. Remember that the rule uses conventional current, which for electrons points opposite to their velocity.

The force is always perpendicular to the velocity. It changes the direction of motion but never the speed. So the electrons move in a circle, with the magnetic force providing the centripetal force:

Because $F$ is perpendicular to the displacement at every instant, the magnetic force does no work on the electron. Its kinetic energy and speed stay constant.

How Thomson measured e/m

In the 1890s J. J. Thomson used a hot cathode to produce a beam, then crossed an electric field and a magnetic field:

1. A voltage across the plates produces an electric force $eE$ on each electron.

2. A magnetic field is applied at right angles, producing a force $Bev$ in the opposite direction.

3. The magnetic field is adjusted until the spot on the screen returns to its undeflected position. The forces are then equal:

So the speed of the beam is found without needing to know $e$ or $m$.

1. The electric field is then switched off. With only $B$, the beam follows a circle of radius $r$ and $r = \dfrac{mv}{eB}$, so

Thomson found that $e/m$ was the same whatever metal the cathode was made of. This showed that the electron is a universal constituent of matter. The accepted value is $e/m = 1.76 \times 10^{11}\ \text{C kg}^{-1}$.

Back to the old television: the "warm-up" was the filament heating up so that it could release electrons. A p.d. of several kilovolts accelerated them, and magnetic fields steered the beam across a fluorescent screen.

Summary

• Thermionic emission: a hot metal releases electrons because heat gives them enough energy to escape the surface. The tube is evacuated so that the electrons are not scattered and the filament does not burn.

• Cathode rays are beams of electrons. They travel in straight lines, cast shadows, cause fluorescence, and are deflected by electric and magnetic fields in the way expected for negative particles.

• Acceleration: $\tfrac{1}{2}mv^2 = eV$, with $1\ \text{eV} = 1.6 \times 10^{-19}\ \text{J}$. Beam current: $I = Q/t$, $N = Q/e$, and beam power $P = IV$.

• Electric field: force $eE$ towards the positive plate gives a parabolic path (constant speed horizontally, constant acceleration sideways).

• Magnetic field: force $Bev$ is perpendicular to the velocity, so the path is a circle, with $r = \dfrac{mv}{eB}$, and the force does no work.

• Thomson's method: balance $eE = Bev$ to get $v = E/B$, then use the radius with $B$ only to find $e/m$.