Thermionic Emission and Cathode Rays – Atomic Models (Senior 5 Physics) | Ganzaa

Atomic Models

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Senior 5 · Physics

Atomic Models

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Lesson 4 of 4

Thermionic Emission and Cathode Rays

25 min

Electrons on demand

Until the early 2000s almost every television and computer screen was a cathode-ray tube: a glass bottle with the air pumped out, in which a beam of electrons swept across a glowing screen hundreds of times a second to paint the picture. Medical X-ray tubes, electron microscopes and the magnetron in a microwave oven still work on the same first step: boiling electrons out of a hot metal. In this lesson you see how electrons are released, how they are accelerated and steered by electric and magnetic fields, and how J. J. Thomson used exactly these effects in 1897 to show that "cathode rays" are streams of tiny charged particles, the electrons of Lesson 1.

Thermionic emission

A metal contains many free electrons, moving at random among the positive ions. They cannot normally leave the surface, because the metal attracts them back: to escape, an electron needs at least a certain energy, called the work function $\phi$ of the metal (a few electronvolts). When the metal is heated, its electrons gain kinetic energy, and at high temperatures (around 1000 °C or more for most cathodes) a significant number have enough energy to escape. This release of electrons from a hot surface is thermionic emission.

The emission rises very steeply with temperature, and it is greater for a metal with a low work function. Practical cathodes are a tungsten filament, or a small metal tube coated with oxides of barium and strontium (low work function) and heated by a filament inside it. The cathode must be in a vacuum, so that the electrons are not stopped by collisions with gas molecules and the hot filament does not burn in air.

The electron gun

The emitted electrons are attracted to a positive anode, a disc with a small hole. A potential difference $V$ between cathode and anode does work $e V$ on each electron, so at the anode

eV = \tfrac{1}{2}mv^2 \quad\Longrightarrow\quad v = \sqrt{\frac{2eV}{m}}

Electrons passing through the hole form a narrow beam. Historically, a beam leaving the cathode of a discharge tube was called a cathode ray, before anyone knew what it was.

Definition

Thermionic emission is the emission of electrons from the surface of a heated metal. Cathode rays are streams of electrons travelling at high speed in a vacuum, produced by thermionic emission (or by a high voltage in a gas at very low pressure). Example: electrons accelerated from a hot cathode through 2000 V reach $v = \sqrt{\frac{2 \times 1.60 \times 10^{-19} \times 2000}{9.11 \times 10^{-31}}} = 2.65 \times 10^7\,\mathrm{m\,s^{-1}}$ , about 9% of the speed of light.

Properties of cathode rays

Observation	What it shows
An object in the beam casts a sharp shadow on the screen	they travel in straight lines
They are deflected towards a positive plate, and by a magnet	they carry negative charge
They heat a metal target and can turn a tiny paddle wheel	they carry energy and momentum
They make glass and certain coatings glow	they cause fluorescence (used in screens)
When stopped suddenly by a metal target, they produce X-rays	basis of the X-ray tube
They are the same whatever cathode metal or gas is used	electrons are part of every atom

Deflection by an electric field

Between two parallel plates with potential difference $V_{p}$ a distance $d$ apart, the field $E = \frac{V_p}{d}$ exerts a constant force $e E$ on each electron, towards the positive plate. An electron entering parallel to the plates at speed $v$ keeps its horizontal speed, and accelerates sideways with $a = \frac{eE}{m}$ . After a horizontal distance $x$ (time $t = \frac{x}{v}$ ) its sideways displacement is

y = \frac{1}{2}\cdot\frac{eE}{m}\cdot\frac{x^2}{v^2}

a parabola, exactly like a ball thrown horizontally under gravity.

Deflection by a magnetic field

A magnetic field $B$ at right angles to the beam exerts a force $e v B$ , always perpendicular to the velocity. It changes the direction of motion but not the speed, so the electrons move in a circle:

evB = \frac{mv^2}{r} \quad\Longrightarrow\quad r = \frac{mv}{eB}

Use Fleming's left-hand rule with the current direction opposite to the electrons' motion, because electrons are negative.

Worked example 1: deflection between plates

Electrons accelerated through 2000 V enter the space between two plates 5.0 cm long where $E = 1.0 \times 10^4\,\mathrm{V\,m^{-1}}$ . Find (a) their speed, (b) the time between the plates, (c) the sideways deflection at the end of the plates.

Solution

(a) $v = \sqrt{\frac{2eV}{m}} = 2.65 \times 10^7\,\mathrm{m\,s^{-1}}$ .

(b) $t = \frac{0.050}{2.65 \times 10^7} = 1.89 \times 10^{-9}\,\mathrm{s}$ .

(c) $a = \frac{eE}{m} = \frac{1.60 \times 10^{-19} \times 1.0 \times 10^4}{9.11 \times 10^{-31}} = 1.76 \times 10^{15}\,\mathrm{m\,s^{-2}}$ ; $y = \frac{1}{2}at^2 = \frac{1}{2} \times 1.76 \times 10^{15} \times (1.89 \times 10^{-9})^2 = 3.1 \times 10^{-3}\,\mathrm{m}$ . The weight of the electron ( $mg \approx 9 \times 10^{-30}\,\mathrm{N}$ ) is less than $10^{-14}$ times the electric force, so gravity can be ignored.

Measuring the specific charge $\frac{e}{m}$

In 1897 Thomson measured the ratio of charge to mass, $\frac{e}{m}$ , for cathode rays. A school version uses a fine-beam tube: a glass bulb containing a little gas, an electron gun, and a pair of Helmholtz coils that give a uniform magnetic field across the bulb. The gas glows faintly where the beam passes, so the circular path can be seen and its diameter measured.

Combining $eV = \frac{1}{2}mv^2$ (the gun) with $r = \frac{mv}{eB}$ (the circle) to eliminate $v$ :

v = \frac{eBr}{m} \quad\Longrightarrow\quad eV = \frac{1}{2}m\left(\frac{eBr}{m}\right)^2 \quad\Longrightarrow\quad \frac{e}{m} = \frac{2V}{B^2r^2}

Thomson himself used crossed fields: an electric field and a magnetic field at right angles to each other and to the beam, adjusted until the beam went straight. Then $e E = e v B$ , giving the speed $v = \frac{E}{B}$ without knowing $\frac{e}{m}$ ; switching off $E$ and measuring the deflection in $B$ alone then gives $\frac{e}{m}$ . Today's accepted value is $\frac{e}{m} = 1.76 \times 10^{11}\,\mathrm{C\,kg^{-1}}$ , about 1800 times the charge-to-mass ratio of a hydrogen ion. Once Robert Millikan measured the charge $e = 1.60 \times 10^{-19}\,\mathrm{C}$ in 1909, the electron's mass followed: $m = 9.11 \times 10^{-31}\,\mathrm{kg}$ .

Worked example 2: a fine-beam tube result

Electrons are accelerated through 250 V in a fine-beam tube and move in a circle of diameter 9.6 cm in a field of 1.10 mT. Find $\frac{e}{m}$ and compare it with the accepted value.

Solution

$r = 0.048\,\mathrm{m}$ . $\frac{e}{m} = \frac{2V}{B^2r^2} = \frac{2 \times 250}{(1.10 \times 10^{-3})^2 \times 0.048^2} = \frac{500}{2.79 \times 10^{-9}} = 1.79 \times 10^{11}\,\mathrm{C\,kg^{-1}}$ , within 2% of $1.76 \times 10^{11}$ . The diameter is the hardest reading to take accurately, and $r$ is squared, so a 1% error in $r$ gives a 2% error in $\frac{e}{m}$ .

Worked example 3: crossed fields as a speed selector

Plates 4.0 cm apart have 800 V across them. What magnetic field, at right angles, lets electrons of speed $2.67 \times 10^7\,\mathrm{m\,s^{-1}}$ pass undeflected?

Solution

$E = \frac{800}{0.040} = 2.0 \times 10^4\,\mathrm{V\,m^{-1}}$ . Undeflected when $e E = e v B$ : $B = \frac{E}{v} = \frac{2.0 \times 10^4}{2.67 \times 10^7} = 7.5 \times 10^{-4}\,\mathrm{T} = 0.75\,\mathrm{mT}$ . Faster electrons are bent one way by the magnetic force, slower ones the other way by the electric force: only one speed gets through.

Uses of electron beams

Cathode-ray oscilloscope and old televisions: two pairs of plates (or coils) steer the beam across a fluorescent screen.
X-ray tubes: electrons from a hot cathode, accelerated through tens of thousands of volts, strike a tungsten target; about 1% of their energy comes out as X-rays for medical imaging, the rest as heat.
Electron microscopes: fast electrons behave as waves of very short wavelength, so they can show details far smaller than light can (Unit 7).
Industry: electron beams weld metals, sterilise medical equipment and some foods, and cure paints and plastics.

Warning

Safety. Electron tubes run at hundreds to thousands of volts: only the teacher should connect them, and nobody should touch the connections while the supply is on. Electrons stopped by a target above a few thousand volts produce X-rays; school tubes are designed and rated to keep this low, so use only the voltages stated by the maker and keep a sensible distance. Glass vacuum tubes can implode if knocked: handle them gently by their base.

Warning

Common mistakes. (1) Forgetting that electrons are deflected towards the positive plate and opposite to the direction Fleming's rule gives for current. (2) Saying a magnetic field speeds electrons up: it only changes their direction. (3) Using the diameter as $r$ . (4) Using the heater voltage (about 6 V) instead of the accelerating voltage in $eV = \frac{1}{2}mv^2$ .

Tip

Electric field across the beam: parabola (constant force, like a thrown ball). Magnetic field across the beam: circle (force always perpendicular). Both at once, balanced: straight line, $v = \frac{E}{B}$ .

Demo: electron charge-to-mass ratio (Caltech's Feynman Lecture Hall). Watch the glowing electron beam bend into a circle in the field of the coils, and how the circle shrinks when the field is increased.

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Practice

Explain what is meant by thermionic emission and why the cathode of an electron gun must be in a vacuum.
Find the speed of electrons accelerated through 5000 V.
State four properties of cathode rays and what each shows.
Electrons of speed $3.0 \times 10^7\,\mathrm{m\,s^{-1}}$ enter a magnetic field of 2.0 mT at right angles. Find the radius of their path.
Derive $\frac{e}{m} = \frac{2V}{B^2r^2}$ for the fine-beam tube.
In crossed fields, $E = 3.0 \times 10^4\,\mathrm{V\,m^{-1}}$ and $B = 1.5\,\mathrm{mT}$ . What speed passes undeflected?
Electrons accelerated through 2000 V pass between plates 5.0 cm long with $E = 1.0 \times 10^4\,\mathrm{V\,m^{-1}}$ . Find the angle at which they leave the plates.
Explain why doubling the accelerating voltage in a fine-beam tube does not double the radius of the circle. By what factor does it change?

Show answersHide answers

The emission of electrons from a hot metal surface, because heating gives some electrons more than the work function. In a vacuum the electrons are not stopped by gas molecules, and the hot cathode does not oxidise.
$v = \sqrt{\frac{2 \times 1.60 \times 10^{-19} \times 5000}{9.11 \times 10^{-31}}} = 4.2 \times 10^7\,\mathrm{m\,s^{-1}}$ .
For example: straight lines (sharp shadows); negative charge (deflected towards a positive plate); energy and momentum (heat a target, turn a paddle wheel); fluorescence (make screens glow).
$r = \frac{mv}{eB} = \frac{9.11 \times 10^{-31} \times 3.0 \times 10^7}{1.60 \times 10^{-19} \times 2.0 \times 10^{-3}} = 0.085\,\mathrm{m}$ .
$r = \frac{mv}{eB}$ gives $v = \frac{eBr}{m}$ ; substitute into $eV = \frac{1}{2}mv^2$ : $eV = \frac{e^2B^2r^2}{2m}$ , so $\frac{e}{m} = \frac{2V}{B^2r^2}$ .
$v = \frac{E}{B} = \frac{3.0 \times 10^4}{1.5 \times 10^{-3}} = 2.0 \times 10^7\,\mathrm{m\,s^{-1}}$ .
Sideways speed $v_y = at = 1.76 \times 10^{15} \times 1.89 \times 10^{-9} = 3.3 \times 10^6\,\mathrm{m\,s^{-1}}$ ; $\tan\theta = \frac{3.3 \times 10^6}{2.65 \times 10^7} = 0.125$ , $\theta = 7.1^\circ$ .
$r = \frac{mv}{eB}$ and $v \propto \sqrt{V}$ , so $r \propto \sqrt{V}$ : doubling $V$ multiplies $r$ by $\sqrt{2} = 1.41$ .

Key points

Thermionic emission: a heated metal releases electrons; an anode at voltage $V$ gives them $v = \sqrt{\frac{2eV}{m}}$ .
Cathode rays are fast electrons: straight lines, negative charge, energy and momentum, fluorescence, X-rays when stopped.
Electric field: parabola, $y = \frac{eEx^2}{2mv^2}$ . Magnetic field: circle, $r = \frac{mv}{eB}$ . Crossed fields: $v = \frac{E}{B}$ .
Fine-beam tube: $\frac{e}{m} = \frac{2V}{B^2r^2} = 1.76 \times 10^{11}\,\mathrm{C\,kg^{-1}}$ .

1Electrons on demand

2Thermionic emission

3The electron gun

4Properties of cathode rays

5Deflection by an electric field

6Deflection by a magnetic field

7Measuring the specific charge em \frac{e}{m} me​

8Uses of electron beams

Electrons on demand

Thermionic emission

The electron gun

Properties of cathode rays

Deflection by an electric field

Deflection by a magnetic field

Measuring the specific charge $\frac{e}{m}$

Uses of electron beams