# Electrons and Photons

### Cathode Ray

Cathode rays are the streams of electrons and were discovered by Sir William Crooke. They are produced by:

(i) A discharge tube containing gas at allow pressure of the order $10^{-3}$ mm of Hg. At this pressure the gas molecules ionize and the emitted electrons travel towards positive potential of anode. The positive ions hit the cathode to cause emission of electrons from cathode. These electrons also move towards anode. Thus the cathode rays in the discharge tube are the electrons produced due to ionization of gas and that emitted by cathode due to collision of positive ions.

(ii) An electron gun containing a filament fitted in a tube having a number of slits. The filament is heated by passing a current which may be controlled by a rheostat. The emitted electrons move towards slits under accelerated potential and emerge in the form of a collimated electron stream.

#### Properties of Cathode Rays:

1. Cathode rays are the streams of electrons. Therefore they carry negative charge.

2. The cathode rays are independent of the nature of the gas or electrodes employed in the discharge tube. Therefore e/m for cathode rays is universal constant:

$= \dfrac{1.6 \times 10^{-19}}{9.1 \times 10^{-31}}$

3. They can be deflected by electric and magnetic fields.

4. They have penetrating power and can penetrate through small thickness of matter (eg. Thin AL foil).

5. On striking the target of high atomic weight and high melting point, they produce X- rays.

6. They produce fluorescence on certain substance and hence affect photographic plate.

7. They have small ionizing power and ionize the gas through which they pass.

8. They travel in straight lines and cast shadow of objects placed in their path.

9. They produce heat when allowed to fall on matter.

10. They exert mechanical pressure so they can rotate a small paddle wheel.

11. They can exhibit interference and diffraction phenomena under suitable arrangements. Thus they may behave as waves.

### Discovery of Electron

The electron was discovered by sir J.J. Thomson in 1897. He showed that cathode rays are simply the stream of electrons. Thomson was awarded Noble prize for this discovery. The mass of electrons is $9.1 \times 10^{-31} kg$ and the charge on electron is $- 1.6 \times 10^{-19}$ coulomb.

#### Determination of e/m

The e/m of cathode rays was determined by Thomson by using crossed electric and magnetic fields.

If E and B are mutually perpendicular electric and magnetic fields and if an electron beam entering perpendicular to both the fields with velocity v remains undeflected, then:

$\overrightarrow{f_e} + \overrightarrow{F_m} = 0$

$e v B = eE \, \, \, or \, \, \, v = \dfrac{E}{B} \cdots equation\, \, 1$

‘E’ being charge of electron.

If ‘r’ is the radius of circular path of electrons in magnetic field only, then:

$r = \dfrac{mv}{eE} \, \, or \, \, \dfrac{e}{m} = \dfrac{v}{rB} = \dfrac{E}{rB^2}$

The value of $\dfrac{e}{m}$ is called the specific charge and for electrons it comes out to be $1.76 \times 10^{11} coul \ kg$ .

### Millikan’s Experiment

The charge on the electron was found by Millikan’s oil drop experiment.

Millikan’s Experiment

In equilibrium of a charged oil drop between the region of plates:

$qE = mg \rightarrow q = \dfrac{mg}{E}$

As q = ne

Therefore,

$\dfrac{mg}{nE}$

Where electric field,

$E = \dfrac{Potential \, \, difference \, \, between \, \, plates}{Seperation \, \, between \, \, plates}$

$= \dfrac{V}{d}$

### Electrical Conduction in Gases

The electrical conduction of gases is studied with the help of discharge tube. If pressure is reduced to zero atmospheric pressure, the following effects are observed.

(a) At atmospheric pressure no discharge takes place.

(b) Above 1 mm of hg: At a pressure more than 1 mm of Hg a luminosity is observed which is confined to each electrode, but major part of tube remains dark. This is called dark discharge.

(c) At 1 mm of Hg: When pressure is reduced to 1 mm oh Hg, along luminous column starting from anode fills the whole space. This is called positive column.

(d) At 0.5 mm of Hg: When the pressure is reduced to 0.5 mm of Hg, a colored glow is seen at cathode, called the cathode glow. Now the positive column breaks into a number of discrete patches of light, called striations. These are separated from each other by dark intervals. The region between negative column and striations remains dark; called Faraday’s dark space.

(e) At 0.1 mm of Hg : When the pressure is reduced to 0.1 mm of Hg, the striations positive column and negative glow move towards anode and so a dark space appears near the cathode, called Crooke’s dark space.

(f) At $10^{-2} - 10^{-3}$ mm of Hg : Finally when the pressure is reduced to $10^{-2} - 10^{-3}$ mm og Hg, the striations disappear and whole tube is filled with dark space. At pressure cathode rays are produced.

Electrical Conduction in Gases

### Quantum (Particle) Nature of Light

Some phenomena like photoelectric effect, Compton effect, Ramen effect could not be explained by Wave theory of light. Therefore quantum theory of light was proposed by Einstein who extended the Planck’s hypothesis to explain Black Body radiations. According to quantum theory of light  “light is propagated in bundles of small energy, each bundle being called a photon and possessing energy.”

$\epsilon = h v = \dfrac{hc}{\gamma} \cdots Equation \, \, 1$

Where ‘v’ is frequency, $\gamma$ is wavelength of light and h is Planck’s constant = $6.62 \times 10^{-34}$ Joule-sec and c= speed of light in vaccum = $3 \times 10^8$ m/s.

Momentum of photon,

$p = \dfrac{hv}{c} = \dfrac{h}{\gamma} \cdots Equation \, \, 2$

Dynamic or Kinetic mass of Photon,

$m = \dfrac{hv}{Pc^2} = \dfrac{h}{c \gamma} \cdots Equation \, \, \, 4$

The number of photons in a source of monochromatic radiation of wavelength $\gamma$ and energy W or power P.

$N = \dfrac{W}{\epsilon} = \dfrac{Pt}{\epsilon} \cdots Equation \, \, \, 5$

### Photoelectric Effect

The phenomenon of emission of electrons from a metallic surface by the use of light (or radiant) energy is called photoelectric effect. Caesium is best metal for photoelectric effect.

#### Characteristics of Photo-electric Effect:

(i) Effect of intensity: Intensity of light means the energy incident per unit area per second. For a given frequency, if intensity of incident light is increased, the photo-electric current increases and with decrease of intensity, the photo-electric current decreases, but the stopping potential remains the same.

Effect of intensity

Effect of intensity

This means that the intensity of incident light affect the photo-electric current but leaves the maximum kinetic energy of photo-electrons unchanged.

(ii) Effect of Frequency: When the intensity of incident light is kept fixed and frequency is increased, the photo-electric current remains the same; but the stopping potential increases.

If the frequency is decreased, the stopping potential decreases and at a particular frequency of incident light, the stopping potential becomes zero.

Effect of Frequency

This value of frequency of incident light for which the stopping potential is zero is called threshold frequency $v_o$. If the frequency of incident (v) is less than the threshold frequency $( V_0 )$, no photoelectric emission takes place.

Thus the increase of frequency increases maximum kinetic energy of photo-electrons but leaves the photo-electric current unchanged.

(iii) Effect of Photo metal: When frequency and intensity of incident light are kept fixed and photo-metal is changed, we observe that stopping potential $( V_s )$ versus frequency (v) graphs are parallel straight lines, cutting frequency axis at different points (Fig.) This shows that threshold frequencies are different for different metals, the slope $( \dfrac{V_S}{v} )$ for all the metals is same and hence universal constant.

(iv) Effect of time: There is no time lag between incidence of light and the emission of photoelectrons.

#### Einstein’s Explanation of Photo-electric Effect

The wave theory of light could not explain the observed characteristics of photo-electric effect. Einstein extended Planck’s quantum idea for light to explain photo-electric effect.

According to his idea, “The energy of electromagnetic radiation is not continuously distributed over the wave front like the energy of water waves but remains concentrated in packets of energy content hv, where v is frequency of radiations and h is universal Planck’s constant $( = 6.625 \times 10^{-34} j -s$ . Each packet of energy is called a photon or quantum and travels with the speed of light.

The assumptions of Einstein’s theory are :

1. The photo electric effect is the result of collision of two particles-one a photon of incident light and the other an electron of photo-metal.

2. The electron of photo-metal is bound with the nucleus by coulomb attractive forces. The minimum energy required to free an electron from its bondage is called work function (W).

3. The incident photon interacts with a single electron and loses its energy in two parts:

(i) in releasing the electron from its bondage, and

(ii) in impairing kinetic energy to emitted electron.

Accordingly if hv is the energy of incident photon, then

$hv = W + E_k$

$Or \, \, \, E_k = \dfrac{1}{2} m v^2 _max = hv - W \cdots \, \, \, Equation \, \, \, 1$

Where ‘W’ is work function and $E_k = \dfrac{1}{2} m v^2 _{max}$ is the maximum kinetic energy of photo-electron.

Equation (1) is referred as Einstein’s photoelectric equation and explains all experimental results of photo-electric effect.

The efficiency of photo-electric effect is less than 1% of photons are capable of ejecting photo-electrons.

### Photocells

A photocell is a device for covering light energy into electrical energy.

There are three main types of photocells:

(i) Photo emissive cells: In vacuum photo emissive cell, current is directly proportional to intensity.

(ii) Photo-voltaic cells

(iii) Photo-conductive cells

Photocells

### De-Broglie Waves

Light exhibits particle aspects in certain phenomena (e.g. photoelectric effect, emission and absorption of radiation) while wave aspects in other phenomena (e.g. interference, diffraction and polarization). That is light has dual nature. In analogy with dual nature of light, Louis de Broglie postulated that the material particles (e.g. electrons, protons, $\alpha$ -particles, atoms etc.) may exhibit wave aspect. The wavelength associated with material particle having momentum p (mass m moving with velocity v) is given by:

$\gamma = \dfrac{h}{p} = \dfrac{h}{mv}$

If $E_K$ is kinetic energy of moving material particle, then $p = \sqrt{2m E_K}$

$\gamma = \dfrac{h}{\sqrt{2m E_K}}$

I.e.

$\gamma = \dfrac{h}{p} = \dfrac{h}{mv} = \dfrac{h}{\sqrt{2m E_k}}$

The wave associated with material particle is called the de-Broglie wave. The de-Broglie hypothesis has been confirmed by diffraction experiments.

For charged particles accelerated through a potential difference of V – volts,

$E_K = q V$

$\gamma = \dfrac{h}{\sqrt{2m q V}}$

For electrons $( q = e = 1.6 \times 10^{-19} coul \, \, m = 9 \times 10^{-31} \, kg$

$\gamma = \dfrac{h}{\sqrt{2meV}} = \dfrac{6.62 \times 10^{-34}}{\sqrt{2 \times 9 \times 10^{-31} \times 1.6 \times 10^{-19} V}}$

$= \dfrac{12.27}{\sqrt{V}} \times 10^{-10} m = \dfrac{12.27}{\sqrt{V}}$

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