**CBSE NOTES CLASS 12 PHYSICS CHAPTER 11**

**DUAL NATURE OF RADIATION AND MATTER**

Effect of intensity of incident light

Photoelectric emission is an instantaneous process

Failure of wave theory in explaining photoelectric effect

Einstein’s photoelectric equation: energy quantum of radiation

Heisenberg’s uncertainty principle

**CBSE NOTES CLASS 12 PHYSICS CHAPTER 11**

**DUAL NATURE OF RADIATION AND MATTER**

**Cathode rays**

Cathode rays are the stream of fast moving electrons.

The minimum energy required for the electron emission from the metal surface can be supplied to the free electrons by

(i) Thermionic emission: By suitably heating, sufficient thermal energy can be imparted to the free electrons to enable them to come out of the metal.

(ii) Field emission: By applying a very strong electric field (of the order of 10^{8} Vm^{–1}) to a metal, electrons can be pulled out of the metal, as in a spark plug.

(iii) Photo-electric emission: When light of suitable frequency illuminates a metal surface, electrons are emitted from the metal surface. These photo(light)-generated electrons are called photoelectrons.

**Properties of cathode rays**

- Cathode rays are not electromagnetic rays.
- Cathode rays are deflected by electric field and magnetic field.
- Cathode rays produce heat when they fall on the metals.
- Cathode rays can pass through thin aluminium or gold foils without puncturing them.
- Cathode rays can produce physical and chemical change.
- Cathode ray travel in straight line with high velocity, momentum and energy and cast shadow of objects placed in their path.
- On striking the target of high atomic weight and high melting point, they produce X-rays.
- Cathode rays produce fluorescence and phosphorescence in certain substance and hence affect photographic plate.
- When any charged particle move in a field where magnetic and electric fields are present, without any deviation, then Magnetic force = Electrostatic force
$$\mathrm{B}\mathrm{e}\mathrm{v}\mathrm{}=\mathrm{}\mathrm{E}\mathrm{e}\mathrm{}\mathrm{o}\mathrm{r}\mathrm{}\mathrm{v}\mathrm{}=\frac{\mathrm{E}}{\mathrm{B}}$$

- Specific charge of cathode rays means the ratio of charge and mass. Specific charge of electron
$$\frac{\mathrm{e}}{\mathrm{m}}=\frac{{\mathrm{E}}^{2}}{2\mathrm{V}{\mathrm{B}}^{2}}$$

where,

E = electric field,

B = magnetic field and

V = potential difference applied across ends of tube.

$$\frac{\mathrm{e}}{\mathrm{m}}\mathrm{}=\mathrm{}\mathrm{}1.7589\mathrm{}\times \mathrm{}{10}^{11}\mathrm{}\mathrm{C}/\mathrm{k}\mathrm{g}.$$

Charge of electron e = -1.602 × 10

^{-19}C.

**Photon**

Photons are the packets of energy emitted by a source of radiation. The energy of each photon is,

E = hν,

where h is Planck’s constant and ν is frequency of radiation.

The rest mass of a photon is zero.

The momentum of a photon

$$\mathrm{p}\mathrm{}=\frac{\mathrm{h}\mathrm{\nu}}{\mathrm{c}}=\frac{\mathrm{h}}{\mathrm{\lambda}}$$

Dynamic or kinetic mass of photon,

$$\mathrm{}\mathrm{m}\mathrm{}=\frac{\mathrm{h}\mathrm{\nu}}{{\mathrm{c}}^{2}}=\frac{\mathrm{h}}{\mathrm{c}\mathrm{\lambda}}\mathrm{}$$

where c is speed of light in vacuum and λ is wavelength of radiation.

Photons are electrically neutral.

A body can radiate or absorb energy in whole number multiples of a quantum, that is hν, 2hν, 3hν …, nhν, where n is positive integer.

**Photoelectric effect**

The phenomenon of emission of electrons from a metal surface, when radiation of suitable frequency is incident on it, is called photoelectric effect.

**Work function**(W_{o}) - The minimum amount of energy required to eject one electron from a metal surface is called its work function.**Threshold frequency**(ν_{o}) - The minimum frequency of light which can eject photo electron from a metal surface is called threshold frequency of that metal.**Threshold wavelength**(λ_{max}) - The maximum wavelength of light which can eject photo electron from a metal surface is called threshold wavelength of that metal.$${\mathrm{W}}_{\mathrm{o}}\mathrm{}=\mathrm{}\mathrm{h}{\mathrm{\nu}}_{\mathrm{o}}\mathrm{}=\frac{\mathrm{h}\mathrm{c}}{{\mathrm{\lambda}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}$$

**Laws of photoelectric effect**

**Effect of intensity of incident light on photoelectric current**For a given photosensitive material and frequency of incident radiation (above the threshold frequency), the photoelectric current is directly proportional to the

**intensity of incident light.**_{}**Effect of potential on photoelectric current**The photoelectric current increases with increase in accelerating (positive) potential. If we keep increasing the accelerating potential current reaches a maximum beyond which it does not increase. This maximum value of the photoelectric current is called

**saturation current.**For a particular frequency of incident radiation, the minimum negative (retarding) potential V

_{0}for which the photocurrent stops or becomes zero is called the cut-off or stopping potential.Maximum kinetic energy of the ejected electron,

K

_{max}= eV_{0}- For a given frequency of the incident radiation, the
**stopping potential**is independent of its intensity.

**Effect of frequency on photoelectric current**

The stopping potentials are in the order V

_{03}> V_{02}> V_{01}if the frequencies are in the order ν_{3}> ν_{2}> ν_{1}. That is greater the frequency of incident light, greater is the maximum kinetic energy of the photoelectrons. Consequently, we need greater retarding potential to stop them completely.**Threshold frequency**For a given photosensitive material, there is a certain minimum cut-off frequency, called

**threshold frequency**, below which there is no emission of photoelectrons takes place, no matter how intense the incident light is.Above the threshold frequency, the stopping potential or equivalently the maximum kinetic energy

**KE**of the emitted photoelectrons increases linearly with the frequency of the incident radiation, but is**independent of its intensity**.**Photoelectric emission is an instantaneous process**

The photoelectric emission is an instantaneous process without any time lag (~10

^{–9 }s or less), even when the incident radiation is made exceedingly dim.- For a given frequency of the incident radiation, the

**Failure of wave theory and photoelectric effect**

- According to the wave theory of light, the free electrons at the surface of the metal (over which the beam of radiation falls) absorb the radiant energy continuously.
The greater the intensity of radiation, the greater is the amplitude of electric and magnetic field. Consequently, the greater the intensity, the greater should be the energy absorbed by each electron.

The maximum kinetic energy of the photoelectrons on the surface is expected to increase with increase in intensity. Also, irrespective of the frequency of radiation, a sufficiently intense beam of radiation (over sufficient time) should be able to impart enough energy to the electrons, so that they exceed the minimum energy needed to escape from the metal surface. A threshold frequency, therefore, should not exist.

These expectations of the wave theory directly contradict observations.

- Also, in the wave theory, the absorption of energy by electron takes place continuously over the entire wavefront of the radiation. Since a large number of electrons absorb energy, the energy absorbed per electron per unit time turns out to be small and it can take hours or more for a single electron to pick up sufficient energy to overcome the work function and come out of the metal. This conclusion is again in striking contrast to observation that the photoelectric emission is instantaneous

**Einstein’s photoelectric equation: energy quantum of radiation**

- Photoelectric emission does not take place by continuous absorption of energy from radiation. Radiation energy is made up of discrete units called
**quanta of energy of radiation**. - Each quantum of radiant energy has energy hν, where h is Planck’s constant and ν the frequency of light. In photoelectric effect, an electron absorbs a quantum of energy (hν of radiation. If this quantum of energy absorbed exceeds the minimum energy needed for the electron to escape from the metal surface (work function W
_{0}), the electron is emitted with maximum kinetic energyK

_{max}= hν – W_{o} - Intensity of radiation is proportional to the number of energy quanta per unit area per unit time. The greater the number of energy quanta available, the greater is the number of electrons absorbing the energy quanta and greater is the number of electrons coming out of the metal (for ν > ν
_{o}). This explains why, for ν > ν_{o}, photoelectric current is proportional to intensity. - Whatever may be the intensity i.e., the number of quanta of radiation per unit area per unit time, photoelectric emission is instantaneous. Low intensity does not mean delay in emission, since the basic elementary process is the same. Intensity only determines how many electrons are able to participate in the elementary process (absorption of a light quantum by a single electron) and, therefore, the photoelectric current.
**This theory accounts for all the observations of the photoelectric effect.**

**Particle nature of light: the photon **

- In interaction of radiation with matter, radiation behaves as if it is made up of particles called photons.
- Each photon has energy E (= h ν) and momentum p (= $\frac{h\nu}{c}$), and speed c, the speed of light.
- All photons of light of a particular frequency ν, or wavelength λ, have the same energy E (= h ν = $\frac{hc}{\lambda}$) and momentum p (= $\frac{h\nu}{c}$ = $\frac{h}{\lambda}$), whatever the intensity of radiation may be. By increasing the intensity of light of given wavelength, there is only an increase in the number of photons per second crossing a given area, with each photon having the same energy. Thus, photon energy is independent of intensity of radiation.
- Photons are electrically neutral and are not deflected by electric and magnetic fields.
- In a photon-particle collision (such as photon-electron collision), the total energy and total momentum are conserved. However, the number of photons may not be conserved in a collision. The photon may be absorbed or a new photon may be created.

**Wave nature of matter**

The wave nature of light shows up in the phenomena of interference, diffraction and polarisation. On the other hand, in photoelectric effect and Compton effect which involve energy and momentum transfer, radiation behaves as if it is made up of a bunch of particles – the photons.

The gathering and focusing mechanism of light by the eye-lens is understood in the wave picture.

But its absorption by the rods and cones (of the retina) requires the photon picture of light.

Louis Victor de Broglie suggested that moving particles of matter should display wave-like properties under suitable conditions. He reasoned that nature was symmetrical and that the two basic physical entities – matter and energy, must have symmetrical character. If radiation shows dual aspects, so should matter.

De Broglie proposed that the wave length λ associated with a particle of momentum p is given as

$$\mathrm{\lambda}\mathrm{}=\frac{\mathrm{h}}{\mathrm{p}}=\mathrm{}\frac{\mathrm{h}}{\mathrm{m}\mathrm{v}}$$

where m is the mass of the particle and v its speed.

Now if V is the magnitude of accelerating potential in volts,

K = eV

$$\mathrm{B}\mathrm{u}\mathrm{t}\mathrm{}\mathrm{}\mathrm{K}\mathrm{}\mathrm{}=\frac{1}{2}\mathrm{}\mathrm{m}{\mathrm{v}}^{2}=\frac{{\mathrm{p}}^{2}}{2\mathrm{m}}$$

$$\Rightarrow \mathrm{}\mathrm{p}=\sqrt{2\mathrm{m}\mathrm{K}}=\sqrt{2\mathrm{m}\mathrm{}\mathrm{e}\mathrm{V}}\mathrm{}$$

$$\Rightarrow \mathrm{\lambda}\mathrm{}\mathrm{}=\frac{\mathrm{h}}{\mathrm{p}}=\frac{\mathrm{h}}{\sqrt{2\mathrm{m}\mathrm{K}}}=\frac{\mathrm{h}}{\sqrt{2\mathrm{m}\mathrm{}\mathrm{e}\mathrm{V}}}=\frac{1.227}{\sqrt{\mathrm{V}}}\mathrm{}\mathrm{n}\mathrm{m}$$

**Heisenberg’s uncertainty principle**

It is not possible to measure both the position and momentum of an electron (or any other particle) at the same time exactly. There is always some uncertainty (Δx) in the specification of position and some uncertainty (Δp) in the specification of momentum. The product of Δx and Δp is of the order of $\frac{h}{2\pi}$ or,

$$\Delta x\Delta p\frac{h}{2\pi}$$