An electron have mass (m) and charge (e). It moves in
electric field (E) . Then,
(i) force on electron, `F = eE`
(ii) when electron is projected in the direction of electric field or opposite direction, then its path is linear line.
(iii) electric field intensity, `E = V/d = text(potentail between plates)/text( distance between plates)`
(iv) applied acceleration on electron,

`a = F/m = (eE)/m`

`text(Motion of Cathode Rays)` (Electron) Perpendicular to Electric Field

An electron have mass (m) and charge (c). lt enters into electric field perpendicularly. Then,
(i) force on electron, `F = eE`.
(ii) with velocity `v_x`, the time required to pass length of electric field, `t = l/v_x`.

(iii) Acceleration produced in perpendicular direction of motion,

` a = F/m = (eM)/m`0

(iv) After covering the length (l), deviation of electron,

`y = 1/2 at^2 = 1/2 ((eE)/m) (l/v_x)^2`

So, the path of electron is parabolic.
(v) If the vertrical component of electron velocity `v_x`,
then
`v_x = at = ((eE)/m) t = (V/d) (e/m) (l/v_x)`
(vi) Resultant velocity, `v = sqrt(v_x^2 + v_y^2 )` and `theta^(-1) (v_y/v_x)`

`text(Uses of Cathode Rays)`

(i) In cathode ray oscilloscope.
(ii) In production of X-rays.

Positive rays are moving positive ions of the gas filled in the discharge tube. The mass of these particles is nearly equal to the mass of the atoms of gas. Positive rays were discovered by Goldstein.

`text(Properties of Positive Rays)`

Some properties of positive rays are given below
(i) These consist of fast moving positively charged particles.
(ii) These rays are deflected in magnetic field.
(iii) These rays are deflected in electric field.
(iv) These rays travel in straight line.
(v) The speed of positive rays is less than that of cathode rays.
(vi) These rays can affect the photographic plate.
(vii) These rays penetrate through the thin aluminium foil.
(viii) These rays can produce fluorescence and phosphorescence.

In case of light some phenomenon like diffraction and interference can be explained on the basis of its wave character. However, the certain other phenomenon such as black body radiation and photoelectric effect. can be explained only on the basis of its particle
nature. Thus, light is said to have a dual character. Such studies on light wave were made by Einstein in 1905. Louis-de-Broglie, in 1942 extended the idea of photons to material particles such as electron and he proposed that matter also has a dual character as wave and as particle.

`text(de-Broglie Relation)`

According to de-Broglie, a wave is associated with energy moving particle. These waves are called de-Broglie waves or
matter waves.
According to quantum theory, energy of photon,
`E = bv` ... (i)
If mass of the photon is taken as m, then as per Einstein's
equation,

`E = mc^2` ... (ii)
From Eqs. (i) and (ii), we get
`bv = mc^2`

`b c/lamda = mc^2 [ ∵ v = c // lamda ]`

where, `lamda =` wavelength of photon

` lamda = b/(mc)`

de-Broglie asserted that the above equation is completely a general function and applies to photon as well as all other moving particles.

So, ` lamda = b/(mv) = b/sqrt(2mE)`

when, m is mass of particle and v is its velocity

Some properties of matter waves are given below
(i) de-Broglie wavelength, `lamda prop 1/v`. If the particle moves faster, then the wavelength will be smaller and vice-versa.
(ii) If the particle is at rest `(v = 0)`, then the de-Broglie wavelength is infinite `(lamda = oo) `. Such a wave cannot be visualised.
(iii) de-Broglie waves cannot be electromagnetic in nature because electromagnetic waves are produced by motion of charged particles.
(iv) The wavelength of a wave associated with moving particle defines a region of uncertainty, within which the whereabouts of the particles are unknown.

For emission of electrons from a metal surface, a minimum amount of energy (`E`) is required to overcome the potential barrier (`W_0` : work function) provided by the metal. When energy is provided in the form of electromagnetic waves of photons, then emitted electrons are called photoelectrons and this effect is called photoelectric effect. As photon energy (`E = hv`) is proportional to its frequency, a minimum frequency is required so that photon has energy greater than or equal to work function. This minimum frequency is called threshold frequency (`v_0` ). for photoemission to take place, either of the following conditions must be satisfied

`E >= W_0 ` or `v >= v_o` or `lamda = lamda_0`
where, `v_o =` threshold frequency, `lamda_o =` threshold wavelength.

(i) There is no time lag between emission of electrons and incidence of light.
(ii) The number of emitted electrons is directly proportional to the intensity of incident light.
(iii) The kinetic energy of emitted electrons does not depend upon the intensity of incident light.
(iv) The kinetic energy of emitted electrons is directly proportional to the frequency of incident light.
(v) The number of emitted electrons does not depend on the frequency of incident light.
(vi) If the frequency of incident light is less than threshold frequency, then no electron will be ejected.
(vii) The photoelectric emission is an instantaneous process. 'The time lag between the incidence of radiations and emission of photoelectrons is very small or less than even `10^(-9)` second.

The minimum frequency of light necessary to eject the electrons from metallic surface is called threshold frequency or cut-off frequency (`v_0` ). For a frequency lower than cut-off frequency, no photoelectric emission is possible even, if the intensity
is large.

`text(Work Function)`

The minimum energy required to eject the electrons from metal is called the work function of metal (W).
`W bv_o` [where, `h =` Planck's constant ]
[and `v_o =` threshold frequency ]

`text(Einstein's Equation of Photoelectric Effect)`

`∵` Kinetic energy, `E_K = b (v - v_0 )`

`=> 1/2 mv_(max)^2 = b(v- v_0 )`

or `1/2 mv_(max)^2 = bv - bv_o`

where, `E_K` = maximum kinetic energy of photoelectron,
`v =` frequency of incident light,
`v_0 =` threshold frequency,
`b =` Planck's constant
and `W =` work function

If electron is retarded through a potentiaI difference of `V_0`, then all kinetic energy will be converted to potential energy.
`=> eV_o = K_(max) = 1/2 mv_(max)^2 = bv - bv_o = bv - W [ ∵ W = bv_o]`
where, `V_0` is called stopping potential

`text(Photoelectric Cell)`

It is a device based or phenomena of photoelectric effect which converts light energy directly into electric energy.
These are of three types
(i) Photoemissivc cell
(ii) Photovoitaic cell
(iii) Photoconductive cell
(ii) Photoconductive cell
♦ Note Saturation current in the photocell varies with distance as ` i alpha 1/d^2`

`text(Applications of Photoelectric Cells)`

• In reproduction of sound in cinema, television and phototelegraphy.
• In measuring the temperature of celestial bodies.
• To control the temperature in furnance and in chemical process.
• In automatic doors.
• In photoelectric counters.
• In automatic switches for street lights.
• In photoelectric sorters.
• In space for obtaining electrical energy from sunlight during space travel, ` v >= v_0` or ` lamda <= lamda_0`

When fast moving cathode rays strike on a metal piece of high
melting point and of high mass number, then some invisible
rays produced. These rays are called X-rays.
These rays were discovered by Roentgen. These rays are
electromagnetic in nature. The device used to produce `X-rays`
is called coolidge tube.

`text(Properties of X-rays)`

Some properties of X-rays are given below
(i) These are electromagnetic in nature.
(ii) X-rays travel in straight line with speed of light.
(iii) These show reflection, refraction, interference, diffraction and polarisation.
(iv) Wavelength of X rays is the order of `1 Å`.
(v) These arc not deflected by electric and magnetic fields.
(vi) These produce illumination on falling on fluorscent substances.
(vii) X-rays ionise the gas in which they pass.
(viii) X-rays penetrate through different depth into different substances.
(ix) X-rays show photoelectric effect.

`text(Applications of X-rays)`

• In surgery • In radiotherapy
• In trading • In laboratory
• In searching

Frequency v of characteristic X -rays spectrum
`sqrtv = n(z - sigma)`
where, `a` and `sigma` are constants and screening constant for `K_alpha`
line, `sigma = 1`, Screening constant for `L_alpha` line, `sigma = 7.4`

`text(Bragg's Law)`

This law states that, when the X -ray is incident onto a crystal surface its angle of incidence `theta`, will reflect back with a same angle of scattering `theta`. And when the path difference `d` is equal to a whole number `n`, of wavelength `lamda`. A constructive interference will occur. Consider the diagram in which beam of an `X`-rays incident at an angle `theta` get diffracted as shown in figure. Clearly, path difference between ray `1` and ray `2` is `2d sin theta`.
For maxima of X-rays diffracted from the crystal. `2d sin theta = n lamda`
where, `n = 1 ,2,3, ... , d` is interatomic gap of the crystal

When X-rays arc scattered by loosely bound electrons in a target some of the scattered X-rays have a long wavelength, then the incident X-rays. This is called compton shift in wavelength. The phenomenon in which the wavelength of the incident X-rays increases and hence the energy decreases due to scattering from an atom is known as Compton effect.

`text(Expressions for Compton Shift)`

(i) This compton shift in wavelength is given by
`Delta lamda = b/(m_oc) (1 - cos phi)`
where, `b =` Planck's constant,
`m_0 =` rest mass of electron,
`phi` = angle of scattering and `c =` speed of light

(ii) Kinetic energy of recoil electron,
` E_K = (bc)/lamda = (bc)/(lamda')`

(iii) Direction of recoil electron,
`tan theta = ( lamda sin phi)/( lamda' - lamda cos phi)`

(iv) Compton wavelength of electron
` b/(m_oc) = 0.024 Å`

The orderly arrangement of `EM` wave in increasing or decreasing order of wavelength (`lamda`) or frequency (`v`) is called electromagnetic spectrum. The range varies from `10^(-12)` m to `10^4 m`, i.e. from `gamma` - rays to radiowaves .

`text(Bohr's Model of Hydrogen Like Atoms)`

Bohr combined classical and early quantum concepts and gave his theory in the form of three postulates as given below First postulate An electron in an atom could revolve in certain stable orbit without the emission of radiant energy, contrary to the predictions of electromagnetic theory.

`text(Second postulate)` The electron revolves around the nucleus only in those orbits for which the angular momentum is some integral multiple of `b//2 pi`, where b is the Planck's constant.

The angular momentum , ` L = (n.b)/(2pi)`

`mvr = (nb)/(2 pi)`
where, `n = 1 ,2 ,3, ...`

`text(Third postulate)` An electron might make a transition from one of its specified non-radiating orbit to another of lower energy. When it does, so a photon is emitted having energy equal to the energy difference between the initial and final states.
`bv = E_i - E_f`
where, `E_i` and `E_f` are the energies of the initial and final
states and `E_i > E_f`

The atoms consist of central nucleus, containing the
entire positive charge and almost all the mass of the
atom. The necessary centripetal force for circular orbits
is provided by coulomb attraction between the electron
and nucleus.
`(mv^2)/r = 1/( 4 pi epsilon_0) (Ze^2)/r^2`

• Condition for quantisation, `mvr = (nb)/(2 pi)`

• Radius of `n` th orbit (from above two equations)

` r_n = (epsilon_0 b^2 n^2)/( pi m Z e^2) = 0.53 n^2/Z Å = n^2/Z r_o`

where, `r_0 = 0.53 Å`
Size of an atom is approx `1 Å`.
Speed of electron in `n` th orbit,

`v_n = e^2/( 2 epsilon_0 b) Z/n = c/(137) Z/n`

• `KE` is `E_K = 1/( 4 pi epsilon_0) . (Ze^2)/(2r)`

• `PE` is `U = - 1/( 4 pi epsilon_0) . (Ze^2)/(r)`

• Total energy `= - 1/( 4 pi epsilon_0) . (Ze^2)/(2r)`

• Condition of transition, `bv = E_1 - E_2`
Frequency of emitted radiation is

` v = (E_1 - E_2)/b = z^2 Rc ( 1/n_1^2 - 1/n_2^2)`

• Wave number ` bar v = 1/lamda = RZ^2 ( 1/n_1^2 - 1/n_2^2)`

The atomic hydrogen emits spectrum consisting of various
series. The frequency of a line in a series can be expressed as a
difference of two terms

(i) Lyman series `1/lamda = R (1/1^2 - 1/n^2),` where `n = 2, 3, 4, 5, ...`
[ultraviolet region]
(ii) Balmer series ` 1/lamda = R (1/2^2 - 1/n^2),` where `n = 3, 4, 5, 6, ...`
[visible region]
(iii) Paschen series `1/lamda = R (1/3^2 - 1/n^2),` where `n = 4, 5, 6, 7, ...`
[ infrared region]
(iv) Brackett series `1/lamda = R (1/4^2 - 1/n^2),` where `n = 5, 6, 7, 8, ...`
[infrared region]
(v) Pfund series `1/lamda = R (1/5^2 - 1/n^2),` where `n = 6, 7, 8, 9, ...`
[infrared region ]