\(E=\frac{12400 \stackrel{o}A}{4100 \stackrel{o}A}=3.02 eV\)

\(\text {K.E }  =\frac{B^2 q^2 r^2}{2 m}  \)
\(\phi  =\frac{h c}{\lambda}-K . E  \)
\(=\frac{h c}{\lambda}-\frac{B^2 q^2 r^2}{2 m}  \)
\(\phi  =\frac{h c}{\lambda}-2 m\left(\frac{B q r}{2 m}\right)^2\)

\(\lambda_{\mathrm{i}}  \frac{1}{\mathrm{mv}}  \)
\(\frac{\lambda_p}{\lambda_e}  =\frac{m_e v_e}{m_P v_P}  \)
\(1  =\frac{9.1 \times 10^{-31} \times 6 \times 10^6}{3 \times 10^{-9} \times v_p}  \)
\(\mathrm{v}_{\mathrm{p}}  =9.1 \times 10^{-16} \times 2  \)
\(\mathrm{v}_{\mathrm{p}}  =18.2 \times 10^{-16}  \)
\(\mathrm{v}_{\mathrm{p}}  =1.82 \times 10^{-15} \mathrm{~m} \mathrm{~s}^{-1}\)

\(\lambda\propto \frac{1}{\sqrt{V}}\)
\(\frac{\lambda_1}{\lambda_2}=\frac{\sqrt{224\times10^3}}{\sqrt{14\times 10^3}}\)
\(=\sqrt{16}=4\)
\(\lambda_{\mathrm{2}}= \frac{\lambda_1}{4}\)

\(\mathrm{E}_{\mathrm{p}}  =\frac{\mathrm{hc}}{\lambda_{\mathrm{p}}}  \)
\(\mathrm{E}_{\mathrm{e}}  =\frac{\mathrm{h}^2}{2 \mathrm{~m} \lambda_{\mathrm{e}}^2}  \)
\(\frac{\mathrm{hc}}{\lambda_{\mathrm{p}}}  =\frac{\mathrm{h}^2}{2 \mathrm{~m} \lambda_{\mathrm{e}}^2}  \)
\(\lambda_{\mathrm{p}}  \propto \lambda_{\mathrm{e}}^{{ }^2}\)

\(\lambda =\frac { h }{ \sqrt { 2mqV } } ;\lambda '=\frac { h }{ \sqrt { 2me(4V) } } =\frac { \lambda }{ 2 } \)

Davisson-Germer experiment was performed to verify wave nature of electrons. It is the first experimental evidence for wave nature of matter.

From Einstein's photoelectric equation hv = W + Ek
E_k = hv - W
Clearly, the smaller the work function, the greater is the K.E. As W₁ > W₂ K.E for metal of work function W₂ will be greater.

The photon of blue light has higher energy as compared to red light; so blue light emits electrons of greater kinetic energy than that of red light.

According to de Broglie's hypothesis, all matter particles like electrons, protons, neutrons in motion are associated with waves. These waves are called de Broglie waves or matter waves.

Sensitive element made of semiconductor is used which generates voltage proportional to the intensity of light or other radiations.

i) According to wave theory, the energy in a light wave is spread out uniformly and continuously over the wavefront.
The energy absorbed by each electron in time t is given by
E = IAt
With this energy absorbed, the most energetic electron is released with K_max by overcoming the surface energy barrier or work function ϕ₀ and this is expressed as
K_max = IAt - ϕ₀    (1)
Thus, wave theory predicts that for a unit time, at low light intensities when IA < ϕ₀ no electrons are emitted. At higher intensities, when IA ≥ ϕ₀, electrons are emitted. This implies that the higher the light intensity, the greater will be K_max.
K_max is dependent only on the intensity under given conditions - that is, by suitably increasing the intensity, one can produce a photoelectric effect even if the frequency is less than the threshold frequency. So the concept of threshold frequency does not even exist in wave theory.
ii) According to wave theory, the intensity of a light wave is proportional to the square of the amplitude of the electric field \(({ E }_{ 0 }^{ 2 })\). The amplitude of this electric field increases with increasing intensity and imparts an increasing acceleration and kinetic energy to an electron.
Now I is replaced with a quantity proportional to \(({ E }_{ 0 }^{ 2 })\) in equation (1). This means that K_max should not depend at all on the frequency of the classical light wave which again contradicts the experimental results.
(iii) If an electron accumulates light energy just enough to overcome the work function, then it is ejected out of the atom with zero kinetic energy. Therefore, from equation (1),
0 = IAt - ϕ₀
t = \(\frac { { \phi }_{ 0 } }{ IA } =\frac { \phi _{ 0 } }{ I(\pi r^{ 2 }) } \)
By taking the atomic radius r = 1.0 x 10^-10 m and substituting the given values of I and ϕ₀, we can estimate the time interval as
t = \(\frac { 2.14\times 1.6\times 10^{ -19 } }{ 1.60\times 10^{ -6 }\times 3.14\times (1\times 10^{ -10 })^{ 2 } } \)
= 0.68 x 10⁷ s ≈ 79 days.
Thus, wave theory predicts that there is a large time gap between the incidence of light and the ejection of photoelectrons but the experiments show that photoemission is an instantaneous process.

P = 50 mW; λ = 640nm = 640 x 10^-9 m
P = 50 x 10^-3W
\(n=\cfrac { hc }{ \lambda } = \frac{6.626 \times10^{-34} \times 3 \times 10^8}{640 \times 10{-9}}=3.106 \times 10^{-19}J\)
\(n=\frac{E}{hv}=\cfrac { 50\times { 10 }^{ -3 } }{ 3.106\times { 10 }^{ -19 } } = 1.61\times 10^{17} s^{-1}\) 
n = 1.61 x 10¹⁷ s^-I

The de Broglie wavelength associated with the kinetic energy k is given as \(\lambda=\frac{h}{\sqrt{2 m k}}\) , where m is the mass of the particle.
Therefore \(\lambda_{\mathrm{e}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{~m}_{\mathrm{e}} \mathrm{k}_{\mathrm{e}}}} \text { and } \lambda_{\alpha}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{~m}_{\alpha} \mathrm{k}_{\alpha}}} \text {. But } \mathrm{k}_{\mathrm{e}}=\mathrm{k}_{\alpha} . \)
Therefore \(\frac{\lambda_{\mathrm{e}}}{\lambda_{\alpha}}=\sqrt{\frac{\mathrm{m}_{\alpha}}{\mathrm{m}_{\mathrm{e}}}} \mathrm{m}_{\alpha}>\mathrm{m}_{\mathrm{e}^{*}} \text {. Therefore, } \lambda_{\mathrm{e}}>\lambda_{\alpha^{\circ}}\)

Due to the large mass of a baseball, the de Broglie wavelength (⋋ = h/mv) associated with a moving baseball is very small. Hence, its wave nature is not visible.

According to quantum concept, intensity of light of given wavelength is defined as the number of energy quanta or photons incident per unit area per unit time, with each photon having same energy. Its unit is Wm^-2.

In metals, the electrons in the outer most shells are loosely bound to the nucleus. Even at room temperature, these large number of electrons which are moving inside the metal in random manner and they cannot leave the surface of metal. So that metals have a large number of free electrons.

(i) The threshold frequency is the frequency of incident light at which kinetic energy of ejected photoelectron is zero.
∴ From fig. threshold frequency,
v₀ = 4.5 x 10¹⁴ Hz
(ii) Work function, W = hv₀
= 6.6 x 10^-34 x 4.5 x 10¹⁴ joule
= \(\frac { 6.6\times { 10 }^{ -34 }\times 4.5\times 10^{ 14 } }{ 1.6\times { 10 }^{ -19 } } \) eV
= 1.85 eV

v₀ = 1.3 volt,
λ = 2271 x 10^-10m
We-know that hv = \({ hv }_{ 0 }+\frac { 1 }{ 2 } { m }_{ max }^{ 2 }\)
\(or\quad hv={ \phi }_{ 0 }+e{ V }_{ 0 }\)
\({ \phi }_{ 0 }=hv-e{ V }_{ 0 }or\quad { \phi }_{ 0 }=\frac { hc }{ \lambda } -e{ V }_{ 0 }\)
\({ \phi }_{ 0 }=\frac { 6.62\times { 10 }^{ -34 }\times 3\times { 10 }^{ 8 } }{ 2271\times { 10 }^{ -10 } } -1.3\times 1.6\times { 10 }^{ -19 }\)
On simplification \({ \phi }_{ 0 }=6.665\times { 10 }^{ -19 }J\)
\(or\quad { \phi }_{ 0 }=\frac { 6.665\times { 10 }^{ -19 } }{ 1.6\times { 10 }^{ -19 } } eV\)
= 4.166 eV
Again \({ \phi }_{ 0 }={ hv }_{ 0 }=\frac { hc }{ { \lambda }_{ 0 } } or{ hv }_{ 0 }=\frac { hc }{ { \phi }_{ 0 } } \)
\(=\frac { 6.62\times { 10 }^{ -34 }\times 3\times { 10 }^{ 8 } }{ 6.665\times { 10 }^{ -19 } } \)
= 2.98 x 10 ^-7 m = 2980 Ă

\(p=\frac { h }{ \lambda } \)
h = 6.6 x 10^-34 Js, λ = 0.1Ӓ = 0.1 x 10^-10m
\(p=\frac { 6.6\times { 10 }^{ -34 } }{ 0.1\times { 10 }^{ -10 } } kg\quad { ms }^{ -1 }\)
= 6.6 x 10^-23 kg ms^-1

de Broglie wavelength associated with electron
\(\lambda =\frac { 12.27 }{ \sqrt { V } } \times { 10 }^{ -10 }m\)
Here V = 50kV = 50 x 10³ V
\(\lambda =\frac { 12.27 }{ \sqrt { 50\times { 10 }^{ 3 } } } =5.5\times { 10 }^{ -12 }m\)
The wavelength of yellow light,
\(\lambda _y\) = 5.9 x 10^-7m
The resolving power of an electron microscope is given by
\(RP=\frac { 1 }{ { d }_{ min } } =\frac { 2\mu sin\beta }{ 1.22\lambda } \)
Where d_min, minimum separation For constant numerical aperture
Resolving power of microscope \(\alpha \frac { 1 }{ \lambda } \)
\(\therefore \frac { Resolving\ power\ of\ electronmicro\ scope }{ Resolving\ power\ of\ optical\ microscope } =\frac { { \lambda }_{ y } }{ \lambda } \)
\(=\frac { 5.9\times { 10 }^{ -7 } }{ 5.5\times { 10 }^{ -12 } } \approx { 10 }^{ 5 }\)
That is, the resolving power of an electron microscope is 10⁵ times the resolving power of an optical microscope.

(i) λ_e = \(\frac { h }{ { p }_{ e } } \) and λ_p = \(\frac { h }{ { p }_{ e } } \) , λ_e-λ_p = 1.00 nm
So, \(\frac { \lambda _{ e } }{ { \lambda }_{ p } } =\frac { { p }_{ e } }{ { p }_{ e } } =\frac { 1 }{ 1 } \Rightarrow \frac { { p }_{ p } }{ { p }_{ e } } =\frac { 1 }{ 1 } \) = 1:1
(ii) From relation K = \(\frac { 1 }{ 2 } { mv }^{ 2 }=\frac { { p }^{ 2 } }{ 2m } \)
K_e = \(\frac { { p }_{ e }^{ 2 } }{ 2m_{ e } } \) and K_p = \(\frac { { p }_{ p }^{ 2 } }{ 2m_{ p } } \)
\(\frac { { K }_{ p } }{ { K }_{ e } } =\frac { { p }_{ p }^{ 2 } }{ { 2m }_{ p } } \times \frac { { 2m }_{ e } }{ { p }_{ e }^{ 2 } } =\frac { m_{ e } }{ { m }_{ p } } \)
Since m_e <<< m_p. So K_p <<< K_e
\(\frac { { K }_{ p } }{ { K }_{ e } } =\frac { 9.1\times 10^{ -31 } }{ 1.67\times 10^{ -27 } } \) = 5.4 x 10^-4.

Davisson - Germer experiment
(i) The filament F is heated by a low tension (L . T) battery. Electrons are emitted from the hot filament by thermionic emission.
(ii) They are then accelerated due to the potential diference between the filament and the anode aluminum cylinder by a high tension (H.T) battery.
(iii) Electron beam is collimated by using two thin aluminum diaphragms and is allowed to strike a single crystal of Nickel.
(iv) The electrons scattered by Niatoms in diflerent directions are received by the electron detector which measures the intensity of scattered electron beam.
(v) The detector is capable of rotation in the plane of the paper, so that the angle (\(\theta\)) between the incident beam and the scattered beam can be changed at our will.
(vi) The intensity of the scattered electron beam is measured as a function of the angle \(\theta\).

(i) Figure shows the variation of intensity of the scattered electrons with the angle \(\theta\) for the accelerating voltage of 54 V.
(ii) For a given accelerating voltage V, the scattered wave shows a peak or maximum at an angle of 50^o to the incident electron beam.
(iii) This peak in intensity is attributed to the constructive interference of electrons diffracted from various atomic layers of the target material.


(iv) From the known value of interplanar spacing of Nickel, the wavelength of the electron wave has been experimentally calculated as 1.65\(\overset { o }{ A }\).
(v) The wavelength can also be calculated from de Broglie relation for V = 54 V from equation as
\(\lambda =\cfrac { 12.27 }{ \sqrt { V } } \overset { o }{ A } =\cfrac { 12.27 }{ \sqrt { 54 } } \) 
\(\lambda =1.67\overset { o }{ A } \)
(vi) This value agrees very well with the experimentally observed wavelength of 1.65 \(\overset { o }{ A }\). Thus this experiment directly verifies de Broglie's hypothesis of the wave nature of moving particles.

Explanation for the photoelectric effect:
The experimentally observed facts of photoelectric effect can be explained with the help of Einstein's photoelectric equation.
(i) As each incident photon liberates one electron, then the increase of intensity of the light (the number of photons per unit area per unit time) increases the number of electrons emitted thereby increasing the photocurrent. The same has been experimentally observed.
(ii) From K_max = hv - Φ_0, it is evident that K_max is proportional to the frequency of the light and is independent of intensity of the light.
(iii) As given in equation \({ hv }_{ o }+\cfrac { 1 }{ 2 } { mv }^{ 2 }\) , there must be minimum energy (equal to the work function of the metal) for incident photons to liberate electrons from the metal surface. Below which, emission of electrons is not possible. Correspondingly, there exists minimum frequency called threshold frequency below which there is no photoelectric emission.
(iv) According to quantum concept, the transfer of photon energy to the electrons is instantaneous so that there is no time lag between incidence of photons and ejection of electrons.
Thus, the photoelectric effect is explained on the basis of quantum concept of light.

(i) To study the effect of potential difference V between the electrodes on photoelectric current, the frequency and intensity of the incident light are kept constant. Initially the potential of A is kept positive with respect to C and the cathode is irradiated with the given light.
(ii) As the potential of A is increased, photocurrent is also increased. However a stage is reached where photo current reaches a saturation value (saturation current) at which all the photoelectrons from C are collected by A.
(iii) This is represented by the flat portion of the graph between potential of A and photocurrent.
(iv) When a negative (retarding) potential is applied to A with respect to C, the current does not immediately drop to zero because the photoelectrons are emitted with some definite and different kinetic energies.
(vi) The kinetic energy of some of the photoelectrons is such that they could overcome the retarding electric field and reach the electrode A.
(vii) When the negative (retarding) potential of A is gradually increased, the photo current starts to decrease because more and more photoelectrons are being repelled away from reaching the electrode A. The photocurrent becomes zero at a particular negative potential V_o' called stopping or cut-off potential.
(viii) Stopping potential is that the value of the negative (retarding) potential given to the collecting electrode A which is just sufficient to stop the most energetic photoelectrons emitted and make the photocurrent zero.
(ix) At the stopping potential, even the most energetic electron is brought to rest. Therefore, the initial kinetic energy of the fastest electron (K_max ) is equal to the work max done by the stopping potential to stop it (eV_o).

\({ K }_{ max }=\cfrac { 1 }{ 2 } { mv }_{ max }^{ 2 }={ ev }_{ o }\) 
\({ v }_{ max }=\sqrt { \cfrac { { 2eV }_{ o } }{ m } } \)  
v_max = \(5.93\times { 10 }^{ 5 }\sqrt { { V }_{ o } } \) 
(xi) From the Figure , When the intensity of the incident light alone is increased, the saturation current also increases but the value of V_o remains constant.
(xii) Thus, for a given frequency of the incident light, the stopping potential is independent of intensity of the incident light.
(xiii) This also implies that the maximum kinetic energy of the photoelectrons is independent of intensity of the incident light.

Hertz observation: 
(i) Maxwell's theory of electromagnetism predicted the existence of electromagnetic waves and concluded that light itself is just an electromagnetic wave. Then, the experimentalists tried to generate and detect electromagnetic waves through various experiments.
(ii) In 1887, Heinrich Hertz first became successful in generating and detecting electromagnetic wave with his high voltage spark discharge between two metallic spheres.
(iii) When a spark is formed, the charges will oscillate back and forth rapidly and the electromagnetic waves are produced.
(iv) The electromagnetic waves thus produced were detected by a detector that has a copper wire bent in the shape of a circle.
(v) Although the detection of waves is successful, there is a problem in observing the tiny spark produced in the detector.
(vi) In order to improve the visibility of the spark, Hertz made many attempts and finally noticed an important thing that small detector spark became more vigorous when it was exposed to ultraviolet light.
(vii) The reason for this behavior of the spark was not known at that time. Later it was found that it is due to the photoelectric emission. whenever ultraviolet light is incident on the metallic sphere, the electrons on the outer surface are emitted which caused the spark to be more vigorous.
Hallwachs' observation:
(i) In 1888, Wilhelm Hallwachs, a German physicist, confirmed that the strange behaviour of the spark is due to the action of ultraviolet light with his simple experiment.
(ii) A clean circular plate of zinc is mounted on an insulating stand and is attached to a gold leaf electroscope by a wire.
(iii) When the uncharged zinc plate is irradiated by ultraviolet light from an arc lamp, it becomes positively charged and the leaves will open.
(iv) Further, if the negatively charged zinc plate is exposed to ultraviolet light, the leaves will close as the charges leaked away quickly.
(v) If the plate is positively charged, it becomes more positive upon UV rays irradiation and the leaves will open further.
(vi) From these observations, it was concluded that negatively charged electrons were emitted from the zinc plate under the action of ultraviolet light.

Lenard's observation:
(i) ln 1902, Lenard studied this electron emission phenomenon in detail. His simple experimental setup is as shown in Figure.
(ii) The apparatus consists of two metallic plates A and C placed in an evacuated quartz bulb. The galvanometer G and battery B are connected in the circuit.


(iii) When ultraviolet light is incident on the negative plate C, an electric current flows in the circuit that is indicated by the deflection in the galvanometer.
(iv) On other hand, if the positive plate is irradiated by the ultraviolet light, no current is observed in the circuit.
(v) From these observations, it is concluded that when ultraviolet light falls on the negative plate, electrons are ejected from it which are attracted by the positive plate A.
(vi) On reaching the positive plate through the evacuated bulb, the circuit is completed and the current flows in it.
(vii) Thus, the ultraviolet light falling on the negative plate causes the electron emission from the surface of the plate.

(i) In metals, the electrons in the outer most shells are loosely bound to the nucleus. Even at room temperature, there are a large number of free electrons which are moving inside the metal in a random manner. Through they move freely inside the metal they cannot leave the surface of the metal. The reason is that when free electrons reach the surface of the metal they are attracted by the positive nuclei of the metal. It is attractive pull which will not allow free electrons to leave the metallic surface at room temperature.
(ii) In order to leave the metallic surface, the free electrons must cross a potential barrier created by the positive nuclei of the metal. The potential barrier. which prevents free electrons from leading the metallic surface is called surface barrier.
(iii) Whenever an additional energy is given to the free electrons, they will have sufficient energy to cross the surface barrier. And they escape from the metallic surface. The liberation of electrons from any surface of a substance is called electron emission.
(iv) The minimum energy needed for an electron to escape from the metal surface is called work function of that metal.
(a) Thermionic emission
(i) When a metal is heated to a high temperature, the free electrons on the surface of the metal get sufficient energy in the form of thermal energy so that they are emitted from the metallic surface. This type of emission is known a thermonic emission.

(ii) The intensity of the thermionic emission (the number of electrons emitted) depends on the metal used and its temperature.
(iii) Examples: cathode ray tubes, electron microscopes, X-ray tubes etc.

(b) Field emission
(i) Electric field emission occurs when a very strong electric field is applied across the metal.
(ii) This strong field pulls the free electrons and helps them to overcome the surface barrier of the metal.
(iii) Ex: Field ermssion scanning electron microscopes, Field-emission display etc.

(c) Photo electric emission
(i) When an electromagnetic radiation of suitable frequency is incident on the surface of the metal, the energy is transferred from the radiation to the free electrons.
(ii) Hence, the free electrons get sufficient energy to cross the surface barrier and the photo electric emission takes place.
(iii) The number of electrons emitted depend on the intensity of the incident radiation.
(iv) Examples: Photo diodes, photo electric cells etc. 

(d) Secondary emission
(i) When a beam of fast-moving electrons strikes the surface of the metal, the kinetic energy of the striking electrons is transferred to the free electrons on the metal surface.
(ii) Thus the free electrons get sufficient kinetic energy so that the secondary emission of electron occurs.
(iii) Examples: Image intensifiers, photo multiplier tubes etc.

Curves a and b have different intensities but the same stopping potential, so curves 'a' and 'b' have the same frequency but different intensities.

a) The number of photons incident normally per unit area per unit time is determined by the intensity of radiations.
b) (i) Red light, because the energy of red light is less than that of blue light
(hv)_R < (hv)_B
(ii) Blue light, because the energy of blue light is greater than that of red light
(hv)_B > (hv)_R

i) Momentum of the electron is
p = \(\sqrt { 2mK } =\sqrt { 2\times 9.1\times { 10 }^{ -31 }\times 2\times 1.6\times 10^{ -19 } } \)
= 7.63 x 10^-25 kg ms^-1
Its de Broglie wavelength is
\(\lambda=\frac { h }{ p } =\frac { 6.626\times { 10 }^{ -34 } }{ 7.63\times { 10 }^{ -25 } } \) = 0.868 x 10^-9 m
= 8.68 \(\mathring { A }\)
ii) Momentum of the bullet is
p = m\({ \upsilon }\) = 0.050 x 200 = 10 kg ms^-1
It's de Broglie wavelength is
\(\lambda=\frac { h }{ p } =\frac { 6.626\times { 10 }^{ -34 } }{ 10 } \) = 6.626 x 10^-35 m
iii) Momentum of the car is
p = mv = 4000 x 50 = 2 x 10⁵ kg ms^-1
Its de Broglie wavelength is
\(\lambda=\frac { h }{ p } =\frac { 6.626\times { 10 }^{ -34 } }{ 2\times { 10 }^{ 5 } } \) = 3.313 x 10^-39 m
From these calculations, we notice that electron has a significant value of de Broglie wavelength (≈10^-9m which can be measured from diffraction studies) but the bullet and car have negligibly small de Broglie wavelengths associated with them (≈10^-33m and 10-³⁹m respectively, which are not measurable by any experiment). This implies that the wave nature of matter is important at the atomic level but it is not really relevant at the macroscopic level.

\(\lambda_{1} =1800 Å=1800 \times 10^{-10} \mathrm{~m} \)
\(\lambda_{2} =4965 Å=4965 \times 10^{-10} \mathrm{~m} \)
\(\mathrm{E} =\mathrm{hc}\left(\frac{1}{\lambda_{1}}-\frac{1}{\lambda_{2}}\right)=\frac{6.626 \times 10^{-34} \times 3 \times 10^{8}}{10^{-10}}\left(\frac{1}{1800}-\frac{1}{4965}\right) \)
\(E=\frac{7.04 \times 10^{-19}}{1.6 \times 10^{-19}}=4.399 \mathrm{eV} \simeq 4.4 \mathrm{eV} \)

(i) For a given frequency of incident light the number of photoelectrons emitted is directly proportional to the intensity of the incident light. The saturation current is also directly proportional to the intensity of incident light.
(ii) Maximum kinetic energy of the photo electrons is independent of intensity 0 the incident light.
(iii) Maximum kinetic energy of the photo electrons from a given metal is directly proportional to the frequency of incident light.
(iv) For a given surface, the emission of photoelectrons takes place only if the frequency of incident light is greater than a certain minimum frequency called the threshold frequency.
(v) There is no time lag between incidence of light and ejection of photo electrons.