The cut-off wavelength of the characteristic x - rays is
\({ \lambda }_{ ° }\frac { 12400 }{ V } \mathring { A } =\frac { 12400 }{ 20000 } \mathring { A }\)
= 0.62 \(\mathring { A } \)
The corresponding frequency is
\({ \upsilon }_{ o }=\frac { c }{ { \lambda }_{ o } } =\frac { 3\times 10^{ 8 } }{ 0.62\times 10^{ -10 } }\) = 4.84 x 10¹⁸ Hz

i) The threshold wavelength is given by
\({ \lambda }=\frac { hc }{ { \phi }_{ 0 } } =\frac { 6.626\times { 10 }^{ -34 }\times 3\times { 10 }^{ 8 } }{ 4.65\times 1.6\times 10^{ -19 } } \)
= 2672 \(\mathring { A }\)
ii) Energy of the photon of wavelength 2200 \(\mathring { A }\) is
E = \(\frac { hc }{ \lambda } =\frac { 6.626\times { 10 }^{ -34 }\times 3\times { 10 }^{ 8 } }{ 2200\times 10^{ -10 } } \)
= 9.035 x 10^-19 J = 5.65 eV
We know that kinetic energy of fastest photo electron is
K_max = hv - ϕ₀ = 5.65 - 4.65
= 1 eV
From equation (7.3), K_max = eV₀
V₀ = \(\frac { { K }_{ max } }{ e } =\frac { 1\times 1.6\times { 10 }^{ -19 } }{ 1.6\times { 10 }^{ -19 } } \)
Therefore, stopping potential = 1 V

Energy of the incident photon is
E = hv = \(\frac { hc }{ \lambda } \) (in joules)
E = \(\frac { hc }{ \lambda e } \) (in eV)
Substituting the known values, we get
E = \(\frac { 6.626\times { 10 }^{ -34 }\times 3\times 10^{ 8 } }{ 300\times { 10 }^{ -9 }\times 1.6\times { 10 }^{ -19 } } \)
E = 4.14 eV
The work function of silver = 4.7 eV. Since the energy of the incident photon is less than the work function of silver, photoelectrons are not observed in this case.

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.

\(P=4 \ \mathrm{cal} \mathrm{} \mathrm{cm}^{-2} \mathrm{~min}^{-1}=4 \times 4.2=16.8 \mathrm{~J} \mathrm{~cm}^{-2} \mathrm{~min}^{-1} \)
\(E=\frac{h c}{\lambda}=\frac{6.626 \times 10^{-34} \times 3 \times 10^{8}}{5500 \times 10^{-10}}=3.6 \times 10^{-19} \mathrm{~J} \)
\(n=\frac{E}{hv}=\frac{16.8}{3.6 \times 10^{-19}}=4.67 \times 10^{19} \)
\(n=4.67 \times 10^{19} \)  per cm² per minute.

\(\lambda_{1}=3310 Å=3310 \times 10^{-10} \mathrm{~m} ; \mathrm{E}_{1}=3 \times 10^{-19} \mathrm{~J} \)
\(\lambda_{2}=5000 Å=5000 \times 10^{-10} \mathrm{~m} ; \mathrm{E}_{2}=0.972 \times 10^{-19} \mathrm{~J} \)
\(\mathrm{E}=\mathrm{E}_{1}-\mathrm{E}_{2}=2.028 \times 10^{-19} J\)
\(\mathrm{hc}\left(\frac{1}{\lambda_{1}}-\frac{1}{\lambda_{2}}\right)=\mathrm{E}_{1}-\mathrm{E}_{2} \)
\(\frac{\mathrm{h} \times 3 \times 10^{8}}{10^{-10}}\left(\frac{1}{3310}-\frac{1}{5000}\right)=2.028 \times 10^{-19} \)
\(\mathrm{~h}=\frac{2.028 \times 10^{-19} \times 10^{-10} \times 3310 \times 5000}{3 \times 10^{8} \times 1690}=6.62 \times 10^{-34} \mathrm{Js} \)
\(\phi_{0} =\frac{\mathrm{hc}}{\lambda}-\mathrm{E}=\frac{6.62 \times 10^{-34} \times 3 \times 10^{8}}{3310 \times 10^{-10}}-3 \times 10^{-19} \)
\(=(6-3) \times 10^{-19}=3 \times 10^{-19} \mathrm{~J} \)
\(\phi_{0} =3 \times 10^{-19} \mathrm{~J} \)
Threshold Wavelength,
\(\lambda_{0}=\frac{\mathrm{hc}}{\phi_{0}}=\frac{6.626 \times 10^{-34} \times 3 \times 10^{8}}{3 \times 10^{-19}}=6.62 \times 10^{-7} \mathrm{~m} \)
\(\lambda_{0}=6620 \stackrel {o}{A}\)

(i) λ_v= 413 nm = 413 x 10^-9 m
\({ E }=\cfrac { hc }{ { \lambda } } =\cfrac { 6.626\times { 10 }^{ -34 }\times 3 \times { 10 }^{ 8 } }{ 413\times { 10 }^{ -9 } m} \)
E = 3.00 eV 
(ii) λ = 0.1 nm = 0.1 x 10^-9 m
\({ E }=\cfrac { hc }{ { \lambda } } =\cfrac { 6.626\times { 10 }^{ -34 }\times 3 \times { 10 }^{ 8 } }{ 0.1\times { 10 }^{ -9 } }=19.878\times10^{-16} \) 
E = \(\cfrac { 19.878\times { 10 }^{ -16 } }{ 1.6\times { 10 }^{ -19 } } =12.4237 \ = eV \ 12424 \ eV\) 
E = 12424 eV.
(iii) λ = 10 m
\({ E }_{ r }=\cfrac { hc }{ { \lambda }_{ r } } =\cfrac { 19.878\times { 10 }^{ -27 } }{ 10 } \) 
= \({ E }=\cfrac { hc }{ { \lambda } } =\cfrac { 6.626\times { 10 }^{ -34 }\times 3 \times { 10 }^{ 8 } }{ 10\times 1.6\times{ 10 }^{ -9 } } =1.2424\times { 10 }^{ -7 }\)
E_r = 1.24 x 10^-7 eV.

V = 81 V
∴ K = eV = 1.6 x 10^-19 x 81 = 1.296 x 10^-17 = 1.3 x 10^-17 J
\({ V }=\sqrt \frac {2K}{m}\sqrt { \cfrac { 2\times 1.3\times { 10 }^{ -17 } }{ 9.1\times { 10 }^{ -31 } } } =5.345 \times 10^6 ms^{-1} \) 
v = 5.345 x 10⁶ m s^-1, K = 1.3 x 10^-17 J

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 \(\frac{\lambda_{\mathrm{e}}}{\lambda_{\alpha}}=\sqrt{\frac{\mathrm{m}_{\alpha}}{\mathrm{m}_{\mathrm{e}}}} \mathrm{m}_{\alpha}>\mathrm{m}_{\mathrm{e}^{*}} \text {. Therefore, } \lambda_{\mathrm{e}}>\lambda_{\alpha^{\circ}}\)

de Broglie wavelength \(\lambda=\frac{\mathrm{h}}{\mathrm{mv}}\)
Kinetic energy of electron K \(=\frac{1}{2} m v^{2} \text { (or) } v=\sqrt{\frac{2 K}{m}}\)
Now de Broglie wavelength is \(\lambda=\frac{h}{m v}=\frac{h}{m \sqrt{2 K / m}} \)
\(\lambda=\frac{h}{\sqrt{2 m K}} \)

The de Broglie wavelength associated with the kinetic energy K is \(\lambda=\frac{h}{\sqrt{2 m k}}\)
Where m is the mass of the particle
Since proton and electron have same KE, the wavelength is inversely proportional to square root of the mass \(\lambda \alpha \frac{1}{\sqrt{m}}\)
Mass of proton is 1840 times greater them that of electron. Therefore, de Broglie's wavelength of electron is greater than the proton.

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 de Broglie hypothesis, if radiation has a dual nature, then the moving particles of matter Iike electrons, protons, neutrons in motion should exhibit wave like character under an appropriate conditions. These waves are called de Broglie waves or matter waves.

For a given metallic Surface, the emission of photo electrons takes place only if the frequency of incident light is greater than a certain minimum frequency called the threshold frequency.

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.

The photocurrent, (ie. the number of electrons emitted per second) is directly proportional to the intensity of the incident light.

The ejection of electrons from the metal plate when illuminated by light or any electromagnetic radiation of suitable wavelength (or frequency) is called photoelectric effect.

The minimum energy needed for an electron to escape from the metal surface is called work function of that metal.
Unit: electron volt (eV).

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.