## Dual Nature of Matter and Radiation

# Wave Theory of Light, Einsteinâ€™s Photoelectric Equation and Particle Nature of Light

- ELNSTEIN’S PHOTO ELECTRIC EQUATION : \tt h\nu=h\nu_{0}\ +\ \frac{1}{2}\ mV_{max}^2
- \tt \frac{1}{2}\ mV_{max}^{2}\ =\ h\left(\nu-\nu_0\right)=hc\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right)
- \tt V_{max}=\sqrt{\frac{2h\left(\nu-\nu_{0}\right)}{m}}=\sqrt{\frac{2\ hc}{m}\frac{\left[\lambda_{0}-\lambda\right]}{\lambda\ \lambda_{0}}}
- \tt \frac{\left[K\ E_{max}\right]_1}{\left[K\ E_{max}\right]_2}=\frac{h\nu_{1}-W}{h\nu_{2}-W}=\frac{\nu_{1}-\nu_{0}}{\nu_{2}-\nu_{0}}
- \tt \frac{\left[K\ E_{max}\right]_1}{\left[K\ E_{max}\right]_2}=\frac{hc\left(\frac{1}{\lambda_{1}}-\frac{1}{\lambda_{0}}\right)}{{hc\left(\frac{1}{\lambda_{2}}-\frac{1}{\lambda_{0}}\right)}}=\frac{\left(\frac{1}{\lambda_{1}}-\frac{1}{\lambda_{0}}\right)}{{\left(\frac{1}{\lambda_{2}}-\frac{1}{\lambda_{0}}\right)}}
- Energy of photon in eV = \tt E=\frac{12400}{\lambda\left(in\ Angstrom\right)} in eV
- Work function = \tt W= \frac{12400}{\lambda_{0}} in e.V
- The graph between maximum kinetic Energy and maximum velocity increases linearly with the frequency of the incident radiation.
- The photoelectric effect is an instantaneous phenomenon. The time lag between incidence of radiation and emissions of photo electron is very small.
- The graph between stopping potential, Vs and incident frequency (ν) is a straight line with slope \tt \frac{h}{e}
- The graph between K.E
_{max}and incident frequency, ν is a straight line with slope h - In photoelectric effect all the emitted photoelectrons do not have same kinetic energy. The K.E ranges from 0 to K.E
_{max}(or) 0 to h (ν – ν_{0}) - Intensity of the radiation obeys inverse square law \tt I\propto\frac{1}{d^{2}}\ and\ i\propto I\Rightarrow\ i\propto\ \frac{1}{d^{2}}
- Photo electric Cells convert light energy to electrical energy.
- Vacuum type of photo emissive cells has instantaneous response.
- In gas filled photo emissive cells, photo current will not be proportional to intensity of the incident radiation.
- In the perspective of De.Broglie, matter behaves both as particles and waves.
- de-BROGLIE wavelength associated with the matter wave of that particle is \tt \lambda=\frac{h}{mv}=\frac{h}{p}=\frac{h}{\sqrt{2mE}}
- de-BROGLIE wave equation: \tt \lambda\propto\frac{1}{\nu} & if ν = 0, λ = ∞
- \tt \lambda\propto\frac{1}{m} {λ = de-Broglie wavelength m = mass}
- \tt \lambda\propto\frac{1}{p} {λ = de-Broglie wavelength P = momentum}
- \tt \lambda\propto\frac{h}{\sqrt{2\ mqv}}{ν = potential difference h = plancks constant q = charge}
- For an electron \tt \lambda\approx\frac{12.27}{\sqrt{v}}\ A^{o}
- For a proton, \tt \lambda=\frac{0.286}{\sqrt{v}}\ A^{o}
- For deuteron, \tt \lambda=\frac{0.202}{\sqrt{E}}\ A^{o}
- For α particle, \tt \lambda=\frac{0.101}{\sqrt{v}}\ A^{o}
- For neutron, \tt \lambda=\frac{0.286}{\sqrt{E}}\ A^{o}
- If the de Broglie waves are associated with electron, then these should be diffracted like x-rays. we can determine the wave length of these waves using the Bragg’s formula, 2d sin θ = n λ
- \tt \theta=\left[\frac{180-\phi}{2}\right] = Bragg’s angle
- de Broglie wave length can also be verified using the formula \tt \lambda=\frac{12.27}{\sqrt{v}}=\frac{12.27}{\sqrt{54}}=1.67\ A^{o}
- X-rays are produced by the deceleration of high energy electrons bombarding a hard metal target.
- The target used to produce x-rays should have a) high atomic weight b) high melting point c) high thermal conductivity.
- X-rays are electromagnetic waves of very short wavelength i.e order of wave length 0.1 Å – 100Å, order of frequency 10
^{16}Hz – 10^{19}Hz, order of energy 124eV – 124 KeV - Intensity of x-rays depends on the number of electron striking the target
- High frequency x-rays are called HARD X-RAYS.
- Low frequency X-rays are called SOFT X-RAYS.
- CONTINUOUS X-RAY SPECTROM is produced when high speed electron are suddenly stopped by a metal target.
- Continuous X-ray spectrum contains all wave lengths above a minimum wavelength λm
- \tt \lambda_{min}=\frac{hc}{eV}=\frac{12400}{v}\ A^{o}
- \tt \nu_{max}=\frac{eV}{h}, maximum frequency of emitted X-ray photon.
- Efficiency of X-ray tube, \tt \eta=\frac{output\ power}{input\ power}\times100
- CHARACTERISTIC X-RAY SPECTRUM is produced due to transition of electrons from higher energy level to lower energy level in target atoms.
- Wavelengths of the X-rays of characteristic X-ray spectrum depend only on atomic number of the target element and independent of target potential.
- In characteristic X-rays, K-series of lines are obtained when transition takes place from higher levels to K-shell.
- E
_{Kα}< E_{Kβ}< E_{Kγ}, E_{Kα}> E_{Lα} - I
_{Kα}> I_{Kβ}> I_{Kγ} - \tt \nu_{k\beta} = \nu_{k\alpha} + \nu_{L\alpha} \Rightarrow \frac{1}{\lambda_{k\beta}} =\frac{1}{\lambda_{k\alpha}} + \frac{1}{\lambda_{L\alpha}}
- \tt E_{k} - E_{L} = h\nu_{k\alpha} = \frac{hc}{\lambda_{k\alpha}}

### View the Topic in this video From 00:19 To 28:33

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1. The balance of the photon energy is used up in giving the electron a kinetic energy of \frac{1}{2}mv^{2}

hv = \phi_{0} + \frac{1}{2}mv^{2}

2. Kinetic energy of photoelectrons is ΔKE = hv − hv_{0} = h(v − v_{0})

= hc \left[\frac{1}{\lambda} - \frac{1}{\lambda_{0}}\right] = 12400 \left[\frac{1}{\lambda (A^0)} - \frac{1}{\lambda_{0}(A^0)}\right]eV