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Dual Nature of Radiation and Matter — Physics STD 12 Science — Question

Bihar BoardEnglish MediumSTD 12 SciencePhysicsDual Nature of Radiation and Matter5 Marks
Question
Give a brief description of the basic elementary process involved in the photoelectric emission in Einstein’s picture.
When a photosensitive material is irradiated with the light of frequency v, the maximum speed of electrons is given by vmax. A plot of v2max is found to vary with frequency ν as shown in the figure.

Use Einstein’s photoelectric equation to find the expressions for:
  1. Planck’s constant.
  2. Work function of the given photosensitive material, in terms of the parameters l, n and mass m of the electron.
✓

Answer

  1. Planck’s constant: $\text{h}=\frac{\text{lm}}{2\text{n}}$
  2.  
  • $\nu_1^2$ and $\nu_2^2$ are the velocities of the emitted electrons for radiations of frequencies v1 > v and v2 > v respectively. So,

$\text{h}\nu_1=\text{h}\nu+\frac{1}{2}\text{mv}^2_1\dots(\text{i})$

 and $\text{h}\nu_2=\text{h}\nu+\frac{1}{2}\text{mv}^2_2\dots(\text{ii})$

From equation (i) and (ii), we get

$\text{h}(\nu_2-\nu_1)=\frac{1}{2}\text{m}(\text{v}^2_2-\text{v}^2_1)$

$\therefore\ \text{h}=\frac{\frac{1}{2}\text{m}(\text{v}^2_2-\text{v}^2_1)}{(\nu_2-\nu_1)}$

Slope of $\text{v}^2_\text{max}$ vs frequency graph is,

$\tan\theta=\frac{\text{v}^2_2-\text{v}^2_1}{(\nu_2-\nu_1)}$

$\therefore\ \text{h}=\frac{1}{2}\text{m}.\tan\theta$

From graph $\tan\theta=\frac{1}{\text{n}}$

So, $\text{h}=\frac{1}{2}\text{m}\Big(\frac{\text{l}}{\text{n}}\Big)\dots(\text{iii})$

  • From graph, the work function of the material is,

w = hn ...(iv)

From equations (iii) and (iv), we get

$\text{w}=\frac{1}{2}\text{m}\Big(\frac{\text{l}}{\text{n}}\Big)\times\text{n}=\frac{1}{2}\text{ml}$​​​​​​​

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