Give a brief description of the basic elementary process involved… - Vidyadip

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 $v_{max}.$ A plot of $v_{2max}$ is found to vary with frequency $ν$ as shown in the figure.

Use Einstein’s photoelectric equation to find the expressions for:

Planck’s constant.
Work function of the given photosensitive material, in terms of the parameters $l, n$ and mass $m$ of the electron.

✓

Answer

Planck’s constant: $\text{h}=\frac{\text{lm}}{2\text{n}}$
$\nu_1^2$ and $\nu_2^2$ are the velocities of the emitted electrons for radiations of frequencies $v_1 > v$ and $v_2 > v$ respectively.

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