Question 60: Dual Nature of Matter and Radiation

Question

When a light of a given wavelength falls on a metallic surface the stopping potential for photoelectrons is 3.2 V . If a second light having wavelength twice of first light is used, the stopping potential drops to 0.7 V . The wavelength of first light is $\_\_\_\_$ m .
$$ \left(\mathrm{h}=6.63 	imes 10^{-34} \mathrm{~J} . \mathrm{s}, \mathrm{e}=1.6 	imes 10^{-19} \mathrm{C}, \mathrm{c}=3 	imes 10^8 \mathrm{~m} / \mathrm{s}ight) $$

Accepted Answer

When a light of wavelength ( $\lambda$ ) is incident on a metal of work function $\phi$ then the maximum kinetic energy of the photo-electron is given by Einstein's photoelectric equation:
$$ \mathrm{K}_{\max }=\mathrm{E}-\phi \Rightarrow \mathrm{E}=\phi+\mathrm{K}_{\max } $$
$$ \Rightarrow \frac{\mathrm{hc}}{\lambda}=\phi+\mathrm{eV}_{\mathrm{s}} $$
where $\mathrm{V}_{\mathrm{s}}$ is the stopping potential and $\phi$ is the work function of the metal.
Case 1 :
Wavelength is $\lambda$, stopping potential is 3.2 V .
$$ \frac{\mathrm{hc}}{\lambda}=\phi+3.2 \mathrm{e} $$.....(i)
Case 2 :
Wavelength is $2 \lambda$, stopping potential is 0.7 V .
$$ \frac{\mathrm{hc}}{2 \lambda}=\phi+0.7 \mathrm{e} \ldots (ii)$$
Subtracting equation (ii) from equation (i) :
$$ \left(\frac{\mathrm{hc}}{\lambda}-\frac{\mathrm{hc}}{2 \lambda}ight)=(3.2 \mathrm{e}-0.7 \mathrm{e}) $$
$$\Rightarrow $$ $\frac{\mathrm{hc}}{2 \lambda}=2.5 \mathrm{e}$
$$\Rightarrow $$ $$ \lambda=\frac{\mathrm{hc}}{5 \mathrm{e}} $$
Substituting the given values :
$$ \lambda=\frac{\left(6.63 	imes 10^{-34}ight) 	imes\left(3 	imes 10^8ight)}{5 	imes\left(1.6 	imes 10^{-19}ight)} $$
$$\Rightarrow $$ $$ \lambda=\frac{19.89 	imes 10^{-26}}{8 	imes 10^{-19}} \mathrm{~m} \approx 2.5 	imes 10^{-7} \mathrm{~m} $$
Therefore, the initial wavelength of light used was $2.5 	imes 10^{-7} \mathrm{~m}$.
Hence, the correct option is (B).

Answer

$2.9 	imes 10^{-8}$

Answer

$2.5 	imes 10^{-7}$

Answer

$3.1 	imes 10^{-7}$

Answer

$2.2 	imes 10^{-8}$

Dual Nature of Matter and Radiation

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