Question 37: Dual Nature of Matter and Radiation

Question

For a certain metal, when monochromatic light of wavelength $\lambda$ is incident, the stopping potential for photoelectrons is $3V_0$. When the same metal is illuminated by light of wavelength $2\lambda$, then the stopping potential becomes $V_0$. The threshold wavelength for photoelectric emission for the given metal is $\alpha \lambda$. The value of $\alpha$ is ______.

Accepted Answer

The power $(P)$ of the laser source is, $P=6 \mathrm{~mW}=6 	imes 10^{-3} \mathrm{~J} / \mathrm{s}$
The wavelength ( $\lambda$ ) of the emitted photon is $\lambda=663 \mathrm{~nm}=663 	imes 10^{-9} \mathrm{~m}$
Planck's constant $\mathrm{h}=6.63 	imes 10^{-34} \mathrm{~J} \cdot \mathrm{~s}$
Speed of light $\mathrm{c}=3 	imes 10^8 \mathrm{~m} / \mathrm{s}$
The energy of a single photon is given by :
$$ \mathrm{E}=\frac{\mathrm{hc}}{\lambda} $$
Total Power is the number of photons per second multiplied by the energy of one photon:
$$ \mathrm{P}=\mathrm{n} 	imes \mathrm{E} \Rightarrow \mathrm{n}=\frac{\mathrm{P}}{\mathrm{E}}=\frac{\mathrm{P} \lambda}{\mathrm{hc}} $$
Substituting the given values into the equation :
$n=\frac{\left(6 	imes 10^{-3}ight) 	imes\left(663 	imes 10^{-9}ight)}{\left(6.63 	imes 10^{-34}ight) 	imes\left(3 	imes 10^8ight)}$
$$\Rightarrow $$ $\mathrm{n}=\frac{600}{3} 	imes \frac{10^{-12}}{10^{-26}}$
$$\Rightarrow $$ $\mathrm{n}=200 	imes 10^{14}$
$$\Rightarrow $$ $$ \mathrm{n}=2 	imes 10^{16} $$
Therefore, the laser source emits $2 	imes 10^{16}$ photons every second. Hence, the correct option is (D).

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Dual Nature of Matter and Radiation

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