Question 62: Dual Nature of Matter and Radiation

Question

Light is incident on a metallic plate having work function $110 	imes 10^{-20} \mathrm{~J}$. If the produced photoelectrons have zero kinetic energy then the angular frequency of the incident light is $\_\_\_\_$ rad/s. $\left(\mathrm{h}=6.63 	imes 10^{-34} \mathrm{~J} . \mathrm{s}ight)$.

Accepted Answer

The energy of the incident photon (E) is used to overcome the work function ( $\Phi$ ) of the metal, and the remaining energy becomes the maximum kinetic energy ( $\mathrm{K}_{\max }$ ) of the emitted photoelectron:
$$ \mathrm{K}_{\max }=\mathrm{E}-\Phi $$
It is given that the produced photoelectrons have exactly zero kinetic energy ( $K_{\max }=0$ ). This means the energy of the incident light is exactly equal to the work function of the metal (this corresponds to the threshold frequency).
$$ \begin{aligned} & 0=E-\Phi \\ & \Rightarrow E=\Phi \ldots 	ext { (i) } \end{aligned} $$
The energy of a photon is given by $E=h f$, where $h$ is Planck's constant and $f$ is the linear frequency.
Using the relationship between linear frequency (f) and angular frequency ( $\omega$ ):
$$ \omega=2 \pi f \Rightarrow f=\frac{\omega}{2 \pi} \ldots $$
So, the expression for the energy of a photon is,
$$ \mathrm{E}=\mathrm{h}\left(\frac{\omega}{2 \pi}ight)=\frac{\mathrm{h} \omega}{2 \pi} \ldots $$
From relations (i) and (iii)
$$ \frac{h \omega}{2 \pi}=\Phi $$
$$ \Rightarrow \omega=\frac{2 \pi \Phi}{h} $$
Work function of metal is, $\Phi=110 	imes 10^{-20} \mathrm{~J}$
The Planck's constant is, $\mathrm{h}=6.63 	imes 10^{-34} \mathrm{~J} \cdot \mathrm{~s}$
Putting the values,
$$ \begin{gathered} \omega=\frac{2 	imes 3.14 	imes 110 	imes 10^{-20}}{6.63 	imes 10^{-34}} \mathrm{rad} / \mathrm{s} \ \omega=\frac{691.1 	imes 10^{-20}}{6.63 	imes 10^{-34}} \ \omega \approx 1.04 	imes 10^{14} \mathrm{rad} / \mathrm{s} \end{gathered} $$
Therefore, the angular frequency of the incident light is $1.04 	imes 10^{14} \mathrm{rad} / \mathrm{s}$. Hence, the correct option is (D).

Answer

$1.04 	imes 10^{13}$

Answer

$1.66 	imes 10^{16}$

Answer

$1.66 	imes 10^{15}$

Answer

$1.04 	imes 10^{16}$

Dual Nature of Matter and Radiation

Light is incident on a metallic plate having work function 110 \times 10^{-20} \mathrm{~J}. If the produced photoelectrons have zero kinetic energy then the angular frequency of the incident light is \_\_\_\_ rad/s. \left(\mathrm{h}=6.63 \times 10^{-34} \mathrm{~J} . \mathrm{s}\right).

Choose the correct answer: