Question 310: Dual Nature of Matter and Radiation

Question

Two identical photocathodes receive light of frequencies ' $\mathrm{n}_1$ ' and ' $\mathrm{n}_2$ '. If the velocities of the emitted photoelectrons of mass ' $m$ ' are ' $\mathrm{V}_1$ ' and ' V , respectively, then ( $\mathrm{h}=$ Planck's constant )

Accepted Answer

To analyze the situation, we will use the photoelectric equation, which describes the kinetic energy of the emitted photoelectrons due to incident light of a certain frequency. The equation is:
$$ K.E. = \frac{1}{2}mV^2 = h
u - \phi $$
where $K.E.$ is the kinetic energy of the emitted photoelectrons, $m$ is the mass of the electrons, $V$ is their velocity, $h$ is Planck's constant, $
u$ is the frequency of the incident light, and $\phi$ is the work function of the photocathode (which is the same for both as they are identical).
For two identical photocathodes receiving light of frequencies $
u_1$ and $
u_2$, the kinetic energies of the emitted electrons are:
$$ \frac{1}{2}mV_1^2 = h
u_1 - \phi $$
$$ \frac{1}{2}mV_2^2 = h
u_2 - \phi $$
Subtracting equation 2 from equation 1 gives:
$$ \frac{1}{2}mV_1^2 - \frac{1}{2}mV_2^2 = h(
u_1 - 
u_2) $$
Simplifying this, we get:
$$ \frac{1}{2}m(V_1^2 - V_2^2) = h(
u_1 - 
u_2) $$
Rearranging terms, the equation becomes:
$$ V_1^2 - V_2^2 = \frac{2h}{m}(
u_1 - 
u_2) $$
Thus, the correct relation for the given scenario is:
Option D
:
$$ V_1^2 - V_2^2 = \frac{2h}{m}(
u_1 - 
u_2) $$

Answer

$\mathrm{V}_1+\mathrm{V}_2=\left[\frac{2 \mathrm{~h}}{\mathrm{~m}}\left(\mathrm{n}_1+\mathrm{n}_2\right)\right]^{1 / 2}$

Answer

$\mathrm{V}_1-\mathrm{V}_2=\left[\frac{2 \mathrm{~h}}{\mathrm{~m}}\left(\mathrm{n}_1-\mathrm{n}_2\right)\right]^{1 / 2}$

Answer

$\mathrm{V}_1^2+\mathrm{V}_2^2=\frac{2 \mathrm{~h}}{\mathrm{~m}}\left(\mathrm{n}_1+\mathrm{n}_2\right)$

Answer

$\mathrm{V}_1^2-\mathrm{V}_2^2=\frac{2 \mathrm{~h}}{\mathrm{~m}}\left(\mathrm{n}_1-\mathrm{n}_2\right)$

Dual Nature of Matter and Radiation

Two identical photocathodes receive light of frequencies ' \mathrm{n}_1 ' and ' \mathrm{n}_2 '. If the velocities of the emitted photoelectrons of mass ' m ' are ' \mathrm{V}_1 ' and ' V , respectively, then ( \mathrm{h}= Planck's constant )

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