Dual Nature of Radiation and Matter

The De-Broglie wavelength of an electron in the $n=2$ state of a hydrogen atom can be calculated as follows:

Firstly, calculate the radius $r$ for the $n=2$ state using the Bohr radius formula:

$ r = 0.052 n^2 $

For $n=2$, the radius is:

$ \begin{aligned} r &= 0.052 \times 4 \\ &= 0.208 \, \text{nm} \end{aligned} $

We use the relationship $M v r = \frac{n h}{2 \pi}$, leading to the expression for the De-Broglie wavelength $\lambda$:

$ \lambda = \frac{h}{M v} = \pi r $

Substituting the calculated $r$ value:

$ \begin{aligned} \lambda &= \pi \times 0.208 \, \text{nm} \\ &= 3.14 \times 0.208 \, \text{nm} \\ &= 0.65317 \, \text{nm} \\ &\approx 0.67 \, \text{nm} \end{aligned} $

Thus, the De-Broglie wavelength of the electron in the $n=2$ state is approximately $0.67$ nm.

Photoelectric current is directly proportional to intensity of light.

To find the ratio of the de Broglie wavelengths of a photon and an electron when they both have the same energy $E$, follow these steps:

Photon Wavelength:

For a photon, the energy is given by:

$ E = \frac{h c}{\lambda_{\mathrm{Ph}}} $

Where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda_{\mathrm{Ph}}$ is the wavelength of the photon. Rearranging this equation gives:

$ \lambda_{\mathrm{Ph}} = \frac{h c}{E} $

Electron Wavelength:

For an electron, the energy related to its momentum is:

$ E = \frac{p^2}{2m} $

Where $m$ is the mass of the electron and $p$ is its momentum. Using the de Broglie wavelength expression $p = \frac{h}{\lambda_e}$, we can write:

$ E = \left( \frac{h}{\lambda_e} \right)^2 \times \frac{1}{2m} $

Solving for the electron's wavelength $\lambda_e$:

$ \lambda_e = \frac{h}{\sqrt{2mE}} $

Ratio of Wavelengths:

Now, calculate the ratio of the wavelengths $\frac{\lambda_{\mathrm{Ph}}}{\lambda_e}$ as follows:

$ \frac{\lambda_{\mathrm{Ph}}}{\lambda_e} = \frac{\frac{h c}{E}}{\frac{h}{\sqrt{2mE}}} = c \sqrt{\frac{2m}{E}} $

This ratio equation defines the relation between the photon and electron wavelengths when both have the same energy $E$.

de-Broglie wavelength is given by

$\lambda=\frac{h}{p}=\frac{h}{\sqrt{2 m q V}}$

For same potential difference

$\begin{gathered} \lambda \propto \frac{1}{\sqrt{m q}} \\ \frac{\lambda_\alpha}{\lambda_e}=\sqrt{\frac{m_e q_e}{m_\alpha q_\alpha}} \\ \because m_\alpha \gg m_e \\ \lambda_e>\lambda_\alpha \end{gathered}$

The energy of the incident light is given by:

$E = h \nu = \frac{hc}{\lambda}$

where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength of the light. Substituting the given value of $\lambda$, we get:

$E = \frac{hc}{hc/e} = e$

The maximum kinetic energy of the photoelectron is given by:

$KE_{max} = E - \phi$

where $\phi$ is the work function. Substituting the values, we get:

$KE_{max} = e - \phi$

Therefore, the correct answer is Option C.

de-Broglie wavelength $\lambda=\frac{h}{P}=\frac{h}{m v}=\frac{h}{\sqrt{2 m E}}$ where $E=\frac{1}{2} m v^2$

Squaring both sides,

$\begin{aligned} & \lambda^2=\frac{h^2}{4 m^2 E} \\ & \Rightarrow \frac{1}{\lambda^2}=\text { (constant) } E \end{aligned}$

Graph passes through origin with constant slope.

A. The energy of a photon is $E = h v$.

This statement is correct. The energy of a photon is given by the equation:

$E = h v$

where $E$ is the energy, $h$ is Planck's constant, and $v$ (also written as $\nu$) is the frequency of the photon.

B. The velocity of a photon is $c$.

This statement is also correct. In free space (vacuum), the velocity of light, and hence the velocity of a photon, is constant and is denoted by $c$.

C. The momentum of a photon, $p = \frac{h v}{c}$.

This statement is correct as well. The momentum $p$ of a photon can be expressed as:

$p = \frac{E}{c} = \frac{h v}{c}$

D. In a photon-electron collision, both total energy and total momentum are conserved.

This statement is correct. During collisions involving photons and electrons, such as Compton scattering, both energy and momentum are conserved.

E. Photon possesses positive charge.

This statement is incorrect. Photons are electrically neutral particles and do not possess any charge.

Based on the analysis, the correct statements are A, B, C, and D. Therefore, the correct answer is:

Option B
A, B, C, and D only

${\lambda _e} = {{12.27} \over {\sqrt V }}\mathop A\limits^o = {{12.27} \over {\sqrt {81} }}\mathop A\limits^o = {{12.27} \over 9}\mathop A\limits^o $

$ = 1.36\mathop A\limits^o $ ($\because$ $1\mathop A\limits^o = {1 \over {10}}$ nm)

$ = 0.136$ nm

Maximum kinetic energy of emitted electron is independent of intensity of radiation.

Given energy of photon $E=2.20 ~\mathrm{eV}$

Work function of $\mathrm{Cs} ~\phi_{0}=2.14 ~\mathrm{eV}, \mathrm{K} ~\phi_{0}=2.30 ~\mathrm{eV}, \mathrm{Na} ~\phi_{0}=2.75 ~\mathrm{eV}$

We know that $e^{-}$ emitts when $h v>\phi_{0}$

here it is clear that energy of photon is more than the work function of $\mathrm{Cs}$ [Caesium] only so Ans. only (Cs).

The minimum wavelength, $\lambda_{\min}$, of X-rays produced by an electron can be determined using the equation derived from the energy of a photon and the energy given to an electron by an accelerating voltage, $V$.

The energy of a photon is given by $E = hc/\lambda$, where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength of the photon.

When an electron is accelerated through a potential difference of $V$ volts, it gains kinetic energy equal to $eV$, where $e$ is the electron charge.

This energy is then converted into a photon's energy when the electron collides with a target in an X-ray tube, resulting in X-rays of wavelength $\lambda$. Setting the kinetic energy equal to the photon energy gives $eV = hc/\lambda$. Solving for $\lambda$ gives $\lambda = hc/(eV)$.

Therefore, the minimum wavelength (corresponding to the maximum energy photon produced when all the kinetic energy is converted into photon energy) is inversely proportional to the voltage, $V$.

Thus, $\lambda_{\min} \propto 1/V$.

We know, $K{E_{\max }} = hv - h{v_0}$

$e{V_0} = hv - h{v_0}$ ($\because$ $K{E_{\max }} = e{V_0}$)

$e{V_0} = 4.2\,eV - 2.2\,eV$

$\therefore$ ${V_0} = 2\,V$

According to Einstein's photoelectric equation

${(K.E)_{\max }} = hv - h{v_0}$

${(K.E)_{\max }} = h(4{v_0}) - h{v_0}$

${(K.E)_{\max }} = 3h{v_0}$

Since ${k_{\max }} = e{V_s} = hv - \phi $

${{e{V_s}} \over 2} = hv - h{v_0}$ ....... (i)

$e{V_s} = {{hv} \over 2} - h{v_0}$ ...... (ii)

${1 \over 2}\left[ {{{hv} \over 2} - h{v_0}} \right] = hv - h{v_0}$

$ \Rightarrow h{v_0} - {{h{v_0}} \over 2} = hv - {{hv} \over 4}$

$ \Rightarrow {{h{v_0}} \over 2} = {{3hv} \over 4}$

${v_0} = {{3v} \over 2}$

* Language of question is wrongly framed. The values of stopping potentials should be interchanged.