Dual Nature of Radiation and Matter Set-2

Q1. An electron is accelerated from rest through a potential difference of $200\ \text{V}$. Using $h=6.63\times10^{-34}\ \text{J\,s}$, $m_e=9.11\times10^{-31}\ \text{kg}$ and $e=1.60\times10^{-19}\ \text{C}$, calculate its de Broglie wavelength $\lambda$ (in nm) using the non‑relativistic relation $\displaystyle \lambda=\frac{h}{\sqrt{2m_e eV}}$.

Q2. For a metal with work function $\phi=2.50\ \text{eV}$, monochromatic light of wavelength $400\ \text{nm}$ is incident. Using $hc=1240\ \text{eV\,nm}$, calculate the stopping potential $V_0$ for emitted electrons (use $eV_0 = \dfrac{hc}{\lambda}-\phi$).

Q3. Assertion (A): In the photoelectric effect, for incident light with frequency above threshold, increasing the intensity increases the photoelectric current but does not change the maximum kinetic energy of emitted electrons.
Reason (R): Increasing intensity increases the number of incident photons per second but does not change the energy per photon; hence energy available to each emitted electron remains unchanged.

Q4. An electron beam accelerated through $V_1=250\ \text{V}$ produces the first‑order diffraction maximum at $\theta_1=30^\circ$ from a crystal (Bragg condition $n\lambda=2d\sin\theta$). If the accelerating voltage is changed to $V_2=1000\ \text{V}$, what is the new first‑order diffraction angle $\theta_2$? (Assume non‑relativistic electrons so $\lambda\propto 1/\sqrt{V}$.)

Q5. In a double‑slit experiment with electrons, the source is attenuated so that at most one electron is inside the apparatus at a time. Individual impacts on the detector are localized, yet after many impacts an interference pattern appears. Which explanation best accounts for these observations?

Q6. A monochromatic X‑ray photon of wavelength $\lambda=0.05\ \text{nm}$ is Compton‑scattered by a (nearly) free electron at scattering angle $\theta=60^\circ$. Using $\Delta\lambda=\lambda_C(1-\cos\theta)$ with $\lambda_C=\dfrac{h}{m_e c}=2.43\times10^{-12}\ \text{m}$, find the change in wavelength $\Delta\lambda$ (in m).

Q7. Assertion (A): In the electron double‑slit experiment, placing a photon detector at the slits that can determine which slit each electron passes through destroys the interference pattern even when the photon imparts negligible energy to the electron.
Reason (R): Destruction of interference requires a significant exchange of energy between photon and electron; negligible energy transfer would not affect the interference.

Q8. In a photoelectric experiment the stopping potential is measured at two frequencies: at $f_1=6.0\times10^{14}\ \text{Hz}$, $V_1=0.20\ \text{V}$; at $f_2=1.0\times10^{15}\ \text{Hz}$, $V_2=1.86\ \text{V}$. Using $h=e\,\dfrac{\Delta V}{\Delta f}$ with $e=1.60\times10^{-19}\ \text{C}$, estimate Planck's constant $h$ (in J·s).

Q9. Two particles of masses $m$ and $4m$ have the same de Broglie wavelength $\lambda$. Let their speeds be $v_1$ and $v_2$ and kinetic energies be $K_1$ and $K_2$ for the particles of mass $m$ and $4m$ respectively. Which relation is correct?

Q10. For two metals A and B the stopping potential vs frequency relations are measured to be

Which of the following statements is/are correct?
I. Metal A has larger work function than metal B.
II. For the same incident frequency $f$ (above thresholds), maximum kinetic energy of photoelectrons from B is greater than that from A.
III. Threshold frequency of A is smaller than that of B.