Dual Nature of Radiation and Matter Set-5

Q1. A monochromatic light of wavelength $400\ \mathrm{nm}$ falls on a metal surface whose work function is $2.3\ \mathrm{eV}$. The stopping potential $V_0$ required to reduce the photoelectric current to zero is approximately:

Q2. A photon and a non-relativistic electron have the same wavelength $\lambda$. The electron was accelerated from rest through a potential $V$. Which of the following expressions for $V$ is correct?

Q3. A photon of wavelength $0.050\ \mathrm{nm}$ is Compton-scattered by a stationary electron at angle $\theta=60^\circ$. Using $\Delta\lambda=\dfrac{h}{m_e c}(1-\cos\theta)$, the kinetic energy of the recoiling electron (approximately) is:

Q4. Two monochromatic light beams of equal intensity $I$ with wavelengths $\lambda_1=300\ \mathrm{nm}$ and $\lambda_2=600\ \mathrm{nm}$ are incident on the same metal whose work function is $\phi=2.0\ \mathrm{eV}$. Assume every photon above threshold produces one photoelectron. Which statement is correct about the maximum kinetic energy $K_{\max}$ of emitted electrons and the saturation current $I_{\text{sat}}$?

Q5. If an electron's de Broglie wavelength equals its Compton wavelength $\lambda_C=\dfrac{h}{m_e c}$, the kinetic energy $K$ of the electron (using relativistic relations and $m_e c^2=511\ \mathrm{keV}$) is approximately:

Q6. A monochromatic light of wavelength $400\ \mathrm{nm}$ falls on a metal surface whose work function is $2.0\ \mathrm{eV}$. Given $hc=1240\ \mathrm{eV\,nm}$, the stopping potential $V_s$ for the photoelectrons is:

Q7. In a Davisson–Germer electron diffraction experiment electrons accelerated through potential $V$ give a first-order diffraction maximum at angle $\theta_1=30^\circ$ (crystal spacing $d$ unchanged). If the accelerating potential is increased to $4V$ (non-relativistic electrons), the new first-order diffraction angle $\theta_2$ will be approximately (use $n\lambda=2d\sin\theta$ and $\lambda\propto1/\sqrt{V}$):

Q8. A photon whose wavelength equals the Compton wavelength of the electron, $\lambda=\lambda_C=\dfrac{h}{mc}$, is scattered by a free electron at rest. Using the Compton relation $\lambda'=\lambda+\lambda_C(1-\cos\theta)$, for which photon scattering angle $\theta$ will the scattered photon have exactly half the initial energy?

Q9. A photon of initial energy $E=2mc^2$ scatters from a stationary electron (Compton scattering). Using the relation $E'=\dfrac{E}{1+\dfrac{E}{mc^2}(1-\cos\theta)}$, for what photon scattering angle $\theta$ will the recoiling electron receive exactly one-half of the initial photon energy?

Q10. Assertion (A): In a double-slit experiment with slit separation $d$, using photons of wavelength $\lambda\le d$ to determine which slit an electron passes through inevitably destroys the interference pattern.
Reason (R): To resolve the two slits the photon must have momentum $p_\gamma\sim h/d$, which imparts to the electron a momentum uncertainty $\Delta p\sim p_\gamma$ large enough to randomize the relative phase between contributions from the two slits.

Q11. Electrons accelerated through a potential $V_1=400\ \mathrm{V}$ produce a first-order ($n=1$) Bragg diffraction peak at angle $\theta_1=30^\circ$ from crystal planes with spacing $d=0.25\ \mathrm{nm}$. Assuming non-relativistic electrons and using $n\lambda=2d\sin\theta$ together with $\lambda\propto 1/\sqrt{V}$, the accelerating potential $V_2$ required to produce the first-order peak at $\theta_2=45^\circ$ for the same planes is:

Q12. An electron beam with central de Broglie wavelength $\lambda=0.05\ \mathrm{nm}$ is prepared as a wavepacket localized to spatial width $\Delta x=1.0\ \mathrm{nm}$. Using the uncertainty relation $\Delta x\,\Delta p\approx \hbar/2$ and $\lambda=h/p$ (with $\hbar=h/2\pi$), estimate the minimum uncertainty $\Delta\lambda$ in its de Broglie wavelength.

Q13. An electron is accelerated from rest through a potential difference $V=1.0\ \mathrm{MV}$ (so kinetic energy $eV\approx 1.0\ \mathrm{MeV}$). Using the non-relativistic de Broglie expression $\lambda_{\mathrm{nr}}=\dfrac{h}{\sqrt{2meV}}$ and the relativistically correct expression $\lambda_{\mathrm{rel}}=\dfrac{h}{p}$ with

estimate approximately by what percentage $\lambda_{\mathrm{nr}}$ overestimates the true relativistic de Broglie wavelength $\lambda_{\mathrm{rel}}$. (Take $mc^2=511\ \mathrm{keV}$.)

Q14. A photon of energy $E$ scatters from a stationary free electron. For which photon scattering angle $\theta$ (measured from the incident direction) will the recoil electron receive exactly half of the incident photon energy? Choose the correct statement.

Q15. A photon of energy $E=511\ \mathrm{keV}$ (equal to $mc^2$ for an electron) undergoes Compton scattering with scattering angle $\theta=180^\circ$. Using

the energy $E'$ of the scattered photon is approximately: