Chapter 11 Dual Nature Of Radiation And Matter

Question 11.1. A particle is dropped from a height H. The de Broglie wavelength of the particle as a function of height is proportional to

(a) H

(b) $H^{1/2}$

(c) $H^0$

(d) $H^{-1/2}$

Question 11.2. The wavelength of a photon needed to remove a proton from a nucleus which is bound to the nucleus with 1 MeV energy is nearly

(a) 1.2 nm

(b) $1.2 \times 10^{-3}$ nm

(c) $1.2 \times 10^{-6}$ nm

(d) $1.2 \times 10^1$ nm

Question 11.3. Consider a beam of electrons (each electron with energy $E_0$) incident on a metal surface kept in an evacuated chamber. Then

(a) no electrons will be emitted as only photons can emit electrons.

(b) electrons can be emitted but all with an energy, $E_0$.

(c) electrons can be emitted with any energy, with a maximum of $E_0 – \phi$ ($\phi$ is the work function).

(d) electrons can be emitted with any energy, with a maximum of $E_0$.

Question 11.4. Consider Fig. 11.7 in the NCERT text book of physics for Class XII. Suppose the voltage applied to A is increased. The diffracted beam will have the maximum at a value of $\theta$ that

(a) will be larger than the earlier value.

(b) will be the same as the earlier value.

(c) will be less than the earlier value.

(d) will depend on the target.

Question 11.5. A proton, a neutron, an electron and an $\alpha$-particle have same energy. Then their de Broglie wavelengths compare as

(a) $\lambda_p = \lambda_n > \lambda_e > \lambda_\alpha$

(b) $\lambda_\alpha < \lambda_p = \lambda_n > \lambda_e$

(c) $\lambda_e < \lambda_p = \lambda_n > \lambda_\alpha$

(d) $\lambda_e = \lambda_p = \lambda_n = \lambda_\alpha$

Question 11.6. An electron is moving with an initial velocity $\vec{v} = v_0 \hat{i}$ and is in a magnetic field $\vec{B} = B_0 \hat{j}$. Then it’s de Broglie wavelength

(a) remains constant.

(b) increases with time.

(c) decreases with time.

(d) increases and decreases periodically.

Question 11.7. An electron (mass m) with an initial velocity $\vec{v} = v_0 \hat{i}$ ($v_0 > 0$) is in an electric field $\vec{E} = –E_0 \hat{i}$ ($E_0 = \text{constant} > 0$). It’s de Broglie wavelength at time t is given by

(a) $\frac{\lambda_0}{1 + \frac{eE_0t}{mv_0}}$

(b) $\lambda_0\left(1 + \frac{eE_0 t}{mv_0}\right)$

(c) $\lambda_0$

(d) $\lambda_0 t$

Question 11.8. An electron (mass m) with an initial velocity $\vec{v} = v_0 \hat{i}$ is in an electric field $\vec{E} = E_0 \hat{j}$. If $\lambda_0 = h/mv_0$, it’s de Broglie wavelength at time t is given by

(a) $\lambda_0$

(b) $\lambda_0 \sqrt{1 + \frac{e^2 E_0^2 t^2}{m^2 v_0^2}}$

(c) $\frac{\lambda_0}{\sqrt{1 + \frac{e^2 E_0^2 t^2}{m^2 v_0^2}}}$

(d) $\frac{\lambda_0}{\left(1 + \frac{e^2 E_0^2 t^2}{m^2 v_0^2}\right)}$

Question 11.9. Relativistic corrections become neccssary when the expression for the kinetic energy $\frac{1}{2}mv^2$, becomes comparable with $mc^2$, where m is the mass of the particle. At what de Broglie wavelength will relativistic corrections become important for an electron?

(a) $\lambda = 10$ nm

(b) $\lambda = 10^{-1}$ nm

(c) $\lambda = 10^{-4}$ nm

(d) $\lambda = 10^{-6}$ nm

Question 11.10. Two particles $A_1$ and $A_2$ of masses $m_1, m_2$ ($m_1 > m_2$) have the same de Broglie wavelength. Then

(a) their momenta are the same.

(b) their energies are the same.

(c) energy of $A_1$ is less than the energy of $A_2$.

(d) energy of $A_1$ is more than the energy of $A_2$.

Question 11.11. The de Broglie wavelength of a photon is twice the de Broglie wavelength of an electron. The speed of the electron is $v_e = \frac{c}{100}$. Then

(a) $\frac{E_e}{E_p} = 10^{-4}$

(b) $\frac{E_e}{E_p} = 10^{-2}$

(c) $\frac{p_e}{m_e c} = 10^{-2}$

(d) $\frac{p_e}{m_e c} = 10^{-4}$

Question 11.12. Photons absorbed in matter are converted to heat. A source emitting n photon/sec of frequency $\nu$ is used to convert 1kg of ice at 0°C to water at 0°C. Then, the time T taken for the conversion

(a) decreases with increasing n, with $\nu$ fixed.

(b) decreases with n fixed, $\nu$ increasing

(c) remains constant with n and $\nu$ changing such that $n\nu = \text{constant}$.

(d) increases when the product $n\nu$ increases.

Question 11.13. A particle moves in a closed orbit around the origin, due to a force which is directed towards the origin. The de Broglie wavelength of the particle varies cyclically between two values $\lambda_1, \lambda_2$ with $\lambda_1 > \lambda_2$. Which of the following statement are true?

(a) The particle could be moving in a circular orbit with origin as centre

(b) The particle could be moving in an elliptic orbit with origin as its focus.

(c) When the de Broglie wave length is $\lambda_1$, the particle is nearer the origin than when its value is $\lambda_2$.

(d) When the de Broglie wavelength is $\lambda_2$, the particle is nearer the origin than when its value is $\lambda_1$.

Question 11.14. A proton and an $\alpha$-particle are accelerated, using the same potential difference. How are the deBroglie wavelengths $\lambda_p$ and $\lambda_\alpha$ related to each other?

Question 11.15. (i) In the explanation of photo electric effect, we asssume one photon of frequency $\nu$ collides with an electron and transfers its energy. This leads to the equation for the maximum energy $E_{max}$ of the emitted electron as $E_{max} = h\nu – \phi_0$ where $\phi_0$ is the work function of the metal. If an electron absorbs 2 photons (each of frequency $\nu$) what will be the maximum energy for the emitted electron?

(ii) Why is this fact (two photon absorption) not taken into consideration in our discussion of the stopping potential?

Question 11.16. There are materials which absorb photons of shorter wavelength and emit photons of longer wavelength. Can there be stable substances which absorb photons of larger wavelength and emit light of shorter wavelength.

Question 11.17. Do all the electrons that absorb a photon come out as photoelectrons?

Question 11.18. There are two sources of light, each emitting with a power of 100 W. One emits X-rays of wavelength 1nm and the other visible light at 500 nm. Find the ratio of number of photons of X-rays to the photons of visible light of the given wavelength?

Question 11.19. Consider Fig.11.1 for photoemission. How would you reconcile with momentum-conservation? Note light (photons) have momentum in a different direction than the emitted electrons.

Question 11.20. Consider a metal exposed to light of wavelength 600 nm. The maximum energy of the electron doubles when light of wavelength 400 nm is used. Find the work function in eV.

Question 11.21. Assuming an electron is confined to a 1nm wide region, find the uncertainty in momentum using Heisenberg Uncertainty principle (Ref Eq 11.12 of NCERT Textbook). You can assume the uncertainty in position $\Delta x$ as 1nm. Assuming $p \approx \Delta p$, find the energy of the electron in electron volts.

Question 11.22. Two monochromatic beams A and B of equal intensity I, hit a screen. The number of photons hitting the screen by beam A is twice that by beam B. Then what inference can you make about their frequencies?

Question 11.23. Two particles A and B of de Broglie wavelengths $\lambda_1$ and $\lambda_2$ combine to form a particle C. The process conserves momentum. Find the de Broglie wavelength of the particle C. (The motion is one dimensional).

Question 11.24. A neutron beam of energy E scatters from atoms on a surface with a spacing d = 0.1nm. The first maximum of intensity in the reflected beam occurs at $\theta = 30^\circ$. What is the kinetic energy E of the beam in eV?

Question 11.25. Consider a thin target ($10^{-2}$m square, $10^{-3}$m thickness) of sodium, which produces a photocurrent of 100$\mu$A when a light of intensity 100W/m$^2$ ($\lambda = 660$nm) falls on it. Find the probability that a photoelectron is produced when a photons strikes a sodium atom. [Take density of Na = 0.97 kg/m$^3$].

Question 11.26. Consider an electron in front of metallic surface at a distance d (treated as an infinite plane surface). Assume the force of attraction by the plate is given as $\frac{1}{4\pi\epsilon_0}\frac{q^2}{4d^2}$. Calculate work in taking the charge to an infinite distance from the plate. Taking d = 0.1nm, find the work done in electron volts. [Such a force law is not valid for d < 0.1nm].

Question 11.27. A student performs an experiment on photoelectric effect, using two materials A and B. A plot of $V_{stop}$ vs $\nu$ is given in Fig. 11.2.

(i) Which material A or B has a higher work function?

(ii) Given the electric charge of an electron = $1.6 \times 10^{-19}$ C, find the value of h obtained from the experiment for both A and B. Comment on whether it is consistent with Einstein’s theory:

Question 11.28. A particle A with a mass $m_A$ is moving with a velocity v and hits a particle B (mass $m_B$) at rest (one dimensional motion). Find the change in the de Broglie wavelength of the particle A. Treat the collision as elastic.

Question 11.29. Consider a 20 W bulb emitting light of wavelength 5000Å and shining on a metal surface kept at a distance 2m. Assume that the metal surface has work function of 2 eV and that each atom on the metal surface can be treated as a circular disk of radius 1.5 Å.

(i) Estimate no. of photons emitted by the bulb per second. [Assume no other losses]

(ii) Will there be photoelectric emission?

(iii) How much time would be required by the atomic disk to receive energy equal to work function (2 eV)?

(iv) How many photons would atomic disk receive within time duration calculated in (iii) above?

(v) Can you explain how photoelectric effect was observed instantaneously? [Hint: Time calculated in part (iii) is from classical consideration and you may further take the target of surface area say 1cm$^2$ and estimate what would happen?]