JEE Physics Dual Nature Of Radiation and Matter MCQs Set 22

Answer: (d) \( \dfrac1{\lambda^2}=\dfrac1{\lambda_1^2}+\dfrac1{\lambda_2^2} \)

Explanation:
1. Momentum is conserved as a vector: \( p^2=p_1^2+p_2^2 \) (perpendicular momenta).
2. With \( p=h/\lambda \): \( \dfrac1{\lambda^2}=\dfrac1{\lambda_1^2}+\dfrac1{\lambda_2^2} \).

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (b) \( 1.45\times10^6\ \text{m/s} \)

Explanation:
1. Photon wavelength \( =\dfrac{3\times10^8}{6\times10^{14}}=5\times10^{-7} \) m, so the electron wavelength is \( 5\times10^{-10} \) m.
2. \( v=\dfrac h{m\lambda}=\dfrac{6.63\times10^{-34}}{9.1\times10^{-31}\times5\times10^{-10}}=1.46\times10^6 \) m/s.

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (d) \( \dfrac{\lambda_d^3}{\lambda^2} \)

Explanation:
1. \( K=\dfrac{hc}\lambda-\varphi_0=\dfrac{h^2}{2m\lambda_d^2} \). Differentiate: \( -\dfrac{hc}{\lambda^2}d\lambda=-\dfrac{h^2}{m\lambda_d^3}d\lambda_d \).
2. \( \dfrac{d\lambda_d}{d\lambda}=\dfrac{mc}{h}\dfrac{\lambda_d^3}{\lambda^2}\propto\dfrac{\lambda_d^3}{\lambda^2} \).

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (b) The cut-off wavelength of the continuous X-rays depends on the atomic number of the target

Explanation:
1. The cut-off wavelength \( \lambda_{min}=\dfrac{hc}{eV} \) depends only on the tube voltage, not on the target.
2. Characteristic wavelengths follow Moseley's law (smaller for larger Z) and their intensity grows with the tube power.

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (d) Z = 6

Explanation:
1. \( \dfrac1\lambda\propto(Z-1)^2 \): \( \dfrac{(Z-1)^2}{(11-1)^2}=\dfrac14 \).
2. \( Z-1=5\Rightarrow Z=6 \).

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (b) \( 2\times10^{16} \)

Explanation:
1. \( N=\dfrac Ie=\dfrac{3.2\times10^{-3}}{1.6\times10^{-19}} \).
2. \( N=2\times10^{16} \) per second.

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (d) 1

Explanation:
1. The parent is at rest, so the two products have equal and opposite momenta.
2. \( \lambda=h/p \) is the same for both: \( \lambda_1/\lambda_2=1 \).

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (b) 59 keV

Explanation:
1. \( \Delta E=\dfrac{hc}\lambda=\dfrac{1240\text{ eV nm}}{0.021\text{ nm}} \).
2. \( \Delta E=5.9\times10^4 \) eV \( =59 \) keV.

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (d) the voltage applied to the tube

Explanation:
1. \( \lambda_{min}=\dfrac{hc}{eV} \).
2. It depends only on the accelerating voltage V.

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (b) 41

Explanation:
1. Moseley: \( \dfrac1\lambda=\tfrac34R(Z-1)^2 \).
2. \( (Z-1)^2=\dfrac{1}{0.75\times1.1\times10^7\times0.76\times10^{-10}}=1595\Rightarrow Z-1=39.9\Rightarrow Z=41 \).

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (d) \( 0.27\ \text{\AA} \)

Explanation:
1. \( \dfrac1{\lambda_\alpha}=RZ^2\left(1-\tfrac14\right) \), \( \dfrac1{\lambda_\beta}=RZ^2\left(1-\tfrac19\right) \).
2. \( \lambda_\beta=\lambda_\alpha\dfrac{3/4}{8/9}=0.32\times\dfrac{27}{32}=0.27\ \text{\AA} \).

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (b) 56

Explanation:
1. Nuclear radius \( r\propto A^{1/3} \): \( \left(\dfrac A4\right)^{1/3}=14^{1/3}\Rightarrow A=56 \).
2. With 30 neutrons, \( Z=26 \) (iron).

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (d) \( \sqrt2 \)

Explanation:
1. Kinetic energy for \( 0\le x\le1 \): \( 2E_0-E_0=E_0 \); for \( x\gt 1 \): \( 2E_0 \).
2. \( \lambda\propto\dfrac1{\sqrt K} \), so \( \dfrac{\lambda_1}{\lambda_2}=\sqrt{\dfrac{2E_0}{E_0}}=\sqrt2 \).

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.

Answer: (b) 150 eV

Explanation:
1. Standing wave: \( d=\dfrac{n\lambda}{2} \). Consecutive allowed spacings: \( \dfrac{n\lambda}2=2 \) and \( \dfrac{(n+1)\lambda}2=2.5 \), so \( \dfrac\lambda2=0.5\ \text{\AA} \) and \( \lambda=1\ \text{\AA} \), n = 4.
2. \( E=\dfrac{h^2}{2m\lambda^2}=\dfrac{150.4}{\lambda^2(\text{\AA})}\) eV \( \approx150 \) eV.

Teacher's Note:
1. Write Einstein's equation or the de Broglie relation first, and keep energies in eV with hc = 1240 eV nm.
2. Check that the answer is consistent with the threshold condition or a limiting case.