A particle is dropped from a height \(H.\) The de-Broglie wavelength of the particle as a function of height is proportional to:
1. \(H\)
2. \(H^{1/2}\)
3. \(H^{0}\)
4. \(H^{-1/2}\)

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

1. 1.2 nm

2. $1.2\times {10}^{-3} \mathrm{nm}$

3. $1.2\times {10}^{-6} \mathrm{nm}$

4. $1.2\times 10 \mathrm{nm}$

Consider a beam of electrons (each electron with energy E₀) incident on a metal surface kept in an evacuated chamber. Then:

Consider the figure given below. Suppose the voltage applied to A is increased. The diffracted beam will have the maximum at a value of $\mathrm{\theta }$ that
                  
1. will be larger than the earlier value

2. will be the same as the earlier value

3. will be less than the earlier value 

4. will depend on the target

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 ${\mathrm{\lambda }}_1, {\mathrm{\lambda }}_2$ with ${\mathrm{\lambda }}_1>{\mathrm{\lambda }}_2$. Which of the following statement/s is/are true?

(a) The particle could be moving in a circular orbit with origin as the centre.
(b) The particle could be moving in an elliptic orbit with origin as its focus.
(c) When the de-Broglie wavelength is ${\mathrm{\lambda }}_1$, the particle is nearer the origin than when its value is ${\mathrm{\lambda }}_2$.
(d) When the de-Broglie wavelength is ${\mathrm{\lambda }}_2$, the particle is nearer the origin than when its value is ${\mathrm{\lambda }}_1$.

1. (b, d)

2. (a, c)

3. (b, c, d)

4. (a, c, d)

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

(a) decreases with increasing n, with $\mathrm{\nu }$ fixed
(b) decreases with n fixed, $\mathrm{\nu }$ increasing
(c) remains constant with n and $\mathrm{\nu }$ changing such that n$\mathrm{\nu }$=constant
(d) increases when the product n$\mathrm{\nu }$ increases

1. (b, d)
2. (a, c, d)
3. (a, d)
4. (a, b, c)

The de-Broglie wavelength of a photon is twice the de-Broglie wavelength of an electron. The speed of the electron is ${\mathrm{v}}_{\mathrm{e}}=\frac{\mathrm{c}}{100}$. Then,

1. \(\frac{E_e}{E_p}=10^{-4}\)

2. \(\frac{E_e}{E_p}=10^{-2}\)

3. \(\frac{P_e}{m_ec}=10^{-2}\)

4. \(\frac{P_e}{m_ec}=10^{-4}\)

Two particles A₁ and A₂ of masses m₁, m₂ (m₁>m₂) have the same de-Broglie wavelength. Then,

(a) their momenta are the same
(b) their energies are the same
(c) energy of A₁ is less than the energy of A₂
(d) energy of A₁ is more than the energy of A₂

1. (b, c)
2. (a, c)
3. (c, d)
4. (b, d)

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

(a) $\mathrm{\lambda }=10 \mathrm{nm}$
(b) $\mathrm{\lambda }={10}^{-1} \mathrm{nm}$
(c) $\mathrm{\lambda }={10}^{-4} \mathrm{nm}$
(d) $\mathrm{\lambda }={10}^{-6} \mathrm{nm}$

1. (a, c)

2. (a, d)

3. (c, d)

4. (a, b)

An electron (mass m) with an initial velocity $\overrightarrow{\mathrm{v}}={\mathrm{v}}_0\hat{\mathrm{i}}$ is in an electric field $\overrightarrow{\mathrm{E}}={\mathrm{E}}_0\hat{\mathrm{j}}$. If ${\mathrm{\lambda }}_0=\frac{\mathrm{h}}{{\mathrm{mv}}_0}$, its de-Broglie wavelength at time t is given by:

1. ${\mathrm{\lambda }}_0$

2. ${\mathrm{\lambda }}_0\sqrt{1+\frac{{\mathrm{e}}^2{\mathrm{E}}_0^2{\mathrm{t}}^2}{{\mathrm{m}}^2{\mathrm{v}}_0^2}}$

3. $\frac{{\mathrm{\lambda }}_0}{\sqrt{1+\frac{{\mathrm{e}}^2{\mathrm{E}}_0^2{\mathrm{t}}^2}{{\mathrm{m}}^2{\mathrm{v}}_0^2}}}$

4. $\frac{{\mathrm{\lambda }}_0}{\left(1+\frac{{\mathrm{e}}^2{\mathrm{E}}_0^2{\mathrm{t}}^2}{{\mathrm{m}}^2{\mathrm{v}}_0^2}\right)}$