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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. \(H^{-1/2}\)

Subtopic: De-broglie Wavelength |

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The wavelength of a photon needed to remove a proton from a nucleus which is bound to the nucleus with \(1~\text{MeV}\) energy is nearly:

1. \(1.2~\text{nm}\)

2. \(1.2\times 10^{-3}~\text{nm}\)

3. \(1.2\times 10^{-6}~\text{nm}\)

4. \(1.2\times 10~\text{nm}\)

Subtopic: Einstein's Photoelectric Equation |

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

1. | no electrons will be emitted as only photons can emit electrons. |

2. | electrons can be emitted but all with energy, \(E_0\)${\mathrm{}}_{}$ |

3. | electrons can be emitted with any energy, with a maximum of \(\mathrm{E}_0-\phi\) (\(\phi\) is the work function). |

4. | electrons can be emitted with any energy, with a maximum \(E_0\). |

Subtopic: Electron Emission |

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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/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 \(λ_1\), the particle is nearer the origin than when its value is \(λ_2\). |

(d) | When the de-Broglie wavelength is \(λ_2\), the particle is nearer the origin than when its value is \(λ_1\). |

Choose the correct option:

1. (b), (d)

2. (a), (c)

3. (b), (c), (d)

4. (a), (c), (d)

Subtopic: De-broglie Wavelength |

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Photons absorbed in matter are converted to heat. A source emitting \(n\) photon/sec of frequency \(\nu\) is used to convert \(1\) kg of ice at \(0^{\circ}\text{C}\) to water at \(0^{\circ}\text{C}\). 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 \(n\nu=\) constant |

(d) | increases when the product \(n\nu\) increases |

Choose the correct option:

1. (b), (d)

2. (a), (c), (d)

3. (a), (d)

4. (a), (b), (c)

Subtopic: Particle Nature of Light |

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The de-Broglie wavelength of a photon is twice the de-Broglie wavelength of an electron. The speed of the electron is \(v_e = \dfrac c {100}\). Then,

1. \(\dfrac{E_e}{E_p}=10^{-4}\)

2. \(\dfrac{E_e}{E_p}=10^{-2}\)

3. \(\dfrac{P_e}{m_ec}=10^{-2}\)

4. \(\dfrac{P_e}{m_ec}=10^{-4}\)

Subtopic: De-broglie Wavelength |

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Two particles \(A_1\) and \(A_2\) of masses \({m_1},m_2~({m_1>m_2})\) have the same de-Broglie wavelength. Then:

Choose the correct option:

1. (b), (c)

2. (a), (c)

3. (c), (d)

4. (b), (d)

(a) | their momenta (magnitude) 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 \(A_2\). |

Choose the correct option:

1. (b), (c)

2. (a), (c)

3. (c), (d)

4. (b), (d)

Subtopic: De-broglie Wavelength |

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Relativistic corrections become necessary when the expression for the kinetic energy \(\dfrac{1}{2} mv^{2}\), 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) | \(\lambda = 10~\text{nm}\) | (b) | \(\lambda = 10^{-1}~\text{nm}\) |

(c) | \(\lambda = 10^{- 4}~\text{nm}\) | (d) | \(\lambda = 10^{- 6}~\text{nm}\) |

Choose the correct option:

1. (a), (c)

2. (a), (d)

3. (c), (d)

4. (a), (b)

Subtopic: De-broglie Wavelength |

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An electron (mass \(m\)) with an initial velocity \(\overset{\rightarrow}{v} = v_{0} \hat{i}\) is in an electric field \(\overset{\rightarrow}{E} = E_{0} \hat{j}\). If \(\lambda_{0} = \dfrac{h}{ {mv}_0}\), its de-Broglie wavelength at time \(t\) is given by:

1. \(\lambda_0\)

2. \(\lambda_{0} \sqrt{1 + \dfrac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}\)

3. \(\dfrac{\lambda_{0}}{\sqrt{1 + \dfrac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}\)

4. \(\dfrac{\lambda_{0}}{\left(1 + \dfrac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}\right)}\)

Subtopic: De-broglie Wavelength |

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An electron (mass \(m\)) with an initial velocity \(\vec{v}={v}_0 \hat{i}\) $\stackrel{}{\mathrm{}}$\(({v}_0>0)\) is in an electric field \(\vec{E}=-{E}_0 \hat{i}\)$({\mathrm{}}_{}$\(E_0\) = constant \(>0\)) . Its de-Broglie wavelength at time \(t\) is given by:

1. | \(\dfrac{\lambda_0}{\left(1+\dfrac{e E_0}{m} \dfrac{t}{{v}_0}\right)}\) | 2. | \(\lambda_0\left(1+\dfrac{e E_0 t}{m {v}_0}\right)\) |

3. | \(\lambda_0 \) | 4. | \(\lambda_0t\) |

Subtopic: De-broglie Wavelength |

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