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)
1. (a), (c)
2. (a), (d)
3. (c), (d)
4. (a), (b)
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. \(\dfrac{\lambda_{0}}{\left(1 + \dfrac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}\right)}\)
| 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. | \(\lambda_0t\) |
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, its de-Broglie wavelength:
1. remains constant
2. increases with time
3. decreases with time
4. increases and decreases periodically
A proton, a neutron, an electron and an \(\alpha\text-\)particle have the same energy. Then, their de-Broglie wavelengths compare as:
1. \(\lambda_p= \lambda_n>\lambda_e>\lambda_\alpha\)
2. \(\lambda_\alpha <\lambda_p = \lambda_n<\lambda_e\)
3. \(\lambda_e<\lambda_p=\lambda_n>\lambda_\alpha\)
4. \(\lambda_e =\lambda_p = \lambda_n=\lambda_\alpha\)
| (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)
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. \(\dfrac{P_e}{m_ec}=10^{-4}\)
| (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)
| (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)
Consider the figure given below. Suppose the voltage applied to A is increased. The diffracted beam will have the maximum at a value of 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
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