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Chapter 17: Problem 2394

The de-Broglie wavelength associated with a particle with rest mass \(\mathrm{m}_{0}\) and moving with speed of light in vacuum is..... (A) \(\left(\mathrm{h} / \mathrm{m}_{0} \mathrm{c}\right)\) (B) 0 (C) \(\infty\) (D) \(\left(\mathrm{m}_{0} \mathrm{c} / \mathrm{h}\right)\)

Short Answer

Expert verified
The correct answer is option (C) \(\infty\) as the de-Broglie wavelength is undefined in this case.

Step by step solution

01

Recall the de-Broglie wavelength formula

The de-Broglie wavelength associated with any particle is given by \[ λ = \frac{h}{mv} \] Where \(h\) is Planck's constant, \(m\) is the mass of the particle, and \(v\) is the speed of the particle. Since the particle is moving with the speed of light, we'll use the relativistic mass formula, which takes into account the increase in mass as the particle's speed increases.
02

Recall the relativistic mass formula and apply it to the problem

The relativistic mass formula is given by \[ m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} \] Where \(m_0\) is the rest mass of the particle, \(v\) is the speed of the particle, and \(c\) is the speed of light in a vacuum. However, since the problem states that the particle is moving with the speed of light in a vacuum, this means that \(v = c\). Substituting this value into the relativistic mass formula, we obtain \[ m = \frac{m_0}{\sqrt{1-\frac{c^2}{c^2}}} \] Simplifying the denominator, we find that \[ m = \frac{m_0}{\sqrt{1-1}} = \frac{m_0}{0} \]
03

Conclude that the de-Broglie wavelength is undefined in this case

Since the mass of the particle is given by \(\frac{m_0}{0}\), the mass becomes undefined as the speed of the particle approaches the speed of light. This means there is an asymptotic relationship between the speed and the mass of the particle, making it impossible to ever reach the speed of light. So, the de-Broglie wavelength formula is also undefined in this situation, as the mass has become undefined. Thus, the correct answer is option (C) \(\infty\) as the de-Broglie wavelength is undefined in this case.

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Most popular questions from this chapter

What should be the ratio of de-Broglie wavelength of an atom of nitrogen gas at \(300 \mathrm{~K}\) and \(1000 \mathrm{~K}\). Mass of nitrogen atom is $4.7 \times 10^{-26} \mathrm{~kg}$ and it is at 1 atm pressure Consider it as an idecal gas. (A) \(2.861\) (B) \(8.216\) (C) \(6.281\) (D) \(1.826\)

Photons of energy \(1 \mathrm{eV}\) and \(2.5 \mathrm{ev}\) successively illuminate a metal, whose work function is \(0.5 \mathrm{eV}\), the ratio of maximum speed of emitted election is \(\ldots \ldots \ldots .\) (A) \(1: 2\) (B) \(2: 1\) (C) \(3: 1\) (D) \(1: 3\)

Kinetic energy of proton accelerated under p.d. \(1 \mathrm{~V}\) will be........ (A) \(1840 \mathrm{eV}\) (B) \(13.6 \mathrm{eV}\) (C) \(1 \mathrm{eV}\) (D) \(0.54 \mathrm{eV}\)

Each of the following questions contains statement given in two columns, which have to be matched. The answers to these questions have to be appropriately dubbed. If the correct matches are $\mathrm{A}-\mathrm{q}, \mathrm{s}, \mathrm{B}-\mathrm{p}, \mathrm{r}, \mathrm{C}-\mathrm{q}, \mathrm{s}$ and \(\mathrm{D}-\mathrm{s}\) then the correctly dabbled matrix will look like the one shown here: Match the statements of column I with that of column II. Column-I Column-II (A) \(\propto\) -particle and proton have same \(\mathrm{K} . \mathrm{E}\). \((\mathrm{p}) \lambda_{\mathrm{p}}=\lambda_{\propto}\) (B) \(\propto\) -particle has one quarter \(\mathrm{K} . \mathrm{E}<\) than that (q) \(\lambda_{\mathrm{p}}>\lambda_{\propto}\) (C) proton has one quarter \(\mathrm{K} . \mathrm{E}\). than that (r) \(p_{p}=p_{\propto}\) (D) \(\propto\) -particle and proton same velocity (s) \(\mathrm{p}_{\propto}>\mathrm{p}_{\mathrm{p}}\)

Wavelength \(\lambda_{\mathrm{A}}\) and \(\lambda_{\mathrm{B}}\) are incident on two identical metal plates and photo electrons are emitted. If \(\lambda_{\mathrm{A}}=2 \lambda_{\mathrm{B}}\), the maximum kinetic energy of photo electrons is \(\ldots \ldots \ldots\) (A) \(2 \mathrm{~K}_{\mathrm{A}}=\mathrm{K}_{\mathrm{B}}\) (B) \(\mathrm{K}_{\mathrm{A}}<\left(\mathrm{K}_{\mathrm{B}} / 2\right)\) (C) \(\mathrm{K}_{\mathrm{A}}=2 \mathrm{~K}_{\mathrm{B}}\) (D) \(\mathrm{K}_{\mathrm{A}}>\left(\mathrm{K}_{\mathrm{B}} / 2\right)\)
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