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Chapter 17: Problem 2439

If \(\propto\) -particle and proton have same velocities, the ratio of de Broglie wavelength of \(\propto\) -particle and proton is \(\ldots \ldots\) (A) \((1 / 4)\) (B) \((1 / 2)\) (C) 1 (D) 2

Short Answer

Expert verified
The ratio of de Broglie wavelength of 𝛼-particle and proton when they have the same velocities is \(\frac{1}{4}\).

Step by step solution

01

Find the de Broglie wavelength formula

The de Broglie wavelength formula states that the wavelength (λ) is related to the momentum (p) of a particle using Planck’s constant (h) as follows: \[λ = \frac{h}{p}\]
02

Express momentum in terms of mass and velocity

We know that the momentum (p) can be expressed as the product of the mass (m) and velocity (v) of a particle: \[p = m \cdot v\]
03

Substitute momentum expression into de Broglie wavelength formula

Using our expression for momentum, we can rewrite the de Broglie wavelength formula as follows: \[λ = \frac{h}{m \cdot v}\]
04

Set up the ratio for the wavelengths of 𝛼-particle and proton

Let λ_α and λ_p be the de Broglie wavelengths of the 𝛼-particle and proton, respectively. We can set up a ratio as follows: \[\frac{λ_α}{λ_p} = \frac{\frac{h}{m_α \cdot v}}{\frac{h}{m_p \cdot v}}\]
05

Simplify the ratio

Since the 𝛼-particle and proton have the same velocity, we can simplify the ratio by canceling out the "h" and "v" terms and solving for the mass ratio: \[\frac{λ_α}{λ_p} = \frac{m_p}{m_α} \]
06

Use given masses for 𝛼-particle and proton to find the ratio

Given that the mass of 𝛼-particle (m_α) is 4 times the mass of a proton (m_p), we can substitute this information into the ratio formula: \[\frac{λ_α}{λ_p} = \frac{m_p}{4 \cdot m_p}\]
07

Calculate the final ratio

By canceling out m_p, we are left with the final ratio of the de Broglie wavelengths of 𝛼-particle and proton: \[\frac{λ_α}{λ_p} = \frac{1}{4}\] Thus, the correct answer is: (A) \((\frac{1}{4})\)

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Most popular questions from this chapter

A proton and an \(\propto\) -particle are passed through same potential difference. If their initial velocity is zero, the ratio of their de Broglie's wavelength after getting accelerated is \(\ldots \ldots\) (A) \(1: 1\) (B) \(1: 2\) (C) \(2: 1\) (D) \(2 \sqrt{2}: 1\)

Ration of momentum of photons having wavelength \(4000 \AA \& 8000 \AA\) is ........... (A) \(2: 1\) (B) \(1: 2\) (C) \(20: 1\) (D) \(1: 20\)

The cathode of a photoelectric cell is changed such that the work function changes from \(\mathrm{W}_{1}\) to \(\mathrm{W}_{2}\left(\mathrm{~W}_{2}>\mathrm{W}_{1}\right)\). If the currents before and after change are \(\mathrm{I}_{1}\) and \(\mathrm{I}_{2}\), all other conditions remaining unchanged, then assuming $\mathrm{hf}>\mathrm{W}_{2} \ldots \ldots$ (A) \(\mathrm{I}_{1}=\mathrm{I}_{2}\) (B) \(I_{1}\mathrm{I}_{2}\) (D) \(\mathrm{I}_{1}<\mathrm{I}_{2}<2 \mathrm{I}_{1}\)

Light of \(4560 \AA 1 \mathrm{~mW}\) is incident on photo-sensitive surface of \(\mathrm{Cs}\) (Cesium). If the quantum efficiency of the surface is \(0.5 \%\) what is the amount of photoelectric current produced? (A) \(1.84 \mathrm{~mA}\) (B) \(4.18 \mu \mathrm{A}\) (C) \(4.18 \mathrm{~mA}\) (D) \(1.84 \mu \mathrm{A}\)

The mass of a particle is 400 times than that of an electron and charge is double. The particle is accelerated by \(5 \mathrm{~V}\). Initially the particle remained at rest, then its final kinetic energy is \(\ldots \ldots \ldots\) (A) \(5 \mathrm{eV}\) (B) \(10 \mathrm{eV}\) (C) \(100 \mathrm{eV}\) (D) \(2000 \mathrm{eV}\)
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