Question

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

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

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: \[λ = \frac{h}{p}\]

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: \[p = m \cdot v\]

Substitute momentum expression into de Broglie wavelength formula

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

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: \[\frac{λ_α}{λ_p} = \frac{\frac{h}{m_α \cdot v}}{\frac{h}{m_p \cdot v}}\]

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: \[\frac{λ_α}{λ_p} = \frac{m_p}{m_α} \]

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: \[\frac{λ_α}{λ_p} = \frac{m_p}{4 \cdot m_p}\]

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) \((\frac{1}{4})\)