Question

The mass of particle X is four times the mass of particle Y. The velocity of particle Y is four times the velocity of X. The ratio of de Broglie wavelengths of X and Y is: 

• A. 1:5
• B. 1:1
• C. 1:3
• D. 1:2

Hint:

In calculating de Broglie wavelength ratios, remember that changes in mass and velocity can offset each other, leading to equal wavelengths even under differing conditions.

Answer 1

The Correct Option is B

The de Broglie wavelength \(\lambda\) for a particle is given by the equation:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is the Planck constant, \( m \) is the mass, and \( v \) is the velocity of the particle.
Given that:
- The mass of particle X (\( m_X \)) is four times the mass of particle Y (\( m_Y \)), \( m_X = 4m_Y \).
- The velocity of particle Y (\( v_Y \)) is four times the velocity of X (\( v_X \)), \( v_Y = 4v_X \).
The de Broglie wavelength of X (\( \lambda_X \)) and Y (\( \lambda_Y \)) can be calculated as follows:
\[ \lambda_X = \frac{h}{m_Xv_X} = \frac{h}{4m_Yv_X} \]
\[ \lambda_Y = \frac{h}{m_Yv_Y} = \frac{h}{m_Y \cdot 4v_X} \]
Thus, the ratio of their wavelengths is:
\[ \frac{\lambda_X}{\lambda_Y} = \frac{\frac{h}{4m_Yv_X}}{\frac{h}{4m_Yv_X}} = 1:1 \]