Question

Find the relation between the wavelength of proton and electron, if both particles have the same kinetic energy.

Hint:

When comparing the de Broglie wavelengths of two particles with the same kinetic energy, use the mass ratio to determine the relationship between their wavelengths.

Answer 1

The de Broglie wavelength \( \lambda \) of a particle is given by the equation: \[ \lambda = \frac{h}{p} \] where \( h \) is Planck's constant and \( p \) is the momentum of the particle. The kinetic energy \( KE \) of a particle is related to its momentum \( p \) as: \[ KE = \frac{p^2}{2m} \] where \( m \) is the mass of the particle. Rearranging the equation for \( p \), we get: \[ p = \sqrt{2mKE} \] Since the kinetic energy is the same for both the proton and the electron, we can write the wavelengths for the proton and electron as: \[ \lambda_e = \frac{h}{\sqrt{2m_e KE}} \quad \text{and} \quad \lambda_p = \frac{h}{\sqrt{2m_p KE}} \] where \( m_e \) and \( m_p \) are the masses of the electron and proton, respectively. The ratio of the wavelengths is: \[ \frac{\lambda_p}{\lambda_e} = \frac{\sqrt{m_e}}{\sqrt{m_p}} \] Since \( m_p \) (mass of proton) is approximately 1831 times greater than \( m_e \) (mass of electron), we have: \[ \frac{\lambda_p}{\lambda_e} = \sqrt{1831} \] Thus, the relation between the wavelengths is: \[ \lambda_p = \sqrt{1831} \, \lambda_e \]