A proton and an alpha particle are accelerated from rest through different potential differences such that they attain the same de Broglie wavelength. If the accelerating potential of the proton is V_ p and that of the alpha particle is V_ , then the ratio V_ p V_ is ________.

Step-by-step solution

The de Broglie wavelength of a particle of mass m and charge q accelerated through a potential difference V is given by: = h 2mqV Since both particles have the same de Broglie wavelength, we can equate their expressions: h 2m_ p q_ p V_ p = h 2m_ q_ V_ Squaring both sides and rearranging gives: m_ p q_ p V_ p = m_ q_ V_ For a proton, mass is m_ p and charge is q_ p = e . For an alpha particle, mass is m_ = 4m_ p and charge is q_ = 2e . Substituting these values into the equation: m_ p (e)V_ p = (4m_ p )(2e)V_ V_ p