A proton and an alpha particle are accelerated from rest through potential differences V_p and V_ respectively. They then enter a region of uniform magnetic field, moving perpendicular to the field direction. If both particles describe circular paths of the exact same radius, the ratio V_p V_ is :

Step-by-step solution

The kinetic energy gained by a particle of charge q accelerated through a potential difference V is K = qV . The radius R of the circular path in a uniform magnetic field B is given by: R = p qB = 2mK qB = 2mqV qB = 1 B 2mV q Since both particles describe paths of the same radius in the same magnetic field, we have: m_p V_p q_p = m_ V_ q_ Rearranging for the ratio of potentials: V_p V_ = ( m_ m_p ) ( q_p q_ ) For a proton and an alpha particle, m_ = 4m_p and q_ = 2q_p . Substituting these values: V_p V_ = (4) ( 1 2