A proton and an alpha particle enter a uniform magnetic field, moving perpendicular to the magnetic field lines. If the ratio of the radii of their circular paths is 1:3 , the ratio of the kinetic energy of the proton to that of the alpha particle is:

Step-by-step solution

Let the mass and charge of a proton be m and e respectively. Then for an alpha particle, the mass is 4m and the charge is 2e . The radius of the circular path of a charged particle in a magnetic field is given by r = 2mK qB , where K is the kinetic energy. Squaring both sides and rearranging for K , we get: K = q^2 B^2 r^2 2m Since both particles are in the same magnetic field B , we have K q^2 r^2 m . Taking the ratio of kinetic energies: K_p K_ = ( q_p q_ )^2 ( r_p r_ )^2 ( m_ m_p ) Substituting the given values: