(Solved): A proton, \( (q=+e) \) and an alpha particle \( (q=+2 e) \) are used in a mass spectrometer. The p ...

A proton, \( (q=+e) \) and an alpha particle \( (q=+2 e) \) are used in a mass spectrometer. The proton has mass \( m \), and the alpha particle has mass \( 4 m \). Both are sent through the same magnetic field \( B \), which is perpendicular to the velocities of both particles so that they both have uniform circular motion in the field. See figure. If both particles enter the magnetic field with the same kinetic energy, find 1) \( \frac{v_{p}}{v_{\alpha}} \), where \( v_{p} \) and \( v_{\alpha} \) are the speeds of the proton and alpha particle, 2) \( \frac{r_{p}}{r_{\alpha}} \), where \( r_{p} \) and \( r_{\alpha} \) are the radii of the proton and alpha particle, and 3) \( \frac{T_{p}}{T_{\alpha}} \), where \( T_{p} \) and \( T_{\alpha} \) are the periods of the proton and alpha particle. The knowns are \( m, e \) and \( B \). (15 points)