A beam of alpha particles and of protons enter a uniform magnetics field B at right angles to the field lines. The particles describe circular path. Calculate the ratio of radii of their paths, when they enter the field with a) same momentum b) same kinetic energy

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Solution

$a) w h e n t h e y h a v e s a m e m o m e n t u m m_{a} v_{a} = m_{p} v_{p} \frac{m_{a}}{m_{p}} = \frac{v_{p}}{v_{a}} w h e r e m_{a}, m_{p} a r e t h e m a s s e s o f a l p h a a n d p r o t o n a n d v_{a} a n d v_{p} a r e t h e i r v e l o c i t y w e a l s o k n o w t h a t r = \frac{m v}{q B} t h e r e f o r e r_{a} = \frac{m_{a} v_{a}}{q_{a} B} . . . . . . . . 1 r_{p} = \frac{m_{p} v_{p}}{q_{p} B} . . . . . . . . . . . . . . . 2 o n d i v i d i n g e q u t i o n 1 b y 2 w e g e t \frac{r_{a}}{r_{p}} = \frac{m_{a}}{m_{p}} \frac{v_{a}}{v_{p}} \frac{q_{p}}{q_{a}} = \frac{v_{p}}{v_{a}} \frac{v_{a}}{v_{p}} \frac{q_{p}}{2 q_{p}} b e c a u s e q_{a} = 2 q_{p} \frac{r_{a}}{r_{p}} = \frac{1}{2} a n s . . . . p a r t b) w h e n t h e y h a v e s a m e k i n e t i c e n e r g y r = \frac{\sqrt{2 m K}}{q B} w h e r e K i s k i n e t i c e n e r g y r_{a} = \frac{\sqrt{2 m_{a} K}}{q_{a} B} . . . . . . . 1 r_{p} = \frac{\sqrt{2 m_{p} K}}{q_{p} B} . . . . . . . . 2 o n d i v i d i n g e q n o . 1 b y 2 w e t g e t \frac{r_{a}}{r_{p}} = \frac{q_{p}}{q_{a}} \sqrt{\frac{m_{a}}{m_{p}}} w e k n o w m_{a} = 4 m_{p} a n d q_{a} = 2 q_{p} \frac{r_{a}}{r_{p}} = \frac{q_{p}}{2 q_{p}} \sqrt{\frac{4 m_{p}}{m_{p}}} \frac{r_{a}}{r_{p}} = \frac{2}{2} = 1 a n s . . . .$

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