Question
Please be very clear and accurate with the solution.
Thank you so much.
A proton of mass mp and charge qp =e deuteron of mass 2mp and charge qd =e and alpha particle of mass 4mp and charge qa =2e moving in the plane xy , are accelerated by the same potential difference V and then enter an uniform magnetic field B along z axis. Find the velocities and derive the radii of the paths for deuteron and alpha particle in term of the mass and charge of proton. Note : No credit will he given if von use formulas for the velocities or radius. You will get credit only for derivations of velocities and radii, not for substitution
Explanation / Answer
Given mass of proton is m p charge on proton is qp = e mass of deutron is m d = 2 m p charge on deutron is q d = e mass of alphe particle m a = 4 m p charge on alpha particle is q a = 2e three particles are accelerated by same potential difference V magnitude of magnetic field is B 1) kinetic energy of deutron is K.E = Ve ( 1/2) (2m p ) v p2 = V e velocity of deutron is v d = V e / m p force on deutron due to magnetic field is F = B q v md vd2 / r d = B e vd radius of path of deutron is r d = 2 mp v d / B e = ( 2 m p / B e ) ( V e / m p ) = ( 2 / B ) ( V mp / e ) 2) kinetic energy of alpha particle is is K.E = Ve ( 1/2) (4m p ) v p2 = V(2 e ) velocity of alpha particle is v a = V e / m p force on alpha particle due to magnetic field is F = B q v ma va 2 / r a = B e va radius of path of deutron is r a = 4 mp v a / B( 2 e) = ( 2 m p / B e ) ( V e / m p ) = ( 2 / B ) ( V mp / e ) mass of deutron is m d = 2 m p charge on deutron is q d = e mass of alphe particle m a = 4 m p charge on alpha particle is q a = 2e three particles are accelerated by same potential difference V magnitude of magnetic field is B 1) kinetic energy of deutron is K.E = Ve ( 1/2) (2m p ) v p2 = V e velocity of deutron is v d = V e / m p force on deutron due to magnetic field is F = B q v md vd2 / r d = B e vd radius of path of deutron is r d = 2 mp v d / B e = ( 2 m p / B e ) ( V e / m p ) = ( 2 / B ) ( V mp / e ) 2) kinetic energy of alpha particle is is K.E = Ve ( 1/2) (4m p ) v p2 = V(2 e ) velocity of alpha particle is v a = V e / m p force on alpha particle due to magnetic field is F = B q v ma va 2 / r a = B e va radius of path of deutron is r a = 4 mp v a / B( 2 e) = ( 2 m p / B e ) ( V e / m p ) = ( 2 / B ) ( V mp / e ) K.E = Ve ( 1/2) (4m p ) v p2 = V(2 e ) velocity of alpha particle is v a = V e / m p force on alpha particle due to magnetic field is F = B q v ma va 2 / r a = B e va radius of path of deutron is r a = 4 mp v a / B( 2 e) = ( 2 m p / B e ) ( V e / m p ) = ( 2 / B ) ( V mp / e )