# A particle and a proton of the same kinetic energy are in turn allowed to pass through a magnetic field →B, acting normal to the direction of motion of the particles. Calculate the ratio of radii of the circular paths described by them.

The force on a particle of charge and velocity making and angle with magnetic field is given by: The magnitude of the force when the magnetic field is normal to velocity is given by:  As the force is perpendicular to the velocity, (resultant of cross product), it can only change the direction and not the speed of the particle. Hence, it will move in a circle. Let r be the radius of this circular path. The centripetal force is given by: Equating (1) and (2) , we get We know that the kinetic energy is given by:  Substituting in (3), we have  Now,

The ratio of mass of alpha particle to the mass of proton is 4:1.

The ratio of the charge on alpha particle to the charge on proton is 2:1.

Taking the ration of the radii, we have   The ratio of the radii of is 1.

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