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  4. Assertion A proton and an alpha particle having the same kinetic energy are moving in circular paths in a uniform magnetic field. The radii of their circular paths will be equal. Reason Any two charged particles having equal kinetic energies and entering a region of uniform magnetic field mathbf vecB in a direction perpendicular to mathbf vecB , will describe circular trajectories of equal radii.

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Q. Assertion A proton and an alpha particle having the same kinetic energy are moving in circular paths in a uniform magnetic field. The radii of their circular paths will be equal. Reason Any two charged particles having equal kinetic energies and entering a region of uniform magnetic field $ \mathbf{\vec{B}} $ in a direction perpendicular to $ \mathbf{\vec{B}} $ , will describe circular trajectories of equal radii.

VMMC MedicalVMMC Medical 2009

Solution:

The radius of the circular path is given by $ r=\frac{mv}{qB}=\frac{\sqrt{2mK}}{qB}; $ where $ K=\frac{1}{2}m{{v}^{2}} $ Since K and B are the same for the t particles, $ r\propto \frac{\sqrt{m}}{q}. $ Now, the charge of an alpha particle is twice that of a proton and its mass is four times the mass of a proton, $ \sqrt{m}/q $ will be the same for both particles. Hence, r will be same for both particles.