A neutron of K. E. 50 ev is describing a circular orbit of radius 0.5 m in a plane perpendicula

step 1 Hello students. In this question we have a deutron particle which is having the kinetic energy K

A neutron of K. E. 50 ev is describing a circular orbit of...

A neutron of $K$. E. 50 ev is describing a circular orbit of radius $0.5 \mathrm{~m}$ in a plane perpendicular to magnetic field $\mathrm{B}^{-}$. The $\mathrm{K} . \mathrm{E}$. of the proton that describe a circular orbit of radius $0.5 \mathrm{~m}$ in the same plane with the same $\mathrm{B}^{-}$ is (a) $200 \mathrm{ev}$ (b) $100 \mathrm{ev}$ (c) $50 \mathrm{ev}$ (d) $25 \mathrm{ev}$

A neutron of $K$. E. 50 ev is describing a circular orbit of radius $0.5 \mathrm{~m}$ in a plane perpendicular to magnetic field $\mathrm{B}^{-}$. The $\mathrm{K} . \mathrm{E}$. of the proton that describe a circular orbit of radius
$0.5 \mathrm{~m}$ in the same plane with the same $\mathrm{B}^{-}$ is
(a) $200 \mathrm{ev}$
(b) $100 \mathrm{ev}$
(c) $50 \mathrm{ev}$
(d) $25 \mathrm{ev}$

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field. It is given by F = q(v × B), where q is the charge, v is the particle’s velocity, and B is the magnetic field. This force is always perpendicular to the velocity, causing the particle to undergo circular motion if the magnetic field is uniform.

Cyclotron Motion

Cyclotron motion refers to the circular motion of a charged particle in a uniform magnetic field. Because the magnetic force is perpendicular to the velocity, it provides the necessary centripetal force for circular motion. The radius of the circle is determined by the balance between the magnetic Lorentz force and the centripetal force required to keep the particle in a circular path.

Charge-to-Mass Ratio

The charge-to-mass ratio (q/m) of a particle plays a critical role in determining the radius of its circular path in a magnetic field. For a given magnetic field strength and orbit radius, the particle’s velocity and kinetic energy depend on how much charge it carries relative to its mass. A higher charge-to-mass ratio results in a tighter circular orbit for the same conditions.

Kinetic Energy in a Magnetic Field

For a charged particle in cyclotron motion, the kinetic energy can be related to the magnetic field, charge, mass, and radius of the orbit. Specifically, the kinetic energy is proportional to (q²B²r²)/m. This shows that for a fixed magnetic field and orbital radius, the kinetic energy is inversely proportional to the mass of the particle, meaning that lighter particles require more kinetic energy to achieve the same radius in the magnetic field.

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