A deuteron of kinetic energy 50 keV is describing a circular orbit of radius 0.5 m in a plane p

step 1 In this problem I can write QVb is equal to mv square by r QBr is equal to mv solving it further

A deuteron of kinetic energy 50 keV is describing a circular...

A deuteron of kinetic energy $50 \mathrm{keV}$ is describing a circular orbit of radius $0.5 \mathrm{~m}$ in a plane perpendicular to the magnetic field $\vec{B}$. The kinetic energy of the proton that describes a circular orbit of radius $0.5 \mathrm{~m}$ in the same plane with the same field $\vec{B}$ is (a) $5 \mathrm{keV}$ (b) $10 \mathrm{keV}$ (c) $50 \mathrm{keV}$ (d) $100 \mathrm{keV}$

A deuteron of kinetic energy $50 \mathrm{keV}$ is describing a circular orbit of radius $0.5 \mathrm{~m}$ in a plane perpendicular to the magnetic field $\vec{B}$. The kinetic energy of the proton that describes a circular orbit of radius $0.5 \mathrm{~m}$ in the same plane with the same field $\vec{B}$ is
(a) $5 \mathrm{keV}$
(b) $10 \mathrm{keV}$
(c) $50 \mathrm{keV}$
(d) $100 \mathrm{keV}$

Key Concepts

Charged Particle Motion in Magnetic Fields

Charged particles, when moving through a magnetic field, experience a force perpendicular to their velocity and the field, which causes them to move in circular or helical paths. This phenomenon is a fundamental concept in electromagnetism and is integral to understanding how particles such as ions and electrons behave in fields found in devices like cyclotrons and magnetic confinement systems.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving in an electromagnetic field, given by the vector equation F = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. This force is responsible for the circular motion observed in charged particles within a magnetic field, as it provides the necessary centripetal force for their curved trajectory.

Cyclotron Motion and Frequency

Cyclotron motion refers to the circular movement of charged particles in a uniform magnetic field. The cyclotron frequency, which is independent of the particle’s speed and radius, is defined as ? = qB/m, where m is the mass of the particle. This relationship is essential in analyzing the motion, as it connects the magnetic field, the charge, and the mass to the particle’s rotational dynamics.

Mass-Charge Ratio

The mass-to-charge ratio (m/q) plays a critical role in determining the behavior of a charged particle in a magnetic field. Different particles, even with the same radius of orbit and magnetic field strength, will have different kinetic energies due to variations in their masses and charges. Understanding this ratio is crucial for predicting how particles like protons and deuterons move and for designing experiments and devices that rely on magnetic confinement and particle manipulation.

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