A positively charged (+q) particle of mass m has kinetic...

A positively charged $(+q)$ particle of mass $m$ has kinetic energy $K$ enters vertically downward in a horizontal field of magnetic induction $\vec{B} .$ The acceleration of the particle is (a) $q B \sqrt{\frac{2 K}{m}}$ (b) $\frac{q B \sqrt{2 K}}{(m)^{3 / 2}}$ (c) $\frac{2 q B}{(m)^{3 / 2}} \sqrt{2 K}$ (d) $2 q B \sqrt{\frac{2 K}{m}}$

A positively charged $(+q)$ particle of mass $m$ has kinetic energy $K$ enters vertically downward in a horizontal field of magnetic induction $\vec{B} .$ The acceleration of the particle is
(a) $q B \sqrt{\frac{2 K}{m}}$
(b) $\frac{q B \sqrt{2 K}}{(m)^{3 / 2}}$
(c) $\frac{2 q B}{(m)^{3 / 2}} \sqrt{2 K}$
(d) $2 q B \sqrt{\frac{2 K}{m}}$

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in an electromagnetic field. In a magnetic field, the force is given by the vector cross product F = q(v × B), which means the force is perpendicular to both the particle’s velocity and the magnetic induction field. This principle is fundamental in understanding the motion of charged particles in magnetic fields, particularly when they move perpendicularly to the field lines.

Kinetic Energy Relation

Kinetic energy is related to the speed of an object through the equation K = ½ mv², where m is the mass and v is the speed of the particle. This relationship allows one to express the speed in terms of kinetic energy, which is crucial when analyzing the dynamics of a particle entering a magnetic field with a given kinetic energy.

Circular Motion in Magnetic Fields

When a charged particle enters a magnetic field with its velocity perpendicular to the field, the Lorentz force acts as a centripetal force, causing the particle to move in a circular path. This concept explains how the magnetic force does not work on the particle but continuously changes the direction of its velocity, leading to curved motion without changing the magnitude of the velocity.

Centripetal Acceleration

Centripetal acceleration is the acceleration required to keep an object moving in a circular path, given by a = v²/r, where v is the velocity and r is the radius of the circle. In the context of a charged particle in a magnetic field, the magnetic Lorentz force provides this centripetal acceleration, and by relating the force to the particle’s motion, one can derive expressions for the acceleration in terms of charge, magnetic field strength, kinetic energy, and mass.

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