If proton and electron are projected at right angle to...

If proton and electron are projected at right angle to uniform magnetic field with the same kinetic energy, then (a) the radius of the circular trajectory of proton $r_{p}$ is smaller than the radius of the circular trajectory of the electron $r_{e}$ (b) $r_{p}>r_{e}$ (c) $r_{P}=r_{e}$ (d) both the particles move in straight line

If proton and electron are projected at right angle to uniform magnetic field with the same kinetic energy, then
(a) the radius of the circular trajectory of proton $r_{p}$ is smaller than the radius of the circular trajectory of the electron $r_{e}$
(b) $r_{p}>r_{e}$
(c) $r_{P}=r_{e}$
(d) both the particles move in straight line

Key Concepts

Lorentz Force

The Lorentz force is the fundamental force experienced by a charged particle moving within a magnetic field. This force, defined by F = q(v × B), acts perpendicularly to both the velocity of the particle and the magnetic field, causing the particle to deviate from a straight-line path, typically moving in a circular trajectory when its velocity is perpendicular to the field.

Circular Motion in Magnetic Fields

When a charged particle is subjected to a uniform magnetic field with its velocity perpendicular to the field, it undergoes uniform circular motion. This happens because the magnetic force acts as a centripetal force, maintaining the circular path of the particle. The radius of this motion is determined by the balance between the magnetic force and the centripetal force required for circular motion.

Kinetic Energy and Velocity Relation

Kinetic energy, given by the formula KE = 1/2 mv², links a particle's mass and its velocity. For particles with equal kinetic energy but differing masses, the lighter particle will have a higher velocity. This relationship plays a crucial role in determining the radius of the circular trajectory when the particles enter a magnetic field.

Radius of Curvature

The radius of the circular trajectory of a charged particle in a uniform magnetic field is determined by the equation r = mv/(qB). This equation indicates that the radius is directly proportional to the particle's mass and velocity, and inversely proportional to its charge and the magnetic field strength, making the mass-to-charge ratio a critical factor in the motion.

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