A proton with a speed u along the positive x-axis at y=0...

A proton with a speed $u$ along the positive $x$-axis at $y=0$ enters a region of uniform magnetic field $B=-B_{0} \hat{k}$ which exists to the right of $y$-axis. The proton exits from the region after some time with the speed $v$ at ordinate $y$, then (A) $v>u, y<0$ (B) $v=u, y>0$ (C) $v>u, y>0$ (D) $v=u, y<0$

A proton with a speed $u$ along the positive $x$-axis at $y=0$ enters a region of uniform magnetic field $B=-B_{0} \hat{k}$ which exists to the right of $y$-axis. The proton exits from the region after some time with the speed $v$ at ordinate $y$, then
(A) $v>u, y<0$
(B) $v=u, y>0$
(C) $v>u, y>0$
(D) $v=u, y<0$

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 x B), where q is the charge, v is the velocity vector, and B is the magnetic field vector. This force is always perpendicular to both the velocity and the magnetic field, which determines the direction of the particle's acceleration.

Magnetic Force Does No Work

Since the magnetic force is always perpendicular to the velocity of the particle, it does no work on the particle. This means that the kinetic energy and therefore the speed of the particle remain constant, irrespective of the change in direction of the motion.

Motion in a Uniform Magnetic Field

When a charged particle enters a uniform magnetic field, it experiences a constant magnitude force that continuously changes the direction of its velocity but not its magnitude. This results in circular or cycloidal motion depending on the entry conditions, as the force provides the necessary centripetal acceleration for curved motion.

Trajectory Deflection

The deflection of the trajectory of a charged particle in a magnetic field is determined by the right-hand rule applied to the cross product in the Lorentz force formula. This rule helps predict the direction in which the particle will be deflected relative to its initial motion and the orientation of the magnetic field.

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