Question

A particle having charge \( +q \) enters a uniform magnetic field \( \vec{B} \) as shown in the figure. The particle will describe:

• A. a circular path in XZ plane
• B. a semicircular path in XY plane
• C. a helical path with its axis parallel to Y-axis
• D. a semicircular path in YZ plane

Hint:

Use the right-hand rule on \( \vec{v} \times \vec{B} \) to determine the direction of force on a positively charged particle. If \( \vec{v} \perp \vec{B} \), motion is circular in the plane defined by those two vectors.

Answer 1

The Correct Option is B

Observing the figure, the particle's initial velocity vector is oriented along the positive Y-axis. The magnetic field \( \vec{B} \) is directed perpendicularly into the plane of the paper, indicated by cross symbols.
The magnetic force acting on a charged particle in motion is quantified by the Lorentz force equation: \[\vec{F} = q\vec{v} \times \vec{B}\]- Given that \( \vec{v} \) is aligned with the \( +Y \) direction and \( \vec{B} \) is directed into the page (equivalent to the \( -Z \) direction). - Applying the right-hand rule to the cross product \( \vec{v} \times \vec{B} \), it is determined that the force \( \vec{F} \) exerts its influence along the \( +X \) direction. - This resultant force functions as the centripetal force, thereby inducing circular motion of the charged particle within the XY plane.
As the particle enters the field region perpendicularly and possesses no velocity component parallel to the magnetic field, its movement remains restricted to the plane of motion. Consequently, the particle traces a semicircular trajectory before potentially exiting the magnetic field's influence.
Final answer: Semicircular path in XY plane