Question

A uniform and constant magnetic field $B=\hat{z}B$ exists in the $\hat{z}$ direction in vacuum. A particle of mass m with a small charge q is introduced into this region with an initial velocity $v=\hat{x}{v}_x+\hat{z}{v}_z$. Given that B, m, q, $v_{x}and{v}_z$ are all non-zero. Which one of the following describes the eventual trajectory of the particle?

• A. Helical motion in the $\hat{z}$ direction.
• B. Circular motion in the xy plane
• C. Linear motion in the $\hat{z}$ direction
• D. Linear motion in the $\hat{x}$ direction.

Answer 1

The correct option is A Helical motion in the $\hat{z}$ direction.

Ans : (a)

$B{a}_z$ = magnetic field
$v_x{a}_x+v_z{a}_z=velocity$
F = Q(v×B) by Lorentz's law
= $Q\left(\right(v_x{a}_x+v_z{a}_z)\times B{a}_z)$

$F_y=Q{V}_x.B(-a_y)$

This results in a circular path in the XY plane with $v_z{a}_z$ component causing a linear path.
Both result in a helical path along Z axis