Question

A charged particle (charge $q$, mass $m$) is placed at the center of a circular region containing a magnetic field. The field is given as $\overrightarrow{B}=\overrightarrow{B}\left(r\right)$ i.e. it is a function of radial distance and it is perpendicular to the plane of the circle. The total magnetic flux through the circle is zero. The particle is given a radial impulse. At a later instant the radial and tangential components of velocity of the particle are $v_r$ and $v_o$ respectively. Which of the following statements is/are correct?

• A. The tangential force on the particle at an instant is $q{v}_{o}B$
• B. The torque about the center on the particle is $qBr{v}_r$
• C. If the particle leaves the circular region then at the instant it leaves, its velocity is only in the radial direction
• D. If the particle leaves the circular region, then at the instant it leaves, its velocity is only in tangential direction

Answer 1

Answer: (B), (C)

The correct options are
B The torque about the center on the particle is $qBr{v}_r$
C If the particle leaves the circular region then at the instant it leaves, its velocity is only in the radial direction

$\varphi =0\Rightarrow \int B2\pi rdr=0\Rightarrow \int rBdr=0.....\left(1\right)$
Tangential Force, $F_T=qB{v}_r$ as the force should be perpendicular to the velocity.
Hence the torque about the center = $F_{T}r=rqB{v}_r$
Angular momentum $\Delta L=\int F_{T}dt=q\int rB{v}_{r}dt=q\int rBdr=0\Rightarrow v_o=0$ i.e the final tangential velocity is same as initial tangential velocity which is zero.