Question

A charged particle (q, m) is projected along y-axis in a uniform magnetic field directed along z-axis. A viscous force $\overrightarrow{f}=-b\overrightarrow{v}$ also acts on particle(where $\overrightarrow{v}$ is the velocity of the particle).

• A. Path of the particle will be spiral inward.
• B. Particle will stop after travelling a distance of $\frac{m{v}_0}{b}$.
• C. Time required to travel the distance of $\frac{m{v}_0}{b}$ is $\frac{zm}{b}$.
• D. Velocity of the particle decreases linearly

Answer 1

Answer: (A), (B)

Particle will stop after travelling a distance of $\frac{m{v}_0}{b}$.

The magnitude of force on the particle can be given as:
$f=ma\Rightarrow -bv=mv\frac{dv}{dr}$
$\Rightarrow \frac{-b}{m}dr=dv$

Integrating:
${\int }_{v_0}^0=-\frac{b}{m}{\int }_0^{r_0}dr\Rightarrow v_0=\frac{b}{m}{r}_0$
$r_0=\frac{m{v}_0}{b}$
which implies that particle will stop after travelling this distance.

Also, $-bv=m\frac{dv}{dt}\Rightarrow {\int }_{v_0}^v\frac{dv}{v}=-\frac{b}{m}{\int }_0^{t}dt$
$ln\frac{v}{v_0}=-\frac{bt}{m}\Rightarrow v=v_0{e}^{\frac{-bt}{m}}$