Question

A block of mass m and charge q is released from rest on incline plane of angle $\theta$ and coefficient of friction $\mu$. $\theta$ is greater than angle of repose. A magnetic field perpendicular to page inwards also exist as shown in figure, then

• A. Speed of block increases linearly.
• B. Maximum speed attained by block is $\frac{mg}{\mu qB}(sin\theta -\mu cos\theta )$
• C. Speed of block increases exponentially and reaches infinity
• D. Time to reach maximum speed is infinity

Answer 1

Answer: (B), (D)

Time to reach maximum speed is infinity


$F_{net}=mgsin\theta -\mu N$
$ma=mgsin\theta -\mu (mgcos\theta +qVB)$
$a=(gsin\theta -\mu gcos\theta )-\frac{\mu qBv}{m}\{\begin{array}{c}P=gsin\theta -\mu gcos\theta \\ r=\frac{\mu qB}{m}\end{array}$
$a=P-rv\Rightarrow \frac{dv}{dt}=P-rv$

${\int }_0^v\frac{dv}{P-rv}={\int }_0^{t}dt\Rightarrow -\frac{1}{r}ln[P-rv{]}_0^v=t$
$ln\frac{P-rv}{p}=-rt\Rightarrow 1-\frac{vr}{p}=e^{-rt}$
$\frac{P}{r}(1-e^{-rt})=v\Rightarrow v_{max}=\frac{P}{r}=\frac{gsin\theta -\mu gcos\theta }{\frac{\mu qB}{m}}att=\infty$