Question

A particle of mass ${10}^{-3}$ kg and charge $1.0$ C, is initially at rest. At time $t=0$, the particle comes under the influence an electric field $\overrightarrow{E}\left(t\right)=E_{0}sin\left(\omega t\right)\hat{i}$ where $E_0=1.0N{C}^{-1}$ and $\omega ={10}^{3}rad{s}^{-1}$. Consider the effect of only the electrical force on the particle. Then the maximum speed, in $m{s}^{-1}$, attained by the particle at subsequent times is __________.

Answer 1

$n={10}^{-3}kg,q=1C,t=0$

E = $E_{0}sin\omega t$

Force on particle will be

$F=qE=q{E}_{0}sin\omega t$

At $v_{max},a,F=0$

$q{E}_{0}sin\omega t=0$

$F=q{E}_{0}sin\omega t$

$\frac{dv}{dt}=q\frac{E_0}{m}\sin \omega t$

${\int }_0^{v}dv={\int }_0^{\pi /\omega }\frac{q{E}_0}{m}sin\omega tdt$

$v-0=\frac{q{E}_0}{m\omega }[-cos\omega t{]}_0^{\frac{\pi }{\omega }}$

$v-0=\frac{q{E}_0}{m\omega }\left[\right(-cos\pi )–(cos0\left)\right]$

$v=\frac{1\times 1}{{10}^{-3}{10}^3}\times 2=2m/s$