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 of an electric field $\vec{E}\left(t\right) = E_{0}$ sin $\omega t \hat{i}$, where $E_{0} = \,1.0\, N C^{-1}$ 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

$n=10^{-3} kg\, q = 1 \,Ct =0$

$E=E_{0} \sin \omega t$
Force on particle will be $ F=q E$
$=q E_{0} \sin \omega t$
At $V_{m a x} a, F=0 $
$ q E_{0} \sin \omega t=0 $
$F=q E_{0} \sin \omega t$
$\frac{d v}{d t}=q \frac{E_{0}}{m} \sin \omega t $
$\int\limits_{0}^{v} d v=\int\limits_{0}^{\pi / \omega} \frac{q E_{0}}{m} \sin \omega t d t$
$V-0=\frac{q E_{0}}{m \omega}[-\cos \omega t]_{0}^{\frac{\pi}{\omega}}$
$ V-0=\frac{q E_{0}}{m \omega}[(-\cos \pi)-(-\cos 0)]$
$V=\frac{1 \times 1}{10^{-3} 10^{3}} \times 2=$
$2 \,m / s =2.00$