Kinematics 6 Question 10
10. 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 $E(t)=E _0 \sin \omega t \hat{i}$, where $E _0=1.0 NC^{-1}$ 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 …….. .
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Answer:
Correct Answer: 10. (2)
Solution:
- $a=\frac{F}{m}=\frac{q E}{m}=10^{3} \sin \left(10^{3} t\right)$
$$ \frac{d v}{d t}=10^{3} \sin \left(10^{3} t\right) \Rightarrow \int _0^{v} d v=\int _0^{t} 10^{3} \sin \left(10^{3} t\right) d t $$
$$ \therefore \quad v=\frac{10^{3}}{10^{3}}\left[1-\cos \left(10^{3} t\right)\right] $$
Velocity will be maximum when $\cos \left(10^{3} t\right)=-1$
$$ v _{\max }=2 m / s $$
