Question 259: Electrostats & Capacitance

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 $$\overrightarrow E \left( t \right) = {E_0}\sin \,\,$$ $$\omega t\widehat i,$$ 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 _______________.

Accepted Answer

Given, mass of particle = 10
$$-$$3
kg, charge on particle = 1.0 C, electric field $$\overrightarrow E (t) = {E_0}\sin \omega t\,\widehat i$$, E
0
= 1.0 N C
$$-$$1
, $$\omega$$ = 10
3
rad s
$$-$$1
Force on particle is given by
$$ \Rightarrow \overrightarrow F  = q{E_0}\sin \omega t\,\widehat i = 1.0 	imes 1.0 	imes \sin ({10^3}t)\widehat i$$
$$ \Rightarrow \overrightarrow F  = \sin ({10^3}t)\widehat i$$
We know that $$\overrightarrow F  = m\overrightarrow a  \Rightarrow \overrightarrow a  = {{\overrightarrow F } \over m}$$
$$ \Rightarrow a = {{\sin ({{10}^3}t)} \over {{{10}^{ - 3}}}} \Rightarrow a = {10^3}\sin ({10^3}t)$$
We know $$a = {{dv} \over {dt}}$$
$$ \Rightarrow {{dv} \over {dt}} = {10^3}\sin ({10^3}t)$$
$$ \Rightarrow dv = {10^3}\sin ({10^3}t)dt$$
Now, integrating it, we get
$$\int\limits_0^v {dv = \int\limits_0^t {{{10}^3}\sin ({{10}^3}t)dt} } $$
$$ \Rightarrow \left. v ight|_0^v = {10^3}\left( {{{ - \left. {\cos ({{10}^3}t)} ight|_0^t} \over {{{10}^3}}}} ight) \Rightarrow \left. v ight|_0^v = \left. { - \cos ({{10}^3}t)} ight|_0^t$$
$$ \Rightarrow v = ( - \cos {10^3}t + \cos 0)$$
$$ \Rightarrow v = (1 - \cos ({10^3}t))$$
We know that cos$$	heta$$ can take values between $$-$$1 and 1. Therefore, maximum speed attained by the particle when cos$$	heta$$ = $$-$$1. Thus,
$${v_{\max }} = 1 - ( - 1) = 2$$ m s
$$-$$1

Electrostats & Capacitance

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