A particle, of mass and charge , is initially at rest. At time , the particle comes under the influence of an electric field where and . Consider the effect of only the electrical force on the particle. Then the maximum speed, in , attained by the part

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 $\vec{E} (t) = E_{0} \sin (ω t) \hat{i},$ where $E_{0} = 1.0 {NC}^{- 1}$ and $ω = 10^{3} {rad s}^{- 1}$ . Consider the effect of only the electrical force on the particle. Then the maximum speed, in ${ms}^{- 1}$ , attained by the particle at subsequent times is _______. [2018]

Smart Achievers · Accepted Answer

(2)

$∵$ $F = m a$ $∴$ $q E = m \frac{d v}{d t}$

$\Rightarrow d v = \frac{q E d t}{m} = \frac{q \sin (1000 t) \hat{i}}{m} d t$

$(∵ E = \sin (10^{3} t) \hat{i} is given)$

$∴$ $\int_{0}^{v} d v = \frac{q}{m} \int_{0}^{π / ω} \sin (1000 t) d t$ $[Maximum speed is attained at \frac{T}{2} = \frac{2 π}{ω \times 2}]$

$∴$ $v = - \frac{q}{m} {[\frac{\cos (1000 t)}{1000}]}_{0}^{π / ω}$ $= - \frac{1}{10^{- 3}} {[\frac{\cos (1000 t)}{1000}]}_{0}^{π / ω}$

$(∵ m = 10^{- 3} kg, q = 1 C, E_{0} = 1 {N C}^{- 1}, ω = 10^{3} {rad s}^{- 1})$ given

$∴$ $v = - [\cos (1000 \times \frac{π}{1000}) - \cos 0]$ $= - [- 1 - 1] = 2 {ms}^{- 1}$

Hence, the maximum speed attained by the particle.

Solution

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