A particle, of mass 10−3kg and charge 1.0C, is initially at rest. At time t=0, the particle comes under the influence of an electric field Et=E0sinωt i⌢, where E0=1.0N/C and ω=103rad/s. Consider the effect of only the electrical force on the particle. Then the maximum speed,(in m/s) , attained by the particle at subsequent times is .

m=10−3kg q=1C t=0 E=E0sin⁡ωt Force on particle will be F=qE=qE0sin⁡ωt At mean position, speed id Vmax, and, F=0⇒qE0sin⁡ωt=0⇒ωt=π⇒t=πω Now, F=qE0sin⁡ωt⇒dVdt=qE0msin⁡ωt⇒∫0v dv=∫0π/ω qE0msin⁡ωtdt⇒V−0=qE0mω[−cos⁡ωt]0π/ω⇒V−0=qE0mω[(−cos⁡π)−(−cos⁡0)]⇒V=1×110−3103×2=2m/s

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A particle, of mass $10^{- 3} k g$ and charge $1.0 C$ , is initially at rest. At time $t = 0$ , the particle comes under the influence of an electric field $E (t) = E_{0} \sin ω t \overset{⌢}{i}$ , where $E_{0} = 1.0 N / C$ and $ω = 10^{3} r a d / s$ . Consider the effect of only the electrical force on the particle. Then the maximum speed,(in m/s) , attained by the particle at subsequent times is .

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Detailed Solution

$m = 10^{- 3} k g q = 1 C t = 0$

$E = E_{0} \sin ω t$

$Force on particle will be F = q E = q E_{0} \sin ω t$

${At mean position, speed id V}_{m a x}, and, F = 0$

$\begin{array}{l} \Rightarrow q E_{0} \sin ω t = 0 \\ \Rightarrow ω t = π \\ \Rightarrow t = \frac{π}{ω} \\ Now, F = q E_{0} \sin ω t \\ \Rightarrow \frac{d V}{d t} = q \frac{E_{0}}{m} \sin ω t \\ \Rightarrow \int_{0}^{v} d v = \int_{0}^{π / ω} \frac{q E_{0}}{m} \sin ω t d t \\ \Rightarrow V - 0 = \frac{q E_{0}}{m ω} [- \cos ω t]_{0}^{π / ω} \\ \Rightarrow V - 0 = \frac{q E_{0}}{m ω} [(- \cos π) - (- \cos 0)] \\ \Rightarrow V = \frac{1 \times 1}{10^{- 3} 10^{3}} \times 2 = 2 m / s \end{array}$