A particle of mass \( m \) and charge \( q \) is fastened to one end \( A \) of a massless string having equilibrium length \( l \), whose other end is fixed at point \( O \). The whole system is placed on a frictionless horizontal plane and is initially at rest. If a uniform electric field is switched on along the direction as shown in the figure, then the speed of the particle when it crosses the x-axis is:
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In problems involving electric fields, apply the work-energy principle to calculate the kinetic energy gained by the charged particle.
The particle is subjected to an electric field. The work-energy principle will be applied to determine its velocity upon crossing the x-axis. Step 1: The electric force on the particle is calculated as: \[ F_{\text{electric}} = qE \], where \( E \) represents the electric field. Step 2: The work performed by this force as the particle traverses a distance \( l \) along the x-axis is: \[ W = F_{\text{electric}} \times l = qEl \]. Step 3: The particle's kinetic energy gain equals the work done: \[ K = \frac{1}{2} m v^2 \]. By equating work and kinetic energy: \[ qEl = \frac{1}{2} m v^2 \]. Step 4: Solving for \( v \): \[ v = \sqrt{\frac{2qEI}{m}} \]. Final Conclusion: The particle's speed at the x-axis crossing is \( \sqrt{\frac{2qEI}{m}} \), corresponding to Option (2).
