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 moves under the influence of an electric field. We will use the work-energy principle to find its speed when it crosses the x-axis. Step 1: The electric force acting on the particle is given by: \[ F_{\text{electric}} = qE \] where \( E \) is the electric field. Step 2: The work done by this force in moving the particle a distance \( l \) along the x-axis is: \[ W = F_{\text{electric}} \times l = qEl \] Step 3: The kinetic energy gained by the particle is equal to the work done: \[ K = \frac{1}{2} m v^2 \] So, equating the work and kinetic energy: \[ qEl = \frac{1}{2} m v^2 \] Step 4: Solve for \( v \): \[ v = \sqrt{\frac{2qEI}{m}} \] Final Conclusion: The speed of the particle when it crosses the x-axis is \( \sqrt{\frac{2qEI}{m}} \), which is Option (2).
