A particle of mass and charge is fastened to one end of a massless string having equilibrium length , whose other end is fixed at point . 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 -axis is:
Correct answer:B
Work-Energy Method
Given: A particle of mass and charge is attached to a massless string of length on a frictionless horizontal plane. A uniform electric field acts along the positive -direction.
Find: The speed of the particle when it crosses the -axis.
The solution uses the work-energy principle.
The electric force on the particle is
When the particle crosses the -axis, the displacement along the field direction is taken as in the provided solution.
Hence, the work done by the electric field is
This work appears as kinetic energy:
Solving for ,
Therefore, the speed of the particle is , so the correct option is B.
Using the tension in the string to calculate work. Tension does no work here because it is always along the string, while the instantaneous displacement is perpendicular to it. Use only the work done by the electric force in the energy equation.
Substituting the wrong displacement in the work term. In a uniform electric field, work depends on displacement along the field direction, not on the curved path length. Use displacement along the -direction while applying .
Confusing force with potential energy change directly without writing the work-energy relation. First compute the work done by the electric field, then equate it to the change in kinetic energy.
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