Final answer:
The kinetic energy of a charged particle in the electric field E = E0(1 - ax²) becomes zero again at a distance √(1/a) from the origin, assuming the particle started from rest at x = 0 with no other forces acting on it. The correct answer is (3) √1/a.
Step-by-step explanation:
The question pertains to the behavior of a charged particle moving in a non-uniform electric field. The electric field is given by E = E0(1 – ax²) in the x-direction, where a and E0 are constants, and the initial position and velocity of the particle are known.
Using conservation of energy, one can determine the positions at which the kinetic energy of the particle becomes zero again.
To find when the kinetic energy becomes zero again, we set the initial potential energy equal to the final potential energy, as no other forces are doing work on the system, and the particle starts from rest. The initial potential energy is U = qE0 at x = 0, and the final potential energy at x is U = qE0(1 – ax²). Setting these equal, as no kinetic energy is lost or gained, we get:
qE0 = qE0(1 – ax²)
Solving this for x2, we obtain:
x2 = 1/a
Therefore, the distance from the origin where the kinetic energy is zero is √(1/a), which corresponds to option (3).