Introduction: In this problem, we are given a particle of charge q, mass m, and initial velocity v (i j) entering a region of uniform magnetic field Bi. We need to find the displacement of the particle after time t=2piem/qB.

Formula: The formula for the displacement of a charged particle in a uniform magnetic field is given as:
Δr = (q/m) * v x B * (1 - cos(θ))
where,
Δr = displacement
v = velocity of the particle
B = magnetic field strength
θ = angle between the velocity and the magnetic field

Calculation:
Here, the time taken by the particle to move in a complete circle is given as:
T = 2πm/qB
Therefore, the angle θ = 2π

Now, let's calculate the displacement using the above formula:
Δr = (q/m) * v x B * (1 - cos(θ))
Δr = (q/m) * v x B * (1 - cos(2π))
Δr = (q/m) * v x B * (1 - 1)
Δr = 0

Therefore, the displacement of the particle after time t=2piem/qB is zero. This means that the particle will complete a circular path and return to its initial position after this time.

Explanation:
When a charged particle enters a uniform magnetic field, it experiences a force perpendicular to both the velocity and the magnetic field. This force causes the particle to move in a circular path with a radius given by:
r = mv/qB

The time taken by the particle to complete one full circle is given as:
T = 2πm/qB

Therefore, after time t=2piem/qB, the particle completes one full circle and returns to its initial position. Hence, the displacement of the particle is zero.

Conclusion:
In this problem, we have calculated the displacement of a charged particle in a uniform magnetic field after a certain time. We have used the formula for the displacement of a charged particle in a uniform magnetic field and calculated the displacement to be zero. This means that the particle completes a circular path and returns to its initial position after this time.