Introduction:
In this scenario, we have a narrow paraxial beam of light that is converging towards a point on a screen. We introduce a plane parallel plate with a thickness of t and a refractive index of u in the path of the beam. We need to determine how the convergent point is shifted.

Explanation:
To understand the shift in the convergent point, we need to consider the behavior of light when it passes through a medium with a different refractive index. When light passes from one medium to another, it changes its direction due to the change in speed caused by the change in refractive index.

Refraction at the First Surface of the Plate:
When the beam of light enters the plane parallel plate, it undergoes refraction at the first surface. The amount of refraction depends on the angle of incidence and the refractive indices of the two media involved. Since the beam is converging towards a point on the screen, the angle of incidence at the first surface of the plate will be small.

Refraction at the Second Surface of the Plate:
After passing through the first surface of the plate, the beam continues to travel through the plate and reaches the second surface. At this surface, it undergoes refraction again before leaving the plate. The angle of incidence at the second surface will be different from the angle at the first surface, depending on the thickness of the plate.

Effect of Thickness and Refractive Index:
The thickness of the plate and its refractive index play a crucial role in determining how the convergent point is shifted. Let's consider the two cases:

1. If the refractive index of the plate is greater than that of the surrounding medium, i.e., u > 1:
- As the beam passes through the plate, it slows down due to the higher refractive index.
- The beam is deviated towards the normal at both surfaces, and its angle of deviation increases with the thickness of the plate.
- Consequently, the convergent point is shifted closer to the plate by a distance of t(1-1/u).

2. If the refractive index of the plate is less than that of the surrounding medium, i.e., u < />
- As the beam passes through the plate, it speeds up due to the lower refractive index.
- The beam is deviated away from the normal at both surfaces, and its angle of deviation increases with the thickness of the plate.
- In this case, the convergent point would be shifted away from the plate, but the exact distance cannot be determined without knowing the specific values of t and u.

Conclusion:
In our scenario, since the refractive index of the plate is not specified, we cannot conclusively determine the exact shift of the convergent point. However, if the refractive index is greater than 1 (u > 1), the convergent point will be shifted closer to the plate by a distance of t(1-1/u).

Option B is the answer...