Optics And Refraction

To solve this problem, we need to determine where an object should be placed along the optical axis of a thin convex lens with a silvered surface (acting as a lens-mirror system) so that a real and inverted image is formed at the same position as the object.

1. **Conceptual Understanding**:

When a lens is silvered on one surface, it acts as a combination of a lens and a mirror. The light passes through the lens twice (forward and backward) and also reflects off the silvered surface. The image formed by the lens is then treated as an object for the mirror formed by the silvered surface, which results in a scenario similar to a "lens-maker's" formula combined with a mirror formula.

2. **Formula Application**:

For a thin lens, the lens maker's formula is given by:

However, since one surface is silvered, we consider the effective focal length if the lens were not silvered:

Let's simplify for the given condition where the image coincides with the object.

3. **Condition for Coincidence**:

The distance where the object should be placed is equal to the equivalent distance called the "radius of curvature" when the lens is treated as a mirror. From optical principles, this needs:

In the lens-mirror system, for a real and inverted image at the same place:

4. **Substitute the given values**:

After analyzing the lens equation and mirror-formula together, the distance for the object placement simplifies to:

This expression aligns with the provided correct answer among the options.

5. **Conclusion**:

Thus, the object must be placed at a distance from the lens given by:

This ensures that a real and inverted image is formed at the same place, thereby confirming the solution.

To determine the lateral shift of a light ray refracted through a parallel-sided glass slab, we need to consider the geometric path taken by the ray as well as the optical principles governing refraction.

Concept: When a light ray passes through a parallel-sided glass slab, it undergoes refraction at both the air-slab interface and the slab-air interface. The ray exits parallel to its original path but displaced sideways. This displacement is what we refer to as the "lateral shift."

The lateral shift is determined by the geometry of the setup, given by the formula:

where:

• is the thickness of the slab,
• is the angle of incidence,
• is the angle of refraction.

Derivation:

1. As the ray enters the slab, it bends towards the normal due to the higher refractive index of the glass compared to air.
2. Within the slab, its horizontal component causes the ray to travel a different path length than if it traveled in air, leading to lateral displacement.
3. To find the shift, we consider the path difference at the exit face of the slab, using trigonometry:
4. The length of the path through the slab in terms of horizontal distance (lateral shift, ) is given by projecting the thickness on the refracted angle:
5. \)

This matches the correct option, ensuring that it accounts for both the angle of incidence and refraction. The ray ultimately emerges parallel to the initial path but laterally shifted.

Conclusion: The correct formula for the lateral shift in a parallel-sided glass slab when placed in an air medium is , and thus the correct answer is the fourth option given:

To find the angle of a prism when the refractive index of its material is given, and the angle of minimum deviation is equal to the angle of the prism, we utilize the formula for the angle of deviation in a prism.

The formula relating the refractive index ( ), the angle of the prism ( ), and the angle of minimum deviation ( ) is given by:

Given:

• Refractive index,
• Angle of minimum deviation,

Substitute in the formula:

Given , the equation becomes:

We need to find the angle such that the above equation holds.

By solving this equation using trigonometric identities and known values, we find:

This solution matches with the provided options, confirming that the angle of the prism is indeed 60°.

Therefore, the correct answer is 60°.

To solve this problem, let's use the formula for refraction at a spherical surface:

where:

• (refractive index of air)
• (refractive index of glass)
• is the object distance
• is the image distance
• is the radius of curvature

In this scenario, the object is placed on the optical axis such that its real image is formed. Also, given that , we have and .

Substitute the values in the formula:

After simplifying:

Cross-multiplying gives:

\)

Therefore, solving for :

\)

Thus, the distance , which is equal to , is .

Therefore, the correct answer is .

Step 1: Understanding the Concept:
We analyze image formation by lens first. The image formed by lens acts as the object for lens . The final position and magnification are obtained by applying lens and magnification formulas sequentially.

Step 2: Key Formula or Approach:
Lens formula: Magnification: Total magnification:
Step 3: Detailed Explanation:
For Lens :
Image is formed 30 cm to the right of lens .

For Lens :
Lens is 15 cm to the right of lens .
Object distance for : (This is a virtual object for lens ).
Final image is real and formed 7.5 cm to the right of lens .

Total Magnification:
So, the image size is equal to the object size.

Step 4: Final Answer:
The final image is real, formed 7.5 cm to the right of lens , and is of the same size as the object.

Step 1: Understanding the Concept:
For thin lenses in contact, the equivalent power is the sum of individual powers: .
Step 2: Detailed Explanation:
Let and .

The combination behaves as a concave lens if is negative ( ).
.
This means .
Similarly, it behaves as a convex lens if .
Step 3: Final Answer:
Statement (A) is correct.