An thin Plano - convex glass lens `(mu=1.5)` has its plane surface silvered and R is the radius of curvature of curved part, then which of the following ray diagram is true for an object placed at O ?
(O is at a distance of 2 R from the lens)

Options

A.
B.
C.
D. None of the above

Answer

To solve the problem, we will analyze the behavior of light rays passing through a thin plano-convex lens with a silvered plane surface. The object is placed at a distance of \(2R\) from the lens, where \(R\) is the radius of curvature of the curved surface of the lens.

### Step-by-Step Solution:

1. **Identify the Lens Properties**:
- The lens is a plano-convex lens with a refractive index (\(\mu\)) of 1.5.
- The plane surface of the lens is silvered, making it act as a mirror.

2. **Determine the Focal Length**:
- For a plano-convex lens, the focal length \(f\) can be calculated using the lens maker's formula:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
- Here, \(R_1 = R\) (convex side) and \(R_2 = \infty\) (plane side). Thus:
\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{R} - 0 \right) = \frac{0.5}{R} \]
- Therefore, the focal length \(f\) is:
\[ f = 2R \]

3. **Analyze the Object Position**:
- The object is placed at a distance of \(2R\) from the lens. Since the focal length is also \(2R\), the object is located at the center of curvature of the lens.

4. **Ray Diagram Construction**:
- When parallel rays of light (from an object at infinity) pass through the lens, they converge at the focal point \(F\) (which is at a distance \(f = 2R\)).
- Since the object is at \(2R\), the rays that pass through the lens will diverge after passing through the focal point and become parallel after reflecting off the silvered surface.

5. **Reflection from the Silvered Surface**:
- The rays that exit the lens will hit the silvered plane surface and reflect back.
- The reflected rays will retrace their path back through the lens.

6. **Conclusion**:
- The correct ray diagram will show rays coming from the object at \(2R\), passing through the lens, becoming parallel after passing through the focal point, and then reflecting back towards the object.

### Final Answer:
The correct option is **Option 2**, which shows the rays reflecting back after becoming parallel.