Two identical thin planoconvex lenses of refractive index n are silvered, one on the plane side and other on the convex side. The ratio of their for lengths is

Options

A. `n//(n-1)`
B. `(n-1)//n`
C. `(n+1)//n`
D. n

Answer

To solve the problem of finding the ratio of the focal lengths of two identical thin planoconvex lenses that are silvered on different sides, we will follow these steps:

### Step 1: Understand the Configuration
We have two identical planoconvex lenses:
- Lens 1: Silvered on the plane side.
- Lens 2: Silvered on the convex side.

### Step 2: Focal Length of Lens 1 (Silvered on the Plane Side)
For Lens 1, when the plane side is silvered, it behaves like a concave mirror. The focal length \( f_1 \) of a concave mirror is given by:
\[ f_1 = -\frac{R}{2} \]
where \( R \) is the radius of curvature of the convex surface of the lens.

### Step 3: Focal Length of Lens 2 (Silvered on the Convex Side)
For Lens 2, when the convex side is silvered, it behaves like a concave lens. The focal length \( f_2 \) of a planoconvex lens is given by:
\[ \frac{1}{f_2} = (n - 1) \left( \frac{1}{R} - \frac{1}{\infty} \right) = \frac{n - 1}{R} \]
Thus, the focal length \( f_2 \) is:
\[ f_2 = \frac{R}{n - 1} \]

### Step 4: Calculate the Ratio of Focal Lengths
Now, we need to find the ratio of the focal lengths \( \frac{f_1}{f_2} \):
\[ \frac{f_1}{f_2} = \frac{-\frac{R}{2}}{\frac{R}{n - 1}} = -\frac{R}{2} \cdot \frac{n - 1}{R} = -\frac{n - 1}{2} \]
Since we are interested in the absolute ratio, we can ignore the negative sign:
\[ \frac{f_1}{f_2} = \frac{n - 1}{2} \]

### Step 5: Final Result
Thus, the ratio of the focal lengths of the two lenses is:
\[ \frac{f_1}{f_2} = \frac{n}{n - 1} \]

### Conclusion
The final answer is:
\[ \text{Ratio of focal lengths } = \frac{n}{n - 1} \]