Question

Given is a thin convex lens of glass (refractive index \( \mu \)) and each side having radius of curvature \( R \). One side is polished for complete reflection. At what distance from the lens, an object placed on the optic axis so that the image gets formed on the object itself.

• A. \( \frac{R}{\mu} \)
• B. \( \frac{R}{2(\mu-3)} \)
• C. \( \mu R \)
• D. \( \frac{R}{2(\mu-1)} \)

Hint:

For a convex lens with one polished surface, the focal length can be calculated by considering both surfaces and using the lens maker's formula.

Answer 1

The Correct Option is D

Given Information: 

$P_{eq} = 2P_l + P_m$ 

$-\frac{1}{f_{eq}} = \frac{2}{f_l} - \frac{1}{f_m}$ 

$\frac{4(\mu-1)}{R} - \frac{2}{-R} = \frac{1}{R}(4\mu - 4 + 2)$ 

Explanation of Steps: Step 1: Simplify the third equation 

$\frac{4(\mu-1)}{R} - \frac{2}{-R} = \frac{1}{R}(4\mu - 4 + 2)$ 

$\frac{4\mu - 4}{R} + \frac{2}{R} = \frac{4\mu - 2}{R}$
 
$\frac{4\mu - 4 + 2}{R} = \frac{4\mu - 2}{R}$ 

$\frac{4\mu - 2}{R} = \frac{4\mu - 2}{R}$ 

Explanation: Here, we expand the terms and simplify the equation. This step shows that the equation holds true. 

Step 2: Substitute into the second equation 

From the given information, we have: $-\frac{1}{f_{eq}} = \frac{2}{f_l} - \frac{1}{f_m}$ We are given that $\frac{2}{f_l} - \frac{1}{f_m} = \frac{4\mu - 2}{R}$ from the third equation. Substituting this into the second equation: $-\frac{1}{f_{eq}} = \frac{4\mu - 2}{R}$ 

Explanation: This step substitutes the simplified expression from the third equation into the second equation to relate the equivalent focal length to $\mu$ and $R$. 

Step 3: Solve for $f_{eq}$ 

Multiply both sides by -1: $\frac{1}{f_{eq}} = -\frac{4\mu - 2}{R}$ 

Take the reciprocal of both sides: $f_{eq} = -\frac{R}{4\mu - 2}$ Also, 

we are given $R = 2f_{eq}$, so $f_{eq} = \frac{R}{2}$. Substituting $f_{eq} = \frac{R}{2}$ into $R = -f_{eq}(4\mu - 2)$: $R = -\frac{R}{2}(4\mu - 2)$ 

$R = 2f_{eq} = -2 \left( \frac{R}{4\mu - 2} \right) = \frac{-R}{2\mu - 1}$ 

$1 = \frac{-1}{2\mu - 1}$ 

$2\mu - 1 = -1$ 

$2\mu = 0$ 

$\mu = 0$ The final expression is: $R = \frac{-R}{2\mu - 1}$ 

Final Result: $R = \frac{-R}{2\mu - 1}$