JEE Physics 2025 Question with Solution

Given: A thin convex lens of refractive index $\mu$ has both surfaces of radius of curvature $R$, and one side is polished for complete reflection.

Find: The object distance from the lens for which the final image is formed on the object itself.

The system acts as a combination of refraction by the lens surface and reflection at the polished surface. From the solution working, the effective focal length is obtained using lens maker's idea as

Hence,

For the image to be formed on the object itself, the required object position is the same as the image position for this effective system. Therefore,

Therefore, the required distance is $\frac{R}{2(\mu-1)}$, so the correct option is D.

Given: A thin convex lens of glass has refractive index $\mu$ and each surface has radius of curvature $R$. One surface is silvered or polished for complete reflection.

Find: The distance $x$ of the object from the lens so that after refraction, reflection, and refraction again, the final image coincides with the object.

Step 1 — Refraction at the first spherical surface (air to glass): For a spherical refracting surface,

Here $n_1=1$, $n_2=\mu$, and if the object is at distance $x$ to the left, then $s=-x$. So,

Thus,

and therefore,

Step 2 — Reflection at the polished spherical face: For a thin lens, the separation of the two vertices is negligible, so the image distance $v_1$ inside the glass may be treated as the object distance for the mirror. Using the mirror formula,

with $s_m=v_1$, we get

Step 3 — Refraction back at the first surface (glass to air): Now the image formed by the mirror serves as a virtual object for refraction back into air. Using the refraction relation with $n_1=\mu$ and $n_2=1$,

For the final image to coincide with the original object, we require

So,

Step 4 — Combine and simplify: Substituting the expressions of $v_1$ and $s_m'$ into the final equation and simplifying gives

Therefore, the object must be placed at $\frac{R}{2(\mu-1)}$ from the lens. Hence, the correct option is D.