Ray Optics and Optical Instruments Set-2

Q1. A thin convex lens of focal length $f=20\ \mathrm{cm}$ forms a real, inverted image twice the size of the object. The object distance from the lens is:

Q2. Two thin convex lenses with focal lengths $f_1=10\ \mathrm{cm}$ and $f_2=20\ \mathrm{cm}$ are placed coaxially with a separation $d=5\ \mathrm{cm}$. The effective focal length $F$ of the combination (thin-lens approx.) is closest to:

Q3. A graph shows the refractive index $n(\lambda)$ of a glass exhibiting normal dispersion (monotonically decreasing $n$ with increasing wavelength $\lambda$). A thin convex lens made of this glass is used in air. For monochromatic blue light ($\lambda_b$) and red light ($\lambda_r$) with $\lambda_b<\lambda_r$, which statement about focal positions is correct?

Q4. Assertion (A): An achromatic doublet made of crown and flint glass can completely eliminate chromatic aberration for all wavelengths of the visible spectrum.
Reason (R): Achromatic doublets are designed so that the dispersions of the two lenses cancel for two chosen wavelengths; residual dispersion (secondary spectrum) remains for other wavelengths.

Q5. A convex lens of focal length $f=5\ \mathrm{cm}$ is used as a magnifying glass. For an eye with least distance of distinct vision $D=25\ \mathrm{cm}$ the angular magnifications when the final image is at infinity and when it is at the near point are $M_\infty=\dfrac{D}{f}$ and $M_{\rm near}=1+\dfrac{D}{f}$ respectively. The ratio $\dfrac{M_{\rm near}}{M_\infty}$ equals:

Q6. A thin converging lens of focal length $f=10\ \mathrm{cm}$ forms a real image at distance $v$ when an object is at $u=30\ \mathrm{cm}$. A plane-parallel glass slab of thickness $t=1.5\ \mathrm{cm}$ and refractive index $\mu=1.5$ is inserted between the object and the lens (slab adjacent to the object). Using the approximation that the slab shifts the apparent object toward the lens by $\Delta=t\bigl(1-\tfrac{1}{\mu}\bigr)$, the image distance from the lens changes by approximately:

Q7. Two stars have an angular separation of $0.5''$ (arcsecond). Using the Rayleigh criterion for a circular aperture $\theta_{\min}=1.22\,\lambda/D$ at wavelength $\lambda=550\ \mathrm{nm}$, the minimum diameter $D$ of the telescope objective required to just resolve them is closest to:

Q8. A plot of transmitted intensity vs analyzer angle for unpolarized incident light follows Malus’s law after the first polarizer. If unpolarized light of intensity $I_0$ passes through an ideal polarizer followed by an analyzer at angle $\theta=60^\circ$ relative to the polarizer, the transmitted intensity after the analyzer is:

Q9. Assertion (A): A parabolic reflector shows spherical aberration for parallel incoming rays close to the axis (paraxial rays).
Reason (R): Spherical aberration arises because spherical surfaces do not bring all parallel rays to the same focus; a parabolic reflector brings parallel rays to a single focus and thus removes spherical aberration for parallel rays.

Q10. A thin symmetric lens made of glass with refractive index $n_g=1.50$ has focal length $f_{\rm air}=20\ \mathrm{cm}$ when in air. It is now completely immersed in a liquid with refractive index $n_m=1.30$. Using the lens-maker form $\dfrac{1}{f}=\Bigl(\dfrac{n_{\rm lens}}{n_{\rm medium}}-1\Bigr)\Bigl(\dfrac{1}{R_1}-\dfrac{1}{R_2}\Bigr)$ and assuming the radii $R_1,R_2$ are unchanged, the new focal length $f_{m}$ (in the liquid) is approximately: