Wave Optics Set-4

Q1. In Young's double-slit experiment monochromatic light of wavelength $\lambda=500\ \text{nm}$ illuminates two slits separated by $d=0.25\ \text{mm}$. The screen is at distance $D=1.6\ \text{m}$. Using $\beta=\dfrac{\lambda D}{d}$, the fringe width $\beta$ is approximately:

Q2. In a Young's double-slit arrangement light of wavelength $\lambda=600\ \text{nm}$ is used. A thin transparent sheet of refractive index $n=1.5$ and thickness $t=1.8\ \mu\text{m}$ is placed in front of one slit (normal incidence). The central fringe shifts by $m$ fringes, where $m=\dfrac{(n-1)t}{\lambda}$. The numerical value of $m$ is:

Q3. Two coherent monochromatic waves produce interference on a screen with individual intensities in the ratio $I_1:I_2=4:1$. Using $I_{\max}=(\sqrt{I_1}+\sqrt{I_2})^2$ and $I_{\min}=(\sqrt{I_1}-\sqrt{I_2})^2$, the fringe visibility $V=\dfrac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}$ equals:

Q4. A thin transparent plate inserted in front of one slit in Young's double-slit experiment shifts the central fringe by 6 fringes. The same plate (same thickness and refractive index), when inserted normally in one arm of a Michelson interferometer, produces how many fringe shifts? (Neglect dispersion and align plate normal to the beam.)

Q5. A transmission diffraction grating has slit width $b$ and slit separation $d$. Principal maxima of the grating occur at $d\sin\theta=m\lambda$ while minima of the single-slit envelope occur at $b\sin\theta=k\lambda$ ($k\in\mathbb{Z}\setminus\{0\}$). If $b=\dfrac{d}{3}$, which grating orders $m$ of the principal maxima will be missing because they coincide with single-slit minima?

Q6. In a Young's double-slit experiment the slit separation is $d=0.50\ \text{mm}$ and the screen is at distance $D=2.0\ \text{m}$ from the slits. The wavelength of light used is $\lambda=600\ \text{nm}$. The fringe width $\beta$ on the screen is

Q7. In a Young's double-slit setup the slit separation is $d=0.50\ \text{mm}$ and the screen distance is $D=1.5\ \text{m}$. Monochromatic light of wavelength $\lambda=500\ \text{nm}$ is used. A thin glass plate of refractive index $\mu=1.6$ is introduced in front of one slit and the central maximum shifts by $3$ fringes. The thickness $t$ of the plate is

Q8. Two slits separated by $d=0.60\ \text{mm}$ each have finite width $a=0.20\ \text{mm}$. Light of wavelength $\lambda=600\ \text{nm}$ produces interference fringes modulated by single-slit diffraction. Which interference maxima (orders $m$) will be completely missing because they coincide with minima of the single-slit envelope?

Q9. Assertion (A): For reflected light from a thin oil film (refractive index $\mu=1.40$) on water ($n_{\text{water}}=1.33$) at normal incidence, constructive interference occurs for wavelengths satisfying $2\mu t=\left(m+\tfrac{1}{2}\right)\lambda$.
Reason (R): At the air–oil interface the reflected wave undergoes a phase change of $\pi$, while at the oil–water interface it does not; this introduces an effective additional half-wavelength phase difference between the two reflected waves.

Q10. In a Young's double-slit experiment each slit alone produces intensity $I_0$ at a particular point on the screen. Initially both slits are equally illuminated (visibility $V=1$). An absorbing transparent sheet is placed in front of one slit so that the intensity from that slit alone becomes $T\,I_0$ (where $T$ is the transmittance, fraction of intensity transmitted). The fringe visibility then reduces to $V=0.60$. The value of $T$ (as percentage of intensity transmitted) is approximately

Q11. In a Young's double-slit experiment the slits are separated by $d=0.30\ \mathrm{mm}$ and the screen is at a distance $D=1.50\ \mathrm{m}$. Monochromatic light of wavelength $\lambda=540\ \mathrm{nm}$ is used. If the entire apparatus is immersed in a liquid of refractive index $n=1.5$, what is the ratio of fringe width in the liquid to that in air?

Q12. In a Young's double-slit setup two closely spaced wavelengths $\lambda_1=600\ \mathrm{nm}$ and $\lambda_2=605\ \mathrm{nm}$ produce overlapping fringe systems on a screen at distance $D=1.50\ \mathrm{m}$ from the slits. The slit separation is $d=0.25\ \mathrm{mm}$. What is the distance on the screen between two successive positions where bright fringes of the two wavelengths coincide?

Q13. A diffraction grating has $150\ \mathrm{lines/mm}$ and is illuminated by white light. At what angle $\theta$ (with respect to the normal) do the 4th order maximum of $\lambda=600\ \mathrm{nm}$ and the 5th order maximum of $\lambda=480\ \mathrm{nm}$ coincide?

Q14. A sodium lamp has two close spectral components centered at $\lambda=589\ \mathrm{nm}$ with separation $\Delta\lambda=0.6\ \mathrm{nm}$. In a Michelson interferometer fringes disappear once the optical path difference exceeds the coherence length. What is the maximum mirror displacement $s_{\max}$ (one mirror moved) for which fringes remain observable? (Use appropriate relation for coherence length.)

Q15. A transparent thin film of refractive index $n$ is coated on a glass substrate of refractive index $n_s$. Monochromatic light of wavelength $\lambda$ in air is normally incident. For the smallest non-zero film thickness $t$ that minimises the reflected intensity, choose the correct pair for the two cases: (I) $1<n<n_s$ and (II) $n>n_s$.