BSc CSIT (TU) Science Physics (BSc CSIT, PHY113) Question Paper 2081 Nepal

Interference of Light

Interference is the modification (redistribution) of light intensity that results when two or more coherent light waves superpose. At points where the waves arrive in phase the resultant intensity is a maximum (constructive interference); where they arrive out of phase by $\pi$ the intensity is a minimum (destructive interference). Energy is merely redistributed, not created or destroyed.

Conditions for sustained interference: the two sources must be coherent (constant phase difference), have the same frequency, and have comparable amplitudes.

Young's Double-Slit Experiment

Monochromatic light of wavelength $\lambda$ illuminates a single slit $S$, which then illuminates two close, parallel slits $S_1$ and $S_2$ separated by a distance $d$. These act as two coherent sources. The light from them overlaps on a screen placed at a distance $D$ ($D \gg d$), producing alternate bright and dark fringes.

Derivation of Fringe Width

Consider a point $P$ on the screen at distance $y$ from the central point $O$ (foot of the perpendicular from the midpoint of the slits).

The path difference between the two waves reaching $P$ is

Bright fringes (maxima): path difference $= n\lambda$,

Dark fringes (minima): path difference $= (2n-1)\frac{\lambda}{2}$,

The fringe width $\beta$ is the separation between two consecutive bright (or dark) fringes:

Thus the fringes are equally spaced and bright and dark fringes have the same width.

Effect of the Medium

If the apparatus is immersed in a medium of refractive index $\mu$, the wavelength changes to $\lambda' = \lambda/\mu$. Hence the fringe width becomes

Since $\mu > 1$, the fringe width decreases when the experiment is performed in a denser medium (the fringes come closer together).

Gauss's Law

Gauss's law states that the total electric flux through any closed surface equals the net charge enclosed divided by the permittivity of free space:

The closed surface is called a Gaussian surface; it is chosen to exploit the symmetry of the charge distribution.

(a) Field due to a Uniformly Charged Infinite Plane Sheet

Let the sheet carry a uniform surface charge density $\sigma$ (C/m²). By symmetry $\vec{E}$ is perpendicular to the sheet and points outward (for $\sigma>0$) on both sides.

Choose a cylindrical (pill-box) Gaussian surface of cross-sectional area $A$ piercing the sheet, with its two flat faces parallel to and equidistant from the sheet.

• Flux through the curved side $= 0$ (field is parallel to that surface).
• Flux through the two flat faces $= E A + E A = 2EA$.
• Charge enclosed $= \sigma A$.

Applying Gauss's law:

The field is uniform and independent of the distance from the sheet.

(b) Field between Two Parallel Charged Plates

Consider two large parallel plates with equal and opposite surface charge densities $+\sigma$ and $-\sigma$ (a parallel-plate capacitor).

Each sheet alone produces a field of magnitude $E_1 = \dfrac{\sigma}{2\varepsilon_0}$.

Between the plates the two fields point in the same direction and add:

Outside the plates the two fields are equal and opposite, so

Thus a parallel-plate arrangement confines a uniform field $E = \sigma/\varepsilon_0$ to the region between the plates.

Principle of Laser Action

LASER = Light Amplification by Stimulated Emission of Radiation. Laser action depends on three processes and two key conditions.

Three radiative processes:

1. Absorption — an atom in the ground state absorbs a photon and jumps to a higher level.
2. Spontaneous emission — an excited atom decays randomly, emitting an incoherent photon.
3. Stimulated emission — an incident photon of energy $h\nu = E_2 - E_1$ triggers an excited atom to emit a second photon identical in frequency, phase, direction and polarization. This multiplication of identical photons is the basis of laser amplification.

Two essential conditions:

• Population inversion — more atoms in the upper level than the lower level, achieved by pumping. A metastable state (long lifetime) helps sustain it.
• Optical resonant cavity — two mirrors (one fully, one partially reflecting) feed photons back through the medium so that stimulated emission builds up a coherent, intense beam emerging through the partial mirror.

Semiconductor (Diode) Laser

Construction

A semiconductor laser is a heavily doped p–n junction diode, typically made of a direct-band-gap material such as gallium arsenide (GaAs).

• The p and n regions are degenerately doped.
• The junction plane forms the active region (a thin layer ~ a few µm thick).
• The two opposite faces perpendicular to the junction are cleaved and polished to act as mirrors, forming the optical cavity (the high refractive index gives ~30% reflectivity, enough for feedback).
• Metal contacts are attached to p and n sides for forward biasing.

(Diagram in words: a small rectangular block; top layer p-type, bottom n-type, with the thin junction/active region in the middle; left and right cleaved faces act as partial mirrors; current flows vertically through the junction and the laser beam emerges horizontally from the active layer.)

Working

1. The diode is forward biased. Electrons are injected from the n-side and holes from the p-side into the junction region.
2. At high current the carrier concentration becomes large enough to create a population inversion in the active region (more electrons in the conduction band than in the valence band).
3. Some electrons recombine spontaneously with holes, emitting photons of energy $h\nu \approx E_g$ (the band gap).
4. These photons stimulate further electron–hole recombinations, producing more identical photons.
5. The cleaved end faces reflect the photons back and forth, amplifying the light, until a coherent, monochromatic laser beam of wavelength $\lambda = hc/E_g$ emerges through the partially reflecting face once the current exceeds the threshold value.

Advantages: very small, efficient, low cost, directly modulated by drive current — widely used in optical-fiber communication, CD/DVD players, and barcode scanners.

Fresnel vs. Fraunhofer Diffraction

Parallel-Plate Capacitor with a Dielectric Slab

Let a capacitor have plates of area $A$ separated by distance $d$, carrying surface charge density $\sigma = Q/A$.

Without dielectric, the uniform field between the plates is $E_0 = \dfrac{\sigma}{\varepsilon_0}$ and the capacitance is

Now insert a dielectric slab of thickness $t$ and dielectric constant $K$ (so $t \le d$). Inside the dielectric the field is reduced by the factor $K$:

while in the remaining air gap of thickness $(d-t)$ the field is $E_0 = \dfrac{\sigma}{\varepsilon_0}$.

The potential difference across the plates is

Since $\sigma = Q/A$,

Special case: if the slab completely fills the gap ($t = d$),

so the capacitance increases by the factor $K$.

Faraday's Laws of Electromagnetic Induction

First law: Whenever the magnetic flux linked with a circuit changes, an e.m.f. is induced in the circuit; the induced e.m.f. lasts only as long as the flux keeps changing.

Second law: The magnitude of the induced e.m.f. is directly proportional to the rate of change of magnetic flux linked with the circuit:

where $N$ is the number of turns and $\Phi_B$ is the magnetic flux through one turn.

Lenz's Law

Lenz's law gives the direction of the induced current: the induced current always flows in such a direction that it opposes the change in flux that produces it. This is expressed by the negative sign in Faraday's equation and is a consequence of the conservation of energy (work must be done against the opposing force to maintain the flux change).

Acceptance Angle and Numerical Aperture of an Optical Fiber

Consider a step-index fiber with core refractive index $n_1$ and cladding index $n_2$ ($n_1 > n_2$), placed in a medium of index $n_0$ (air, $n_0 = 1$).

Acceptance angle ($\theta_a$): the maximum angle that a ray entering the fiber end face can make with the fiber axis and still undergo total internal reflection at the core–cladding boundary, so that it is guided along the fiber. Rays entering within this angle are propagated; rays outside it leak into the cladding.

The cone of half-angle $\theta_a$ is the acceptance cone.

Numerical Aperture (NA): the light-gathering capacity of the fiber, defined as the sine of the acceptance angle:

In air ($n_0 = 1$):

Defining the fractional index difference $\Delta = \dfrac{n_1 - n_2}{n_1}$, this can be written as $\mathrm{NA} \approx n_1\sqrt{2\Delta}$. A larger NA means the fiber accepts light over a wider cone.

Simple Harmonic Motion (SHM)

Definition: Simple harmonic motion is a type of periodic oscillatory motion in which the restoring force (or acceleration) acting on the body is directly proportional to its displacement from the mean (equilibrium) position and is always directed towards that mean position.

Mathematically,

where $x$ is the displacement, $k$ the force constant, and $\omega = \sqrt{k/m}$ the angular frequency. The negative sign shows the restoring nature. Its solution is $x = A\sin(\omega t + \phi)$.

Two examples:

1. A mass attached to a spring oscillating on a frictionless surface (spring–mass system).
2. The oscillation of a simple pendulum for small angular displacements.

(Other valid examples: oscillation of a liquid column in a U-tube, vibration of a tuning fork.)

Coulomb's Law

Statement: The electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them, and acts along the line joining them.

For two point charges $q_1$ and $q_2$ separated by a distance $r$,

where $\dfrac{1}{4\pi\varepsilon_0} \approx 9\times10^{9}\ \text{N·m}^2/\text{C}^2$ in free space.

Vector Form

Let $\hat{r}_{12}$ be the unit vector pointing from $q_1$ to $q_2$. The force on $q_2$ due to $q_1$ is

By Newton's third law $\vec{F}_{21} = -\vec{F}_{12}$. For like charges the product $q_1 q_2 > 0$ and the force is repulsive (along $\hat{r}_{12}$); for unlike charges it is attractive.

Fraunhofer Diffraction at a Single Slit

A parallel beam of monochromatic light of wavelength $\lambda$ falls normally on a narrow slit of width $a$. The diffracted light is focused by a converging lens onto a screen in its focal plane.

Each point of the slit acts as a source of secondary wavelets (Huygens' principle). For light diffracted at an angle $\theta$, the path difference between the wavelets from the two edges of the slit is $a\sin\theta$.

Resultant amplitude: treating the slit as $N$ infinitesimal sources, the resultant intensity is

Central maximum: at $\theta = 0$, $\beta\to0$ and $I = I_0$ — a bright, broad central maximum.

Minima (dark fringes): intensity is zero when $\sin\beta = 0$ but $\beta\neq0$, i.e.

Secondary maxima: occur approximately midway between the minima, at $a\sin\theta = (2n+1)\dfrac{\lambda}{2}$, with rapidly decreasing intensity (first secondary maximum ≈ 4.5% of the central one).

Thus the pattern consists of a bright central maximum twice as wide as the others, flanked by alternate dark and progressively fainter bright fringes. The angular half-width of the central maximum is $\theta \approx \lambda/a$.

Single-Mode vs. Multi-Mode Optical Fibers

Properties of Laser Light

Laser light differs sharply from ordinary light because of stimulated emission. Its main properties are:

1. Monochromaticity — laser light is essentially of a single wavelength/frequency (very narrow spectral line width), because all photons originate from the same energy-level transition.

2. Coherence — laser light has a high degree of both temporal (phase relation over time) and spatial (phase relation across the beam) coherence, since stimulated photons are emitted in phase.

3. Directionality — the beam is highly collimated and travels long distances with very little divergence (a few milliradians), because the resonant cavity selects axial photons.

4. High intensity / brightness — a large number of coherent photons are concentrated in a narrow beam, giving very high intensity and power per unit area even at moderate total power.

(These properties make lasers ideal for optical communication, holography, surgery, cutting/welding, and metrology.)