BSc CSIT (TU) Science Physics (BSc CSIT, PHY113) Question Paper 2074 Nepal

Interference of Light

Interference is the modification in the distribution of light intensity that results when two or more coherent light waves superpose. By the principle of superposition, the resultant displacement at a point is the algebraic sum of the displacements of the individual waves. Where the waves arrive in phase, intensity is maximum (constructive interference); where they arrive out of phase, intensity is minimum (destructive interference). Coherent sources (constant phase difference, same frequency) are required to obtain a steady interference pattern.

Young's Double-Slit Experiment

Light from a monochromatic source falls on two narrow slits $S_1$ and $S_2$ separated by a distance $d$. These act as coherent sources. The interference pattern is observed on a screen at distance $D$ ($D \gg d$).

Consider a point $P$ on the screen at distance $y$ from the central axis $O$. The path difference between waves from the two slits is:

for small angles ($\tan\theta \approx \sin\theta = y/D$).

Condition for Constructive Interference (Bright Fringes)

Waves arrive in phase when the path difference is an integral multiple of the wavelength:

Hence the position of the $n$-th bright fringe is:

Condition for Destructive Interference (Dark Fringes)

Waves arrive in opposite phase when the path difference is an odd multiple of half a wavelength:

Hence the position of the $n$-th dark fringe is:

Fringe Width

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

Thus the fringe width is directly proportional to the wavelength $\lambda$ and slit-to-screen distance $D$, and inversely proportional to the slit separation $d$. The fringes are equally spaced and of equal width.

Gauss's Law

Statement: The total electric flux through any closed surface is equal to $1/\varepsilon_0$ times the net charge enclosed by that surface.

where $\vec{E}$ is the electric field, $d\vec{A}$ is the outward area element, and $q_{enc}$ is the enclosed charge. The closed surface is called a Gaussian surface.

(a) Infinite Uniformly Charged Plane Sheet

Let the sheet have a uniform surface charge density $\sigma$ (C/m²). By symmetry, $\vec{E}$ is perpendicular to the sheet and points away from it (for $\sigma > 0$).

Choose a Gaussian cylinder (pillbox) of cross-sectional area $A$ piercing the sheet, with its two flat faces at equal distances on either side.

• Flux through the curved surface = 0 (field is parallel to it).
• Flux through each flat face = $EA$, so total flux = $2EA$.
• Charge enclosed = $\sigma A$.

Applying Gauss's law:

The field is independent of distance from the sheet (uniform field).

(b) Uniformly Charged Spherical Shell

Let a shell of radius $R$ carry total charge $Q$ distributed uniformly. By spherical symmetry, $\vec{E}$ is radial and depends only on $r$. Choose a concentric spherical Gaussian surface of radius $r$.

Outside the shell ($r > R$): Enclosed charge $= Q$.

The shell behaves as if all charge were concentrated at its centre.

On the surface ($r = R$): $\displaystyle E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{R^2} = \frac{\sigma}{\varepsilon_0}$.

Inside the shell ($r < R$): Enclosed charge $= 0$.

The electric field inside a uniformly charged spherical shell is zero everywhere.

Laser

LASER stands for Light Amplification by Stimulated Emission of Radiation. It is a device that produces an intense, highly monochromatic, coherent, directional and collimated beam of light by the process of stimulated emission.

Stimulated Emission

An atom in an excited state $E_2$ can return to a lower state $E_1$ in two ways:

• Spontaneous emission: the atom decays on its own, emitting a photon of energy $h\nu = E_2 - E_1$ in a random direction and phase.
• Stimulated emission: an incident photon of energy $h\nu = E_2 - E_1$ induces the excited atom to drop to $E_1$, emitting a second photon identical to the first — same frequency, phase, direction and polarization.

This gives two coherent photons from one, producing optical amplification. This is the fundamental process behind laser action.

Population Inversion

In thermal equilibrium most atoms are in the ground state (Boltzmann distribution), so absorption dominates. Population inversion is the non-equilibrium condition in which the number of atoms in the higher energy level exceeds that in the lower level ($N_2 > N_1$). Only then does stimulated emission outweigh absorption and light gets amplified. It is achieved by pumping (optical or electrical) and requires a metastable state (long lifetime) to accumulate atoms. A three- or four-level system is needed.

He-Ne Laser

Construction

A narrow glass discharge tube (~30 cm long) is filled with a mixture of helium and neon in the ratio about 10:1 at low pressure (~1 torr). Electrodes connected to a high-voltage supply produce a discharge. The tube ends are sealed with two mirrors forming an optical resonant cavity: one fully reflecting and one partially reflecting (output coupler). Brewster windows give a polarized output.

Working

1. The electric discharge excites He atoms by electron collisions to metastable levels (about 20.61 eV and 19.81 eV).
2. These He levels are very close in energy to excited levels of Ne. By resonant collision transfer, He atoms hand their energy to Ne atoms, exciting Ne and creating population inversion in Ne.
3. Lasing transitions occur in Ne, the principal output being the red line at 632.8 nm (visible). Other lines at 1.15 µm and 3.39 µm also exist.
4. The resonant cavity provides feedback; stimulated emission builds up a coherent beam emerging through the partial mirror.

Helium serves only to pump; the actual laser transitions occur in neon.

Characteristics

Continuous-wave operation, highly monochromatic and coherent red beam at 632.8 nm, low power (a few mW), widely used in alignment, holography, and barcode scanners.

Fresnel vs. Fraunhofer Diffraction

In short: Fresnel (near-field) diffraction deals with finite distances and curved wavefronts, whereas Fraunhofer (far-field) diffraction deals with effectively infinite distances and plane wavefronts produced using lenses.

Capacitance

Capacitance is the ability of a conductor (or system of conductors) to store electric charge. It is defined as the ratio of the charge $Q$ stored to the potential difference $V$ across the conductor:

Its SI unit is the farad (F): $1\text{ F} = 1\text{ C/V}$.

Parallel-Plate Capacitor with a Dielectric

Consider two parallel plates each of area $A$ separated by distance $d$, the gap completely filled with a dielectric of relative permittivity (dielectric constant) $\varepsilon_r$. Let the plates carry charge $+Q$ and $-Q$, giving surface charge density $\sigma = Q/A$.

The electric field between the plates in the presence of the dielectric is:

The potential difference between the plates is:

Therefore the capacitance is:

Without the dielectric ($\varepsilon_r = 1$), $C_0 = \varepsilon_0 A / d$, so inserting a dielectric increases the capacitance by the factor $\varepsilon_r$: $C = \varepsilon_r C_0$.

Biot-Savart Law

Statement: The magnetic field $d\vec{B}$ produced at a point by a small current element $I\,d\vec{l}$ is:

• directly proportional to the current $I$,
• directly proportional to the length of the element $dl$,
• directly proportional to $\sin\theta$, where $\theta$ is the angle between $d\vec{l}$ and the line joining the element to the point,
• inversely proportional to the square of the distance $r$ from the element to the point.

Mathematical Form

In vector form:

where $\mu_0 = 4\pi\times10^{-7}\ \text{T·m/A}$ is the permeability of free space and $\hat{r}$ is the unit vector from the element toward the point.

Explanation

The direction of $d\vec{B}$ is perpendicular to the plane containing $d\vec{l}$ and $\vec{r}$, given by the right-hand rule (direction of $d\vec{l}\times\hat{r}$). The total field of a complete circuit is obtained by integrating over the whole conductor:

It is the magnetic analogue of Coulomb's law and is used to compute fields of current-carrying wires, loops and solenoids.

Optical Fiber

An optical fiber is a thin, flexible, transparent dielectric waveguide (made of glass or plastic) that transmits light signals from one end to the other by repeated total internal reflection. It consists of:

• a central core of refractive index $n_1$,
• surrounded by a cladding of slightly lower refractive index $n_2$ ($n_1 > n_2$),
• protected by an outer buffer/jacket.

Principle: Total Internal Reflection (TIR)

When light travels from a denser medium (core, $n_1$) to a rarer medium (cladding, $n_2$) and strikes the interface at an angle of incidence greater than the critical angle $\theta_c$, it is completely reflected back into the denser medium instead of refracting out. This is total internal reflection.

The critical angle is given by Snell's law at the condition where the refraction angle is $90°$:

Conditions for TIR:

1. Light must travel from a denser to a rarer medium ($n_1 > n_2$).
2. The angle of incidence must exceed the critical angle ($\theta > \theta_c$).

Light launched into the core within the acceptance cone strikes the core–cladding boundary at angles greater than $\theta_c$ and is guided along the fiber by successive total internal reflections with almost no loss, allowing low-loss, high-bandwidth transmission of optical signals.

Simple Harmonic Motion (SHM)

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

Mathematically, the restoring force is:

where $x$ is the displacement, $k$ is the force constant, and the negative sign shows that the force opposes the displacement.

Differential Equation

By Newton's second law $F = ma = m\dfrac{d^2x}{dt^2}$, so:

Writing $\omega^2 = k/m$ (angular frequency), the standard differential equation of SHM is:

Its solution is $x = A\sin(\omega t + \phi)$, where $A$ is the amplitude and $\phi$ the initial phase. The time period is $T = 2\pi/\omega = 2\pi\sqrt{m/k}$.

Coulomb's Law (Vector Form)

Coulomb's law states that the electrostatic force 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, acting along the line joining them.

Let two point charges $q_1$ and $q_2$ be separated by a distance $r$, with $\hat{r}_{12}$ the unit vector pointing from $q_1$ to $q_2$. The force on $q_2$ due to $q_1$ is:

or equivalently, using the displacement vector $\vec{r}_{12}$ (with $r = |\vec{r}_{12}|$):

where $\dfrac{1}{4\pi\varepsilon_0} = 9\times10^9\ \text{N·m}^2/\text{C}^2$ and $\varepsilon_0$ is the permittivity of free space.

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

Newton's Rings

Newton's rings are a pattern of concentric bright and dark circular interference fringes produced when a plano-convex lens of large radius of curvature is placed with its convex surface on a flat glass plate. The thin air film of gradually increasing thickness between the lens and the plate produces interference by division of amplitude when illuminated with monochromatic light (viewed in reflection).

Due to the $\pi$ phase change on reflection at the air–glass (denser) boundary, the centre is dark. The radii of the dark rings are given by:

and the bright rings by $r_n^2 = \dfrac{(2n-1)\lambda R}{2}$, where $R$ is the radius of curvature of the lens and $\lambda$ the wavelength. The radii are proportional to $\sqrt{n}$, so rings get closer together outward.

Applications

1. Determination of the wavelength $\lambda$ of monochromatic light (knowing $R$).
2. Determination of the radius of curvature $R$ of a lens.
3. Measurement of the refractive index of a liquid placed in the film.
4. Testing optical flatness and surface quality of lenses and plates.

Acceptance Angle

The acceptance angle ($\theta_a$) of an optical fiber is the maximum angle that an incident ray can make with the fiber axis at the input end and still be guided along the fiber by total internal reflection. Rays entering within the cone of half-angle $\theta_a$ (the acceptance cone) are propagated; rays outside it leak into the cladding and are lost.

If the fiber has core index $n_1$, cladding index $n_2$, and is placed in a medium of index $n_0$, then applying Snell's law at the entrance face together with the critical-angle condition at the core–cladding boundary gives:

For a fiber in air ($n_0 = 1$): $\;\sin\theta_a = \sqrt{n_1^2 - n_2^2}$.

Numerical Aperture

The numerical aperture (NA) is a dimensionless number that measures the light-gathering capacity of the fiber. It equals the sine of the acceptance angle:

Using the fractional index difference $\Delta = \dfrac{n_1 - n_2}{n_1}$, this can be written as:

A larger NA means the fiber accepts light over a wider cone (better coupling efficiency) but generally allows more modal dispersion.

Polarization of Light

Light is a transverse electromagnetic wave in which the electric field vector $\vec{E}$ vibrates perpendicular to the direction of propagation. In ordinary (unpolarized) light the $\vec{E}$ vibrations occur randomly in all directions perpendicular to the ray. Polarization is the process of confining these vibrations to a single plane.

Types of Polarization

• Plane (linear) polarized light: vibrations confined to one fixed plane.
• Circularly polarized light: the tip of $\vec{E}$ traces a circle as the wave advances (two perpendicular components of equal amplitude, $90°$ out of phase).
• Elliptically polarized light: $\vec{E}$ traces an ellipse (unequal amplitudes or arbitrary phase difference).

Methods of Producing Polarized Light

1. Polarization by reflection — at the Brewster (polarizing) angle $\theta_p$, where $\tan\theta_p = n$ (Brewster's law), reflected light is fully plane-polarized.
2. Polarization by refraction / a pile of plates.
3. Polarization by selective absorption (dichroism) — using Polaroid sheets.
4. Polarization by double refraction — in birefringent crystals (e.g., calcite) using Nicol prisms.
5. Polarization by scattering — scattered sunlight is partially polarized.

Importance / Applications

• Proves the transverse nature of light.
• Used in Polaroid sunglasses, LCD displays, photoelasticity, sugar concentration measurement (polarimetry), and stress analysis.

The fact that light can be polarized (whereas sound, a longitudinal wave, cannot) is direct evidence that light is a transverse wave.