BSc CSIT (TU) Science Physics (BSc CSIT, PHY113) Question Paper 2078 Nepal

Fraunhofer Diffraction at a Single Slit

Setup

A plane monochromatic wavefront of wavelength $\lambda$ is incident normally on a slit of width $a$. A converging lens focuses the diffracted light onto a screen. In Fraunhofer diffraction, both source and screen are effectively at infinity (parallel rays).

Divide the slit into a large number of imaginary point sources (Huygens secondary wavelets). Consider rays diffracted at an angle $\theta$.

Path difference and resultant amplitude

The path difference between wavelets from the two edges of the slit is $a\sin\theta$. Let the phase difference between extreme rays be

Summing the wavelets (vector/phasor addition) gives the resultant amplitude:

Intensity distribution

where $I_0$ is the intensity at the centre ($\theta=0$, where $\frac{\sin\alpha}{\alpha}\to 1$).

Condition for minima

$I=0$ when $\sin\alpha=0$ but $\alpha\neq 0$, i.e. $\alpha = m\pi$ ($m=\pm1,\pm2,\dots$):

These are the dark fringes.

Condition for maxima

• Central (principal) maximum at $\theta=0$ ($\alpha=0$), with intensity $I_0$.
• Secondary maxima occur between consecutive minima, approximately where $\tan\alpha=\alpha$, giving

The secondary maxima are weak; the first has intensity $\approx I_0/22 \;(\approx 4.5\%)$, the next $\approx I_0/61$, decreasing rapidly.

Intensity pattern

The pattern is a broad bright central maximum (angular half-width $\sin\theta=\lambda/a$) flanked by symmetric, rapidly diminishing secondary maxima separated by minima. Most of the energy is concentrated in the central maximum.

Electric Potential

The electric potential at a point is the work done per unit positive charge in bringing a small test charge from infinity to that point against the electric field, without acceleration.

For a point charge $q$ at distance $r$: $\;V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r}.$ It is a scalar quantity.

Potential due to an Electric Dipole

A dipole consists of charges $+q$ and $-q$ separated by distance $2l$, with dipole moment $p = q(2l)$, directed from $-q$ to $+q$.

Let $P$ be a point at distance $r$ from the centre $O$, making angle $\theta$ with the dipole axis. Let $r_1$ and $r_2$ be the distances of $P$ from $+q$ and $-q$ respectively.

The net potential is

For $r \gg l$, using geometry (the foot of perpendiculars):

Then

For $r\gg l$, $l^2\cos^2\theta$ is negligible and $q(2l)=p$:

Special cases

• Axial point ($\theta=0$): $\;V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^2}$ (maximum, positive end).
• Equatorial point ($\theta=90^\circ$): $\;V = 0$.

Note the dipole potential falls off as $1/r^2$, faster than the $1/r$ of a point charge.

Optical Fiber as a Communication Medium

Principle

An optical fiber guides light by total internal reflection (TIR). When light travelling in a denser medium (core, refractive index $n_1$) strikes the boundary with a rarer medium (cladding, $n_2 < n_1$) at an angle greater than the critical angle $\theta_c$, it is totally reflected back into the core. Repeated TIR carries the signal along the fiber with little loss.

The maximum acceptance angle is given by the numerical aperture:

Construction

1. Core – central thin glass/silica cylinder of higher refractive index $n_1$; carries the light.
2. Cladding – surrounding layer of lower index $n_2$; confines light to the core by TIR.
3. Buffer/Jacket – protective polymer coating giving mechanical strength and protection from moisture and damage.

Types: step-index (sharp core–cladding boundary) and graded-index (refractive index varies gradually), and single-mode vs multi-mode depending on core diameter.

Working in communication

• The electrical message signal modulates a light source (LED or LASER diode) at the transmitter.
• The optical signal travels along the fiber by TIR.
• At the receiver, a photodetector (PIN/avalanche photodiode) converts light back into an electrical signal, which is amplified and demodulated.

Advantages over conventional (metallic) systems

• Very large bandwidth / high data rate (uses optical frequencies).
• Low transmission loss (attenuation) → long distances between repeaters.
• Immune to electromagnetic and radio-frequency interference; no crosstalk.
• Electrical isolation and safety (no spark hazard, no short-circuit).
• Light weight and small size, easy to handle.
• High security (difficult to tap) and uses cheap, abundant raw material (silica).

Conditions for Sustained Interference of Light

For a steady, observable (sustained) interference pattern the following conditions must hold:

1. Coherent sources – The two sources must maintain a constant phase difference with time (obtained from a single source by division of wavefront or amplitude).
2. Same frequency / wavelength – The interfering waves must be monochromatic (or of the same colour) so the fringes do not wash out.
3. Same or comparable amplitude – For good contrast, the amplitudes should be nearly equal so that minima are (nearly) dark and contrast is maximum.
4. Same polarization plane – The waves should be polarized in the same plane (or be unpolarized similarly) to superpose effectively.
5. Small path/source separation – The two sources should be narrow and close together, and the screen sufficiently far, so that fringes are wide enough to be resolved.

If these are satisfied, the resultant intensity $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$ gives stable bright fringes (constructive, $\phi=2m\pi$) and dark fringes (destructive, $\phi=(2m+1)\pi$).

Dielectric Constant and Polarization

Dielectric constant (relative permittivity, $\varepsilon_r$ or $K$): It is the ratio of the permittivity of a medium to that of free space:

Equivalently, it is the factor by which the capacitance of a capacitor increases when the gap is filled with the dielectric, $\varepsilon_r = C/C_0$, or the factor by which the field between charges is reduced. It is a dimensionless number ($\varepsilon_r=1$ for vacuum, $\approx 81$ for water).

Polarization ($\vec P$): When a dielectric is placed in an electric field, its molecules develop (or align their) electric dipole moments. Polarization is the electric dipole moment induced per unit volume of the dielectric:

It is related to the field by $\vec P = \varepsilon_0(\varepsilon_r - 1)\vec E = \varepsilon_0\chi_e\vec E$, where $\chi_e$ is the electric susceptibility.

Lenz's Law

Statement: The direction of an induced current (or induced EMF) is always such that it opposes the change in magnetic flux that produces it.

Mathematically it gives the negative sign in Faraday's law:

For example, when a north pole is pushed toward a coil, the near face of the coil becomes a north pole to repel the magnet (opposing the increase in flux); when withdrawn, it becomes a south pole to attract it (opposing the decrease).

Significance

• It gives the direction of induced current/EMF (Faraday's law gives only the magnitude).
• It is a direct consequence of the law of conservation of energy — the opposition means external work must be done to change the flux, and this work appears as electrical energy. Without it, energy could be created indefinitely.
• It explains effects such as eddy-current braking and the back-EMF in motors.

Distinction between LED and LASER

Summary: A LED emits incoherent, divergent, polychromatic light by spontaneous emission, while a LASER emits coherent, highly directional, monochromatic, intense light by stimulated emission in a resonant cavity.

Damped Oscillation and Quality Factor

Differential equation

For a body of mass $m$ executing oscillation under a restoring force $-kx$ and a damping (resistive) force $-b\dfrac{dx}{dt}$ proportional to velocity:

or, dividing by $m$,

where $\beta = \dfrac{b}{2m}$ is the damping constant and $\omega_0 = \sqrt{k/m}$ is the natural angular frequency.

For light (under-)damping the solution is

an oscillation whose amplitude decays exponentially.

Quality factor (Q-factor)

The quality factor measures how lightly a system is damped — the ratio of energy stored to energy lost per radian of oscillation:

A high $Q$ means low damping, a sharp resonance, and many oscillations before dying out; a low $Q$ means heavy damping.

Gauss's Divergence Theorem

Statement: The total outward flux of a vector field $\vec A$ through a closed surface $S$ equals the volume integral of the divergence of $\vec A$ over the volume $V$ enclosed by that surface.

It converts a closed surface integral into a volume integral. Physically, it states that the net flux leaving a closed region equals the total amount of "source" of the field distributed throughout the enclosed volume. It is used, for example, to derive the differential form of Gauss's law of electrostatics, $\nabla\cdot\vec E = \rho/\varepsilon_0$, from its integral form.

Formation of Colours in Thin Films

When white light falls on a thin transparent film (e.g. a soap bubble or oil layer on water of thickness $t$ and refractive index $\mu$), part of the light is reflected from the top surface and part from the bottom surface. These two reflected beams are coherent (derived from the same incident beam) and superpose, producing interference.

The effective path difference for light reflected at angle $r$ inside the film is

where the extra $\lambda/2$ arises because reflection at the denser (top) surface causes a phase change of $\pi$.

• Constructive interference (bright) for a wavelength when $2\mu t\cos r = (2n+1)\dfrac{\lambda}{2}$.
• Destructive interference (dark) when $2\mu t\cos r = n\lambda$.

Since white light contains all wavelengths, for a given thickness $t$ and viewing angle, only certain colours interfere constructively while others cancel. The film therefore appears coloured, and as $t$ or the angle of viewing changes, the colour seen changes. This is why soap bubbles and oil films show brilliant, shifting colours.

Attenuation and Dispersion in Optical Fibers

Attenuation (signal loss): It is the reduction in optical power (intensity) of the signal as it propagates along the fiber, expressed in decibels per kilometre (dB/km):

It is caused mainly by absorption (by impurities and the material itself), scattering (Rayleigh scattering by density fluctuations), and bending losses. Attenuation limits the distance a signal can travel before it must be regenerated.

Dispersion (pulse broadening): It is the spreading (broadening) of light pulses as they travel along the fiber, so that pulses overlap at the output. Main types:

• Modal (intermodal) dispersion – different modes (ray paths) take different times in multimode fiber.
• Material (chromatic) dispersion – refractive index varies with wavelength, so different wavelengths travel at different speeds.
• Waveguide dispersion – arises from the fiber geometry.

Dispersion limits the bandwidth and maximum data rate of the fiber, because overlapping pulses cause intersymbol interference.

Maxwell's Equations

Maxwell's equations are the four fundamental equations that completely describe classical electromagnetism, linking electric field $\vec E$ and magnetic field $\vec B$ to charges and currents.

Significance: Together they predict electromagnetic waves propagating in vacuum at speed

showing that light is an electromagnetic wave. The term $\mu_0\varepsilon_0\,\partial\vec E/\partial t$ (the displacement current) was Maxwell's key addition.