BSc CSIT (TU) Science Physics (BSc CSIT, PHY113) Question Paper 2075 Nepal

Diffraction Grating

A diffraction grating is an optical component consisting of a large number of equidistant, parallel slits (or rulings) of equal width ruled on a transparent or reflecting surface. If $a$ is the width of each transparent slit and $b$ the width of each opaque space, the quantity $(a+b)$ is called the grating element (or grating constant).

Derivation of the Grating Equation

Consider a plane wavefront of monochromatic light of wavelength $\lambda$ incident normally on a grating. Light diffracted at an angle $\theta$ from successive slits is brought to focus by a lens. The path difference between rays from two consecutive corresponding points (separated by the grating element) is:

For constructive interference (principal maxima), this path difference must be an integral multiple of the wavelength:

This is the grating equation, where $n$ is the order of the spectrum. If $N = \dfrac{1}{a+b}$ is the number of lines per unit length of the grating, then:

Determination of Wavelength of Light

1. The grating is mounted on the prism table of a spectrometer with its plane normal to the incident parallel beam from the collimator.
2. The grating element $(a+b)$ is found from the known number of lines per cm ruled by the manufacturer: $(a+b)=\dfrac{1}{N}$.
3. The telescope is rotated to locate a spectral line of a chosen order $n$, and the angle of diffraction $\theta$ is measured accurately using the vernier scales.
4. The wavelength is then computed from:

By measuring $\theta$ for several orders and taking the mean, the wavelength of the light is determined precisely.

Energy Stored in a Charged Capacitor

Consider a capacitor of capacitance $C$ being charged. Let $q$ be the charge on it at some instant and $V=q/C$ the corresponding potential difference. To transfer an additional small charge $dq$ against this potential, the work done is:

The total work done in charging the capacitor from $0$ to a final charge $Q$ is:

This work is stored as electrostatic potential energy $U$. Using $Q = CV$, the energy stored can be written in three equivalent forms:

Energy Density of the Electric Field

Consider a parallel-plate capacitor with plate area $A$ and separation $d$. Its capacitance is:

If $E$ is the uniform field between the plates, the potential difference is $V = Ed$. Substituting in $U = \tfrac{1}{2}CV^2$:

Since $Ad$ is the volume of the region containing the field, the energy per unit volume (energy density) is:

This shows that energy is stored in the electric field itself, with a density proportional to the square of the field strength. (In a dielectric medium of permittivity $\varepsilon$, $u = \tfrac{1}{2}\varepsilon E^2$.)

Formation of Newton's Rings

When a plano-convex lens of large radius of curvature is placed with its convex surface on a flat glass plate, a thin air film of gradually increasing thickness is formed between them. When monochromatic light falls normally on this arrangement, light reflected from the top and bottom surfaces of the air film interferes. The locus of points of equal film thickness are concentric circles, so the interference fringes appear as alternate bright and dark concentric rings with a dark spot at the centre. These are called Newton's rings.

The condition for interference in a thin film of air ($\mu = 1$) at normal incidence, including the extra phase change of $\pi$ on reflection at the glass surface, gives:

• Dark ring: $2t = n\lambda$
• Bright ring: $2t = \left(n+\tfrac{1}{2}\right)\lambda$

Diameters of the Rings

Let $R$ be the radius of curvature of the lens and $r$ the radius of a ring where the film thickness is $t$. By the geometry of the circle (chord property):

Dark rings: Using $2t = n\lambda$,

With diameter $D_n = 2r_n$:

Bright rings: Using $2t = (n+\tfrac12)\lambda$,

Thus the diameters of dark rings are proportional to the square roots of natural numbers, and of bright rings to the square roots of odd numbers.

Determination of Wavelength

For dark rings, $D_n^2 = 4nR\lambda$. Taking two different rings, the $n$-th and $m$-th:

Therefore:

The diameters $D_n, D_m$ are measured with a travelling microscope, $R$ is found separately (e.g. with a spherometer), and $\lambda$ is computed. Taking the difference eliminates errors due to imperfect contact at the centre.

Huygens' Principle states that:

1. Every point on a given wavefront acts as a fresh source of secondary wavelets, which spread out in all forward directions with the speed of the wave in the medium.
2. The new position of the wavefront after a time $t$ is given by the forward common tangent (envelope) to all these secondary wavelets.

This principle geometrically explains the propagation of waves and accounts for reflection, refraction, and diffraction of light.

Relation Between Electric Field and Potential

Consider a charge $q_0$ moved through a small displacement $d\vec{l}$ in an electric field $\vec{E}$. The work done by the field is $dW = q_0\vec{E}\cdot d\vec{l}$. The change in potential energy is $dU = -dW$, and since potential $V = U/q_0$:

For a displacement along the $x$-direction this gives the component of the field as the negative gradient of potential:

In three dimensions, generalising over all directions:

Thus the electric field equals the negative potential gradient. The negative sign shows that the field points in the direction of decreasing potential, and its magnitude equals the rate of change of potential with distance (units: V/m).

Ampere's Circuital Law

Statement: The line integral of the magnetic field $\vec{B}$ around any closed loop is equal to $\mu_0$ times the total (net) current $I_{enc}$ enclosed by that loop.

where $\mu_0 = 4\pi\times10^{-7}\ \text{T·m/A}$ is the permeability of free space.

Explanation / Example: For an infinitely long straight wire carrying current $I$, choose a circular Amperian loop of radius $r$ concentric with the wire. By symmetry $B$ is constant along the loop and parallel to $d\vec{l}$, so:

Ampere's law is the magnetic analogue of Gauss's law and is most useful for computing fields in highly symmetric situations (long wire, solenoid, toroid). Maxwell later generalised it by adding the displacement-current term $\mu_0\varepsilon_0\,d\Phi_E/dt$ to make it valid for time-varying fields.

Single-mode vs Multi-mode Optical Fibers

Summary: A single-mode fiber has a narrow core that confines light to a single path, giving very low dispersion and high bandwidth over long distances, whereas a multi-mode fiber has a wider core allowing many paths, which is cheaper but limited by modal dispersion to shorter links.

Damped Harmonic Oscillation

Definition: A damped harmonic oscillation is a periodic motion in which the amplitude of vibration decreases gradually with time because the oscillating system continuously loses energy to a resistive (frictional/viscous) force. This damping force is generally proportional to the velocity of the body, $F_d = -b\dfrac{dx}{dt}$, where $b$ is the damping constant.

Equation of motion: Combining the restoring force $-kx$ and the damping force $-b\,\dot{x}$, Newton's second law gives:

where $2\gamma = b/m$ (damping coefficient) and $\omega_0^2 = k/m$ is the natural angular frequency.

For the under-damped case ($\gamma < \omega_0$) the solution is:

The factor $e^{-\gamma t}$ shows the exponential decay of amplitude with time.

Gauss's Law in Dielectrics

In the presence of a dielectric, both free charges and bound (polarisation) charges contribute to the field. To handle this, the electric displacement vector $\vec{D} = \varepsilon_0\vec{E} + \vec{P}$ is introduced, where $\vec{P}$ is the polarisation.

Statement: The surface integral of the electric displacement $\vec{D}$ over any closed surface equals the total free charge $q_{free}$ enclosed by that surface (bound charges do not appear):

Equivalently, in terms of the field $\vec{E}$ in a linear dielectric of permittivity $\varepsilon = \varepsilon_0\varepsilon_r$:

Thus the law shows that inside a dielectric the field is reduced by the factor $\varepsilon_r$ (the relative permittivity) compared with vacuum, because of the opposing field produced by induced bound charges.

Properties of Laser Light

Laser (Light Amplification by Stimulated Emission of Radiation) light has the following characteristic properties:

1. Monochromaticity — It is highly single-coloured, consisting of essentially one wavelength (very narrow spectral linewidth).
2. Coherence — All the waves are in phase, both spatially and temporally, because they arise from stimulated emission. This gives a constant phase relationship.
3. Directionality — The beam is highly collimated and travels in a single, well-defined direction with very little angular spread (low divergence).
4. High Intensity / Brightness — Because the energy is concentrated into a narrow, parallel beam of one wavelength, the intensity is extremely high.
5. Low Divergence — The beam remains narrow even over long distances.

These properties arise from the process of stimulated emission in a medium with a population inversion inside an optical resonator.

Working of a Step-Index Optical Fiber

A step-index fiber consists of a cylindrical central core of uniform (constant) refractive index $n_1$, surrounded by a cladding of slightly lower refractive index $n_2$ ($n_1 > n_2$). The refractive index changes abruptly (in a step) at the core–cladding boundary — hence the name.

Principle of operation — Total Internal Reflection (TIR): Light launched into the core at an angle within the acceptance cone strikes the core–cladding interface at an angle greater than the critical angle $\theta_c$, where:

The ray is therefore totally internally reflected repeatedly and is guided along the length of the fiber with very little loss, zig-zagging down the core.

The light-gathering capacity is described by the numerical aperture:

where $\theta_a$ is the maximum acceptance angle. Only rays entering within this cone are guided. In a multimode step-index fiber different rays take paths of different lengths, causing modal dispersion (pulse broadening), which limits its bandwidth.

Resolving Power of a Grating

The resolving power of a diffraction grating is its ability to separate two spectral lines of nearly equal wavelengths $\lambda$ and $\lambda + d\lambda$ that are just distinguishable as separate. It is defined quantitatively as:

Expression: Applying the Rayleigh criterion to the grating equation $(a+b)\sin\theta = n\lambda$, the resolving power of a grating is found to be:

where

• $n$ = order of the spectrum, and
• $N$ = total number of lines (rulings) illuminated on the grating.

Key points:

• Resolving power is independent of the grating element; it depends only on the total number of lines used and the order.
• It increases with higher orders ($n$) and with a larger illuminated width (more lines $N$).
• Hence a wide grating with many rulings observed in a high order gives the best resolution of closely spaced spectral lines.