BSc CSIT (TU) Science Physics (BSc CSIT, PHY113) Question Paper 2079 Nepal

Polarization of Light

Polarization is the property of a transverse wave by which the vibrations of the electric field vector $\vec{E}$ are confined to a single plane (or follow a definite pattern) perpendicular to the direction of propagation. Ordinary (unpolarized) light has $\vec{E}$ vibrating randomly in all directions perpendicular to the ray; in polarized light the vibrations are restricted. Polarization confirms that light is a transverse wave.

Plane (Linearly) Polarized Light

When the tip of the electric vector traces a straight line as the wave advances, the light is plane-polarized. It is produced by passing unpolarized light through a polaroid or by reflection at the Brewster angle ($\tan\theta_B = n$). A single linear vibration can be written as:

Circularly and Elliptically Polarized Light

These are produced by superposing two mutually perpendicular plane-polarized waves of the same frequency with a phase difference $\delta$, e.g. by passing plane-polarized light through a birefringent crystal plate (quarter- or half-wave plate):

• Circularly polarized: equal amplitudes ($a=b$) and phase difference $\delta = \pi/2$ (a quarter-wave plate). The tip of $\vec{E}$ traces a circle.
• Elliptically polarized: unequal amplitudes ($a \neq b$), or $\delta \neq 0,\pi/2,\pi$. The tip of $\vec{E}$ traces an ellipse. (Plane and circular polarization are special cases of the general ellipse.)

Law of Malus

When plane-polarized light of intensity $I_0$ passes through an analyser whose transmission axis makes an angle $\theta$ with the plane of polarization, the transmitted intensity is:

Thus $I$ is maximum when $\theta = 0^\circ$ (axes parallel) and zero when $\theta = 90^\circ$ (axes crossed). This follows because only the component $E_0\cos\theta$ of the amplitude is transmitted, and intensity $\propto$ amplitude$^2$.

Maxwell's Equations in Free Space

In free space ($\rho = 0$, $\vec{J}=0$) Maxwell's equations are:

Wave Equation for $\vec{E}$

Take the curl of equation (3):

Using the vector identity $\nabla\times(\nabla\times\vec{E}) = \nabla(\nabla\cdot\vec{E}) - \nabla^2\vec{E}$ and substituting (1) ($\nabla\cdot\vec{E}=0$) and (4):

Similarly, taking the curl of (4) and using (2) gives:

Speed of the Wave

These are standard three-dimensional wave equations of the form $\nabla^2\psi = \dfrac{1}{v^2}\dfrac{\partial^2\psi}{\partial t^2}$. Comparing coefficients:

Substituting $\mu_0 = 4\pi\times10^{-7}\,\text{H/m}$ and $\varepsilon_0 = 8.854\times10^{-12}\,\text{F/m}$:

Since this equals the measured speed of light $c$, Maxwell concluded that light is an electromagnetic wave and that all electromagnetic waves travel at speed $c$ in vacuum.

He-Ne Laser

Construction

A He-Ne laser consists of a long, narrow discharge tube (typically 30-50 cm long, ~few mm bore) filled with a mixture of helium and neon gases in the ratio about 10:1 at low pressure (~1 torr). A high-voltage DC supply drives a discharge between electrodes. The tube is closed by an optical resonant cavity formed by two mirrors: one fully reflecting and one partially reflecting (output mirror), aligned parallel and often with Brewster-angle windows to give polarized output.

Working (Population Inversion)

1. The electric discharge excites He atoms to the metastable states $2^1S$ and $2^3S$ by electron collisions.
2. These He atoms transfer their energy to Ne atoms by resonant (collisional) energy transfer, because Ne has energy levels almost coincident with the He metastable levels. This selectively populates the Ne upper laser levels, producing population inversion.
3. Stimulated emission between Ne levels gives laser output, the strongest line being 632.8 nm (red) (others at 1.15 µm and 3.39 µm).
4. The cavity mirrors provide optical feedback, amplifying the beam; a continuous (CW), highly monochromatic, coherent beam emerges through the partial mirror.

He-Ne is a four-level gas laser giving continuous, low-power (~mW), highly coherent and monochromatic output.

Comparison with Semiconductor (Diode) Laser

Constructive vs Destructive Interference

When two coherent waves superpose, the resultant depends on their phase/path difference.

For equal amplitudes $a$, constructive gives $A=2a$ (intensity $4a^2$) while destructive gives $A=0$ (intensity $0$). Energy is not destroyed but redistributed from dark to bright regions.

Capacitance of a Cylindrical Capacitor

Consider two coaxial conducting cylinders of length $L$, inner radius $a$ and outer radius $b$, with $L \gg b$ (fringing neglected). Let the inner cylinder carry charge $+Q$ and the outer $-Q$, so the linear charge density is $\lambda = Q/L$.

Field between the cylinders (Gauss's law)

Using a coaxial Gaussian cylinder of radius $r$ ($a < r < b$) and length $L$:

Potential difference

Capacitance

The capacitance depends only on the geometry ($L$, $a$, $b$). If a dielectric of permittivity $\varepsilon_r$ fills the gap, replace $\varepsilon_0$ by $\varepsilon_r\varepsilon_0$.

Biot-Savart Law

The Biot-Savart law gives the magnetic field produced by a small current element. A current element $I\,d\vec{l}$ produces, at a point P located at displacement $\vec{r}$ from the element, a magnetic field $d\vec{B}$ given by:

where $\theta$ is the angle between $d\vec{l}$ and $\vec{r}$, and $\mu_0 = 4\pi\times10^{-7}\,\text{T·m/A}$ is the permeability of free space.

Key points

• $dB \propto I$, $dB \propto dl$, $dB \propto \sin\theta$, and $dB \propto 1/r^2$.
• The direction of $d\vec{B}$ is perpendicular to the plane containing $d\vec{l}$ and $\vec{r}$, given by the right-hand rule (the cross product $d\vec{l}\times\hat{r}$).
• $dB = 0$ along the axis of the element ($\theta=0$ or $\pi$) and is maximum at $\theta = 90^\circ$.

The total field of an extended conductor is obtained by integration: $\vec{B} = \dfrac{\mu_0 I}{4\pi}\displaystyle\int \dfrac{d\vec{l}\times\hat{r}}{r^2}$. For example, for an infinite straight wire it gives $B = \dfrac{\mu_0 I}{2\pi r}$.

V-Number (Normalized Frequency) of an Optical Fiber

The V-number (or normalized frequency / $V$-parameter) is a dimensionless quantity that determines the number of modes an optical fiber can support. It is defined as:

where $a$ is the core radius, $\lambda$ is the free-space wavelength, $n_1$ and $n_2$ are the refractive indices of core and cladding, and $\text{NA} = \sqrt{n_1^2-n_2^2}$ is the numerical aperture.

Significance

• If $V \le 2.405$, the fiber is single-mode (only the fundamental $\text{LM}_{01}$ mode propagates). $V=2.405$ is the cut-off (first zero of the Bessel function $J_0$).
• If $V > 2.405$, the fiber is multimode.
• For a multimode step-index fiber the number of supported modes is approximately:

Thus $V$ increases with larger core radius and numerical aperture, and decreases with longer wavelength.

Time Period and Frequency of SHM

Simple harmonic motion (SHM) is periodic motion in which the restoring force is proportional to displacement and directed toward the mean position, described by $x = A\sin(\omega t + \phi)$ with angular frequency $\omega = \sqrt{k/m}$.

Time Period ($T$)

The time period is the time taken to complete one full oscillation (one complete cycle). It is given by:

SI unit: second (s).

Frequency ($f$)

The frequency is the number of complete oscillations per unit time. It is the reciprocal of the time period:

SI unit: hertz (Hz), i.e. cycles per second.

The two are related by $\omega = 2\pi f$ and $T = 1/f$. Both depend only on $m$ and $k$, not on amplitude (isochronism).

Stokes' Theorem

Stokes' theorem relates the surface integral of the curl of a vector field over an open surface $S$ to the line integral of the field around the closed boundary curve $C$ of that surface:

where:

• $\vec{F}$ is a continuously differentiable vector field,
• $d\vec{l}$ is the line element along the closed curve $C$,
• $d\vec{S} = \hat{n}\,dS$ is the outward area element of the surface $S$ bounded by $C$,
• the direction of traversal of $C$ and the normal $\hat{n}$ follow the right-hand rule.

In words: the circulation of a vector field around a closed loop equals the flux of its curl through any surface bounded by that loop. It is used in electromagnetism to convert Ampère's and Faraday's laws between integral and differential forms.

Missing Orders in Double-Slit Diffraction

In a double-slit pattern, the resultant intensity is the product of the single-slit diffraction envelope and the double-slit interference pattern. Two conditions operate simultaneously (slit width $e$, opaque gap so spacing between slit centres $= e + d$, often written $(a+b)$):

Interference maxima:

Diffraction minima (single slit):

Missing orders

A missing order occurs when the direction of an interference maximum coincides with a diffraction minimum — the diffraction envelope is zero there, so although interference predicts a bright fringe, no light reaches that point and the order is absent. Dividing the two equations:

So the orders $n$ that are missing depend on the ratio $(e+d)/e$. For example:

• If $e + d = 2e$ (i.e. $d = e$): orders $n = 2, 4, 6, \dots$ are missing.
• If $e + d = 3e$ (i.e. $d = 2e$): orders $n = 3, 6, 9, \dots$ are missing.

Thus missing orders arise purely from the geometric ratio of slit spacing to slit width.

Step-Index vs Graded-Index Fiber

In short, the key distinction is the index profile: abrupt (step) versus gradual (graded), which makes graded-index fibers superior in reducing intermodal dispersion.

Displacement Current

Displacement current is the term introduced by Maxwell to account for a changing electric field acting as a source of magnetic field, just like a conduction current. It is not a flow of charge but arises from a time-varying electric flux.

Need for it

The original Ampère's law, $\oint\vec{B}\cdot d\vec{l} = \mu_0 I$, fails for a charging capacitor: between the plates no conduction current flows, yet a magnetic field exists. Maxwell resolved this by adding a displacement current $I_d$.

Expression

The displacement current is defined through the rate of change of electric flux $\Phi_E$:

Modified Ampère-Maxwell law

or in differential form $\nabla\times\vec{B} = \mu_0\vec{J} + \mu_0\varepsilon_0\dfrac{\partial\vec{E}}{\partial t}$.

Significance

• It makes the set of Maxwell's equations consistent (continuity of current).
• It predicts the existence of electromagnetic waves, since a changing $\vec{E}$ generates $\vec{B}$ and vice versa.