Q: Find the lowest energy of an electron confirmed to move in one dimensional po…

The energy of an electron in a 1D potential box of width $1\text{ \u00c5}$ in its ground state ($n=1$) is approximately $37.6\text{ eV}$.

Q: The carrier concentration in n-type semiconductor $10^{19} \text{ m}^{-3}$. W…

For a carrier density of $10^{19}\text{ m}^{-3}$, the Hall coefficient ($R_H$) is calculated as $6.25 \times 10^{-1}\text{ m}^3/\text{C}$.

Question 1

Give the physical significance of divergence and curl of a field.

Accepted Answer

Divergence measures the net outward flux of a vector field from a point (source/sink), while Curl measures the rotation or 'churn' of the field at that point.

Question 2

Write all four Maxwell's equations in integral form for free space.

Accepted Answer

Maxwell's equations in integral form describe how electric and magnetic fields are generated and altered by each other and by charges and currents over volumes and closed loops.

Question 3

What do you mean by eigenfunctions and eigenvalues?

Accepted Answer

Eigenfunctions are specific state functions that remain proportional to themselves after an operator is applied; the proportionality constant is the eigenvalue.

Question 4

When the movable mirror of Michelson's interferometer is shifted by $0.030 	…

Accepted Answer

Using Michelson's interferometer formula $\Delta d = n(\lambda/2)$, the wavelength of light for a shift of $0.030	ext{ mm}$ and $100$ fringes is $6000	ext{ \u00c5}$.

Question 5

State Rayleigh's criterion of resolution.

Accepted Answer

Rayleigh's criterion states that two point sources are just resolved when the principal maximum of the diffraction pattern of one coincides with the first minimum of the other.

Question 6

Find the lowest energy of an electron confirmed to move in one dimensional po…

Accepted Answer

The energy of an electron in a 1D potential box of width $1	ext{ \u00c5}$ in its ground state ($n=1$) is approximately $37.6	ext{ eV}$.

Question 7

Calculate the numerical aperture and acceptance angle of an optical fiber. Gi…

Accepted Answer

For an optical fiber with $n_1 = 1.62$ and $n_2 = 1.52$, the Numerical Aperture ($NA$) is $0.56$ and the Acceptance Angle is $34.05^{\circ}$.

Question 8

Define spatial and temporal coherence.

Accepted Answer

Spatial coherence refers to the phase correlation between different points in space, while temporal coherence refers to the phase correlation at a single point over a period of time.

Question 9

What do you mean by stimulated emission and spontaneous emission?

Accepted Answer

Spontaneous emission is a random decay process, while stimulated emission occurs when an incident photon triggers a transition, producing a second coherent photon.

Question 10

The carrier concentration in n-type semiconductor $10^{19} 	ext{ m}^{-3}$. W…

Accepted Answer

For a carrier density of $10^{19}	ext{ m}^{-3}$, the Hall coefficient ($R_H$) is calculated as $6.25 	imes 10^{-1}	ext{ m}^3/	ext{C}$.

Question 11

Give the construction and theory of plane transmission grating and explain th…

Accepted Answer

A plane transmission grating consists of thousands of identical, parallel, equidistant slits. It functions by diffracting incident light waves, which then undergo multi-slit interference to produce extremely sharp, distinct principal maxima (spectral lines) governed by the grating equation $(a+b)\sin	heta = n\lambda$.

Question 12

Prove that in high frequency region laser action is not possible.

Accepted Answer

Creating lasers at very high frequencies, such as X-rays, is extremely challenging. The rate of spontaneous emission scales as the cube of the frequency ($
u^3$), causing high-energy states to decay instantly and making the critical population inversion virtually impossible to sustain without astronomical pumping power.

Question 13

For intrinsic semiconductor with a band gap $E_g = 0.7 	ext{ eV}$, calculate…

Accepted Answer

The intrinsic carrier density $n_i$ of a semiconductor is a measure of the thermally generated electron-hole pairs. For a material with a bandgap of $0.7	ext{ eV}$ (like Germanium) at $300	ext{ K}$, it is calculated using the mass action law formula to be approximately $2.5 	imes 10^{19}	ext{ m}^{-3}$.

Question 14

A ray of light entres from air into fiber. The refractive index of air is one…

Accepted Answer

For a step-index optical fiber with core index $n_1 = 1.5$ and cladding index $n_2 = 1.48$, the Numerical Aperture (NA) is calculated as $0.244$, the relative refractive index difference ($\Delta$) is $1.33\%$, and the acceptance angle is $14.12^{\circ}$.

Question 15

A plane transmission grating of length $6 	ext{ cm}$ has $5000 	ext{ lines/…

Accepted Answer

The resolving power of a diffraction grating measures its ability to separate closely spaced spectral lines. For a $6	ext{ cm}$ wide grating with $5000	ext{ lines/cm}$, the total number of lines is $30,000$, yielding a resolving power of $30,000$ in the first order and $60,000$ in the second order.

Question 16

If a potential function is given by the expression, $\phi = xyz$, determine t…

Accepted Answer

For a scalar potential field given by $\phi = xyz$, the gradient yields the vector field $\vec{V} = yz\hat{i} + xz\hat{j} + xy\hat{k}$. Evaluating the curl of this gradient yields identically zero, mathematically proving that the field is fundamentally irrotational (conservative).

Question 17

Calculate the angles at which the first dark band and the next bright band ar…

Accepted Answer

Using the Fraunhofer single-slit diffraction minimum condition $a \sin	heta = n\lambda$, for a slit width of $0.3	ext{ mm}$ and a sodium light wavelength of $5890	ext{ \AA}$, the angular position of the very first dark fringe ($n=1$) is calculated to be highly narrow at approximately $0.113^{\circ}$.

Question 18

In a Newton's ring arrangement with air film observed with light of wavelengt…

Accepted Answer

In Newton's rings experiment, the difference between the squares of the diameters of any two consecutive dark rings is a constant equal to $4\lambda R$. This phenomenon provides a highly accurate interferometric method for determining the unknown wavelength of monochromatic light.

Question 19

Explain the terms: Population inversion and optical pumping. Discuss with sui…

Accepted Answer

The Helium-Neon (He-Ne) laser is a highly prevalent four-level continuous-wave gas laser. It utilizes an electrical discharge to excite Helium atoms, which then collisionally transfer their energy to Neon atoms to achieve population inversion and emit a stable, coherent red beam at 632.8 nm.

Question 20

The Hall voltage for the sodium metal is $0.001 	ext{ mV}$, measured at $I =…

Accepted Answer

The Hall effect is the production of a transverse voltage across a conductor carrying current in a magnetic field. In sodium, it proves electrons are the charge carriers, and it allows precise calculation of carrier density ($n$) and mobility ($\mu$).

Question 21

State and prove Poynting theorem for the rate of flow of energy in electromag…

Accepted Answer

Poynting's theorem establishes the law of conservation of energy for electromagnetic fields. It states that the rate of energy loss from a volume equals the outward energy flux (given by the Poynting vector) plus the ohmic heat dissipated within that volume.

Question 22

Give physical significance of wave function. Derive time dependent and time i…

Accepted Answer

Schrödinger's wave equations form the mathematical foundation of non-relativistic quantum mechanics. The time-dependent equation describes how a quantum state evolves over time, while the time-independent equation determines the allowed stationary energy states of a bound system.

Engineering Physics