GTU BE01000021 Physics Summer 2026 Solved Paper & Solutions

The stress-strain diagram shows how a wire stretches under a growing load. Stress is the force per unit area, and strain is the change in length per unit length.

• O to P (proportional limit): the graph is a straight line. Stress is proportional to strain, which is Hooke's law.
• Up to E (elastic limit): the material goes back to its original size when the load is removed.
• Y (yield point): the strain grows quickly with little extra stress, and the deformation becomes permanent (plastic).
• U (ultimate tensile strength): the highest stress that the material can bear.
• F (fracture point): the wire narrows (necking) and breaks.

• (i) Young's modulus $Y$: the ratio of the longitudinal stress to the longitudinal strain, within the elastic limit. $Y = \dfrac{F/A}{\Delta l/l}$. Unit: N/m².
• (ii) Bulk modulus $K$: the ratio of the volume stress (pressure) to the volume strain. $K = \dfrac{-\Delta P}{\Delta V/V}$. It tells how hard a material is to compress. Unit: N/m².
• (iii) Shear modulus (rigidity modulus) $\eta$: the ratio of the shear stress to the shear strain. $\eta = \dfrac{F/A}{\theta}$, where $\theta$ is the angle of shear. Unit: N/m².
• (iv) Poisson's ratio $\sigma$: the ratio of the lateral strain to the longitudinal strain. $\sigma = \dfrac{\text{lateral strain}}{\text{longitudinal strain}}$. It is a pure number with no unit, and its value for most materials is between 0 and 0.5.

A cantilever is a beam or a bar that is fixed (clamped) at one end and free at the other end. Examples are a diving board and a balcony.

Setup

A cantilever of length $l$ is fixed at $A$, and a load $W$ hangs at the free end $B$. Let $Y$ be the Young's modulus of the material and $I_g$ the geometrical moment of inertia of the cross section. Take $x$ along the beam from the fixed end, and $y$ as the depression downwards.

Derivation

Integrate again. At $x = 0$, $y = 0$.

For a rectangular beam of breadth $b$ and depth $d$, $I_g = \dfrac{bd^3}{12}$, so $\delta = \dfrac{4Wl^3}{Ybd^3}$. For a circular rod of radius $r$, $I_g = \dfrac{\pi r^4}{4}$, so $\delta = \dfrac{4Wl^3}{3\pi r^4 Y}$.

Reverberation is the persistence of sound in a hall after the source has stopped. The reverberation time $T$ is the time in which the sound intensity falls to one-millionth of its first value.

Sabine's formula:

Here $V$ is the volume of the hall in m³, $S$ is the area of each surface in m², and $a$ is its absorption coefficient. The sum is over all the surfaces of the hall.

Usage: the formula is used to find the reverberation time of a hall, and to decide how much absorbing material is needed to give a good value for speech or music.

Given: frequency $f = 0.09$ MHz $= 9 \times 10^4$ Hz, time of the round trip $t = 0.55$ s, velocity $v = 1800$ m/s.

Ultrasonic flaw detection is a non-destructive testing (NDT) method that finds cracks, voids and other hidden defects inside a material, without damaging it. It uses ultrasonic waves of 0.5 to 10 MHz.

Principle

An ultrasonic pulse sent into a material is reflected by a boundary. A defect inside is a boundary, so it reflects part of the pulse, and the rest goes on to the back wall. This is the pulse-echo method.

Working

1. A piezoelectric transducer is placed on the surface of the specimen, with a thin layer of oil or gel (the couplant) between them to remove air.
2. A pulse generator gives short pulses, and the transducer sends them into the material. It also receives the echoes.
3. The echoes are amplified and shown on an oscilloscope screen (CRO). The first peak is the initial pulse, then a peak from the defect, and the last from the back wall.
4. The time between the pulse and the echo gives the depth of the defect: $d = \dfrac{vt}{2}$. A defect gives a peak between the first and the last peak.

Advantages

• It finds internal defects, and gives their depth and size.
• It has a high sensitivity, and needs access to one side only.
• It is safe, because there is no harmful radiation, and the equipment is portable.

Disadvantages

• It needs a skilled operator to read the signals.
• It is difficult for rough, thin or complex shapes.
• A couplant is needed.
• A defect that lies parallel to the beam may not be detected.

Interference is the superposition of two or more coherent waves in the same region, which gives a new pattern of intensity: maximum at some points and minimum at others.

• Constructive interference: the waves meet in the same phase. The path difference is $n\lambda$ ($n = 0, 1, 2, \dots$), and the phase difference is $2n\pi$. The resultant amplitude is the sum, $a_1 + a_2$, and the intensity is maximum (a bright fringe).
• Destructive interference: the waves meet in opposite phase. The path difference is $(2n+1)\dfrac{\lambda}{2}$, and the phase difference is $(2n+1)\pi$. The amplitude is $a_1 - a_2$, and the intensity is minimum (a dark fringe).

The Michelson interferometer is an instrument that makes interference by division of amplitude of a light beam.

Construction

• Source: an extended monochromatic source.
• Beam splitter $G_1$: a glass plate, half-silvered on its back, set at $45^\circ$ to the beam.
• Compensating plate $G_2$: a glass plate of the same thickness, parallel to $G_1$. It makes the two paths equal in glass.
• Mirrors $M_1$ and $M_2$: $M_1$ is movable with a micrometer screw, and $M_2$ is fixed, and it has tilting screws.
• Telescope or screen to view the fringes.

Working

1. The beam from the source falls on $G_1$. It is divided into a reflected beam and a transmitted beam.
2. The reflected beam goes to $M_2$, and the transmitted beam goes to $M_1$. Both are reflected back, and they meet again at $G_1$.
3. The two beams are coherent, since they come from the same source. They interfere, and the fringes are seen in the telescope.
4. If $M_1$ and $M_2$ are exactly perpendicular, the fringes are circular. If they are slightly tilted, the fringes are straight. The path difference is $2d\cos\theta$, where $d$ is the distance between $M_1$ and the image of $M_2$.

• Bright fringe: $2d\cos\theta = n\lambda$.
• Dark fringe: $2d\cos\theta = (2n+1)\dfrac{\lambda}{2}$.

5. If $M_1$ is moved by $\dfrac{\lambda}{2}$, one fringe moves across the field of view. So by counting $m$ fringes for a distance $d$, the wavelength is

Uses

The interferometer is very sensitive. It can measure a movement of a small fraction of a wavelength. It is used to measure wavelength, the thickness of thin films, the refractive index, and the difference of wavelength of close lines. It is also used to measure small distances very accurately.

A black body is an ideal body that absorbs all the radiation that falls on it, at all wavelengths. When heated, it emits radiation called black-body radiation, and the spectrum depends only on its temperature and not on its material.

• The intensity at each wavelength rises with temperature, and the peak moves to shorter wavelengths (Wien's displacement law, $\lambda_m T =$ constant).
• The total energy is proportional to $T^4$ (Stefan's law).

Classical physics (Rayleigh-Jeans law) failed at short wavelengths, which is the ultraviolet catastrophe. Planck explained the spectrum by assuming that energy is emitted in packets (quanta) of $E = h\nu$.

An energy distribution function tells how the particles of a system, in thermal equilibrium at temperature $T$, are shared among the available energy states. It gives the probability $f(E)$ that a state of energy $E$ is occupied.

There are three main distribution functions.

Here $k$ is the Boltzmann constant, $T$ is the absolute temperature and $E_F$ is the Fermi energy. At high energies, both quantum functions become the same as the Maxwell-Boltzmann function. The Fermi-Dirac function follows the Pauli exclusion principle, so it never exceeds 1. It is used for the free electrons in a metal.

The time-independent Schrodinger equation gives the wave function $\psi$ of a particle whose total energy $E$ does not change with time.

Derivation (one dimension)

This is the Schrodinger time-independent equation in one dimension.

or in the form $\hat{H}\psi = E\psi$, where $\hat{H} = -\dfrac{\hbar^2}{2m}\nabla^2 + V$ is the Hamiltonian.

Laser stands for Light Amplification by Stimulated Emission of Radiation. It is a source of an intense, coherent and highly directional beam of light.

• Spontaneous emission: an atom in an excited state $E_2$ comes down to a lower state $E_1$ by itself, after a short time, and emits a photon of energy $h\nu = E_2 - E_1$. The photons from different atoms have random phase and direction, so the light is not coherent.
• Stimulated emission: an incoming photon of energy $h\nu = E_2 - E_1$ forces an excited atom to come down and emit a second photon. The new photon has the same phase, frequency, direction and polarisation as the first one. So the light is amplified and coherent.

Science

• In spectroscopy and in the study of atoms and molecules.
• To measure long distances (the distance to the Moon, and LIDAR), and in holography (three-dimensional images).

Engineering

• To cut, weld and drill metals and other hard materials with great accuracy.
• In optical fibre communication, to carry data.
• In barcode scanners, laser printers, CD and DVD players.
• In surveying and for measuring distances.

Medicine

• In eye surgery (LASIK) to correct vision, and to treat the retina.
• To remove tumours, and for bloodless surgery, since the laser seals small blood vessels.
• In dentistry and in skin treatment, such as the removal of marks.

A PN junction diode is a semiconductor device formed by joining a p-type semiconductor (many holes) and an n-type semiconductor (many free electrons).

Formation of the junction

Electrons from the n-side diffuse into the p-side, and holes from the p-side diffuse into the n-side. They recombine near the junction. This leaves a layer without free carriers, called the depletion region, which has fixed positive ions on the n-side and negative ions on the p-side. It produces an electric field and a barrier potential, about $0.7$ V for silicon and $0.3$ V for germanium.

Working

• Forward bias: the p-side is connected to the positive terminal. The external voltage opposes the barrier, so the depletion region becomes narrow. When the voltage is above the barrier (the knee voltage), a large current flows.
• Reverse bias: the p-side is connected to the negative terminal. The depletion region becomes wider, and only a very small reverse saturation current (due to minority carriers) flows.
• If the reverse voltage is made very large, the diode reaches breakdown, and the current rises suddenly.

I-V characteristics

• In the forward direction, the current stays very small up to the knee voltage and then rises steeply.
• In the reverse direction, the current is almost constant and very small, until breakdown.

The diode current follows

where $I_0$ is the reverse saturation current. A diode conducts in one direction only, so it is used as a rectifier.