Download Question Paper (Link Below)

Question Paper Solution

1. (a) What are Einstein coefficients? Derive Einstein relation.

   Einstein's Coefficients & Relation

   (b) Explain with the help of suitable diagram the principle, Construction and working of He-Ne laser.

   Helium-Neon Laser

2. (a) Deduce Einstein's mass- energy relation $E = mc^2$ . Give some evidence showing its validity.

   Einstein's Mass- Energy Relation

   (b) Explain time dilation and length contraction.

   Time dilation ,Length contraction

3. (a) Obtain expression for energy of a simple harmonic oscillator and show that total energy of oscillator remains constant.

   Simple Harmonic Motion

   (b) A particle executes S.H.M. motion of period 31.4 seconds and amplitude 5 cm. Calculate its maximum velocity and maximum acceleration.

   For a particle executing simple harmonic motion (S.H.M), the maximum velocity and maximum acceleration can be calculated using the following formulas:

   1. Maximum Velocity $v_{\text{max}}$
   2. Maximum Acceleration $a_{\text{max}}$

   Given:

   • Period $T = 31.4$ seconds
   • Amplitude $A = 5 cm$ = 0.05 m

   Maximum Velocity - The maximum velocity $v_{\text{max}}$ is given by:

   $$v_{\text{max}} = A \omega$$

   where $\omega$ (angular frequency) is:

   $$\omega = \frac{2\pi}{T}$$

   Substituting the given period $T$:

   $$\omega = \frac{2\pi}{31.4} \approx 0.2 \, \text{rad/s}$$

   Now, calculating $v_{\text{max}}$:

   $$v_{\text{max}} = 0.05 \times 0.2 = 0.01 \, \text{m/s}$$

   Maximum Acceleration - The maximum acceleration $a_{\text{max}}$ is given by:

   $$a_{\text{max}} = A \omega^2$$

   Using the previously calculated $\omega$:

   $$a_{\text{max}} = 0.05 \times (0.2)^2$$ $$a_{\text{max}} = 0.05 \times 0.04 = 0.002 \, \text{m/s}^2$$

4. (a) Explain basic structure of an optical fiber with suitable diagram. Explain function of each block.

   Optical Fibre Structure

   Function of each block of optical fibre:-

   • Core
     • Function: The core is the central part of the optical fiber through which light signals are transmitted. It carries the light signals from the transmitter to the receiver.
     • Material: Typically made of high-purity glass or plastic with a high refractive index.
     • Importance: The core's refractive index is higher than that of the cladding, allowing for total internal reflection, which keeps the light within the core as it travels down the fiber.
   • Cladding
     • Function: The cladding surrounds the core and has a lower refractive index. This difference in refractive indices between the core and the cladding ensures that light signals are confined within the core by the process of total internal reflection.
     • Material: Made of glass or plastic with a slightly lower refractive index than the core.
     • Importance: The cladding's role is crucial for guiding the light along the fiber and minimizing signal loss. It helps maintain the integrity of the signal over long distances.
   • Sheath (Outer Jacket)
     • Function: The sheath, or outer jacket, is the protective layer that encases the cladding. It shields the optical fiber from physical damage, environmental factors, moisture, and chemical exposure.
     • Material: Typically made of durable plastic or other protective materials.
     • Importance: The sheath provides mechanical protection and flexibility, allowing the fiber to be installed in various environments without damage. It also prevents abrasion and enhances the fiber's durability.

   (b) What are the various modes of an optical fiber? Give their importance and applications.

   Modes of Optical Fiber ,

   Importance and Their Applications

5. (a) Discuss uncertanity principle. How does it explain the absence of electrons inside the nucleus?

   Uncertainity Principle

   Non-Existence of Electron in the Nucleus

   The radius of the nucleus of an atom is of the order of 10–14 m. If an electron is confined within the nucleus, the uncertainty in its position must not be greater than 10–14 m. According to uncertainty principle for the lowest limit of accuracy

   $$\Delta x \Delta p = \frac{h}{2\pi} \quad \text{(i)}$$

   where $\Delta x$ is uncertainty in the position and $\Delta p$ is the uncertainty in the momentum.

   From Eq. (i),

   $\Delta p = \frac{h}{2\pi \Delta x} = \frac{6.625 \times 10^{-34}}{2 \times 3.14 \times 2 \times 10^{-14}}$

   $$\text{(as } \Delta x = \text{diameter of nucleus)}$$ $$\Delta p = 5.275 \times 10^{-21} \, \text{kg m/sec}$$

   This is the uncertainty in the momentum of the electron. It means the momentum of the electron would not be less than $\Delta p$, rather it could be comparable to $\Delta p$. Thus

   $$p = 5.275 \times 10^{-21} \, \text{kg m/sec}$$

   The kinetic energy of the electron can be obtained in terms of momentum as

   $$T = \frac{1}{2} mv^2 = \frac{p^2}{2m}$$ $$= \frac{(5.275 \times 10^{-21})^2}{2 \times 9.1 \times 10^{-31}} \, \text{J}$$ $$= \frac{(5.275 \times 10^{-21})^2}{2 \times 9.1 \times 10^{-31} \times 1.6 \times 10^{-19}} \, \text{eV}$$ $$= 95.55 \times 10^6 \, \text{eV}$$ $$\approx 96 \, \text{MeV}$$

   From the above result, it is clear that the electrons inside the nucleus may exist only when it possesses the energy of the order of 96 MeV. However, the maximum possible kinetic energy of an electron emitted by radioactive nuclei has been found about 4 MeV. Hence, it is concluded that the electron cannot reside inside the nucleus.

   (b) Define group velocity and phase velocity. Obtain relation between them.

   Phase Velocity ,

   Group Velocity , Relation between Phase and Group velocity

6. (a) What are X-rays? Explain the production and properties of X-rays.

   X-rays , Production of X-rays , Properties of X-rays

   (b) An X-ray tube operates on 80V. Find the maximum speed of electron with which it strikes the target.

   To find the maximum speed of the electrons, follow these steps:

   1. Calculate the Kinetic Energy (KE):

      The kinetic energy of the electrons is given by:

      $$KE = eV$$

      where $e$ is the charge of an electron $e \approx 1.602 \times 10^{-19}$ $coulombs$ and $V$ is the voltage (80 V).

      Therefore:

      $KE = (1.602 \times 10^{-19} \, \text{C}) \times (80 \, \text{V})$

      $$= 1.2816 \times 10^{-18} \, \text{J}$$

   2. Relate Kinetic Energy to Speed:

      The kinetic energy of an electron can also be expressed as:

      $$KE = \frac{1}{2} mv^2$$

      where $m$ is the mass of the electron $m \approx 9.109 \times 10^{-31}$ kg and $v$ is the speed of the electron.

      Rearranging for $v$:

      $$v = \sqrt{\frac{2 \cdot KE}{m}}$$

   3. Substitute the Values:

      $$v = \sqrt{\frac{2 \cdot 1.2816 \times 10^{-18} \, \text{J}}{9.109 \times 10^{-31} \, \text{kg}}}$$ $$v = \sqrt{\frac{2.5632 \times 10^{-18}}{9.109 \times 10^{-31}}}$$ $$v \approx \sqrt{2.81 \times 10^{12}}$$ $$v \approx 1.68 \times 10^6 \, \text{m/s}$$

   So, the maximum speed of the electrons striking the target is approximately $1.68 \times 10^6$ meters per second.

7. (a) Using Maxwell's equations, obtain electromagnetic wave equation in vacuum.

   Wave equation in Vacuum

   (b) What is displacement current? Obtain expression for it.

   Displacement Current

   Expression for Displacement Current

   To derive the expression for the displacement current density, follow these steps:

   1. Starting from Ampère's Law (with Maxwell's Correction):

      Ampère's Law in its modified form (with Maxwell's correction) is:

      $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

      Here, $\mathbf{J}$ is the conduction current density, and the term $\mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$ represents the displacement current density.

   2. Relate Displacement Current Density to Electric Field:

      The displacement current density $\mathbf{J}_D$ is given by:

      $$\mathbf{J}_D = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

      where $\epsilon_0$ is the permittivity of free space.

   3. Integral Form for Displacement Current:

      The total displacement current $I_D$ through a surface $S$ is obtained by integrating the displacement current density over the surface:

      $$I_D = \int_S \mathbf{J}_D \cdot d\mathbf{A}$$

      Substituting $\mathbf{J}_D = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$:

      $$I_D = \int_S \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \cdot d\mathbf{A}$$ $$I_D = \epsilon_0 \frac{\partial}{\partial t} \int_S \mathbf{E} \cdot d\mathbf{A}$$

      The term $\int_S \mathbf{E} \cdot d\mathbf{A}$ is the electric flux $\Phi_E$ through the surface $S$ is:

      $$\Phi_E = \int_S \mathbf{E} \cdot d\mathbf{A}$$

      Therefore:

      $$I_D = \epsilon_0 \frac{\partial \Phi_E}{\partial t}$$

8. (a) What is superconductivity ? Discuss BCS theory of superconductivity.

   Superconductivity , BCS Theory

   (b) Distinguish between type I and type II superconductors.

   Types of Superconductors

9. (a) Why wave nature of matter is not apparent in our daily life observations?

   The wave nature of matter is not apparent in our daily life observations primarily because the wavelengths associated with macroscopic objects are extraordinarily small. This can be explained using the de Broglie wavelength, which is given by the formula:

   $$\lambda = \frac{h}{p}$$

   (b) what are Bremsstrahlung radiations?

   Bremsstrahlung radiations

   (c) Why should wave function be single valued everywhere?

   The wave function must be single-valued everywhere to ensure the consistency and physical meaningfulness of quantum mechanics.

   • Continuity and Smoothness: A single-valued wave function ensures smooth and continuous changes in physical quantities like position and momentum, which aligns with our physical expectations of these properties.

   • Unique Solutions to Schrödinger Equation: The Schrödinger equation, a fundamental equation in quantum mechanics, requires a single-valued wave function to yield unique and physically meaningful solutions.

   • Boundary Conditions and Normalization: Single-valuedness is essential for the wave function to meet boundary conditions and for the probability to be normalized (i.e., the total probability of finding the particle somewhere in space is 1).

   (d) Define Poynting vector. What are its units?

   Poynting vector

   (e) Why light waves travel through vacuum, whereas sound waves cannot?

   Light waves can travel through a vacuum because they are electromagnetic waves that do not require a medium to propagate. Sound waves cannot travel through a vacuum because they are mechanical waves that require the vibration of particles in a medium to transmit their energy.

   (f) Explain whether earth is inertial or non-inertial frame of reference.

   Is earth an inertial frame of reference?

   (g) What is transition temprature in superconductivity?

   In superconductivity, the transition temperature, often denoted as $T_c$, is the critical temperature below which a material transitions from a normal conductive state to a superconducting state. In the superconducting state, the material exhibits zero electrical resistance and expels magnetic fields (the Meissner effect).

   Key Characteristics of Transition Temperature ($T_c$):

   1. Zero Electrical Resistance: Below $T_c$, a superconductor allows the flow of electric current without any resistance. This means that an electric current can persist indefinitely without an external power source.

   2. Meissner Effect: When a material becomes superconducting below $T_c$, it expels magnetic fields from its interior, a phenomenon known as the Meissner effect. This is a definitive characteristic of superconductors.

   3. Material Dependence: The value of $T_c$ varies widely among different materials. Some materials have very low transition temperatures (close to absolute zero), while others (such as high-temperature superconductors) can have $T_c$ values above the boiling point of liquid nitrogen (77 K).

   4. Second-Order Phase Transition: The transition from the normal state to the superconducting state is a second-order phase transition. This means there is a smooth, continuous change in properties such as specific heat and magnetic susceptibility at $T_c$.

   (h) Explain single mode and multi-mode fibers?

   Singlemode fibers , Multimode fibers

   (i) At what displacement from mean position, the total energy of a particle is half kinetic energy and half potential energy.

   For a particle in simple harmonic motion (SHM), the total energy $E$ is conserved and is the sum of its kinetic energy $K$ and potential energy $U$. The total energy in SHM is given by:

   $$E = \frac{1}{2} k A^2$$

   where:

   • $k$ is the spring constant,
   • $A$ is the amplitude of the motion.

   The kinetic energy $K$ and potential energy $U$ at a displacement $x$ from the mean position are given by:

   $$K = \frac{1}{2} k (A^2 - x^2)$$ $$U = \frac{1}{2} k x^2$$

   We want to find the displacement $x$ where the kinetic energy $K$ and potential energy $U$ are equal, each being half of the total energy:

   $$K = U = \frac{1}{2} E$$

   Given that:

   $$E = K + U = \frac{1}{2} k A^2$$

   When $K = U$:

   $$\frac{1}{2} k (A^2 - x^2) = \frac{1}{2} k x^2$$

   Simplifying this equation:

   $$A^2 - x^2 = x^2$$ $$A^2 = 2x^2$$ $$x^2 = \frac{A^2}{2}$$ $$x = \frac{A}{\sqrt{2}}$$

   Therefore, the displacement $x$ from the mean position where the total energy is equally divided between kinetic energy and potential energy is:

   $$x = \frac{A}{\sqrt{2}} \approx 0.707A$$

   This is approximately 70.7% of the amplitude $A$.

   (j) What do you mean by inductive coupling?

   Inductive coupling refers to the process by which energy is transferred between two electrical circuits through mutual inductance. This occurs when the magnetic field created by a current flowing through one coil induces a voltage in a nearby coil. It's a fundamental concept in electromagnetism and is widely used in various applications.

   The voltage induced in the secondary coil ($V_s$) is related to the rate of change of current in the primary coil ($I_p$) by:

   $$V_s = - M \frac{dI_p}{dt}$$

   Where $M$ is the mutual inductance between the two coils.