First Year (FE)
- Write the properties of matter waves. Properties of matter waves (de Broglie waves): Wavelength λ=h/p=h/mv : lighter particles have larger wavelengths, and so do slower ones. If... 1
- Derive the Schrodinger’s Time Dependent differential equation for matter waves A free particle moving along x is described by the wave function psi(x,t)=A e^(-i(Et-px)/hbar) using E=hbarω and p=hbar k . Step 1 (time der... 1
- Calculate velocity and de-Broglie wavelength of alpha particle of energy 1[unit prefix partly illegible in the source scan; read as k]eV. Given: mass of alpha particle as 6.68x10-27 kg. Assumption: The energy unit is partly obscured in the scan. We read it as E=1 keV. The results for 1 MeV are given at the end. Given: E=1 ke... 1
- Write Fermi Dirac distribution function and explain the terms involved in it The Fermi–Dirac distribution function gives the probability that an available energy state E is occupied by an electron at absolute temperat... 1
- Predict the effect on the electrical properties of a silicon at room temperature if every millionth silicon atom is replaced by an atom of indium. Given: Concentration of silicon atoms are 5 x 1028 /m3, intrinsic conductivity of silicon is 4.4 x 10-4… Given: N_Si=5×10²⁸ m ⁻³ , σ_i=4.4×10⁻⁴ (Ω m) ⁻¹ , μ_h=0.048 m²/V s. Indium is trivalent, so it acts as an acceptor. Replacing one Si atom in... 1
- Show that for an intrinsic semiconductor, fermi level lies midway of conduction band and valence band. In an intrinsic semiconductor, the electron concentration in the conduction band is n=N_c e^(-(E_c-E_F)/kT) and the hole concentration in th... 1
- Show that the divergence of a curl of a vector is zero. Let vec A=A_xhat i+A_yhat j+A_zhat k . nabla×vec A=Big(∂ A_z/∂ y-∂ A_y/∂ zBig)hat i+Big(∂ A_x/∂ z-∂ A_z/∂ xBig)hat j+Big(∂ A_y/∂ x-∂ A_x/∂ y... 1
- What is gradient? If \(\phi=3(x^2y-y^2x)\), find \(\nabla\phi\) at the point (1, -2, -1). Gradient: The gradient of a scalar field φ(x,y,z) is the vector nablaφ=∂φ/∂ xhat i+∂φ/∂ yhat j+∂φ/∂ zhat k Its magnitude is the maximum rate... 1
- State and explain Maxwell’s fourth equation. Statement (Ampere–Maxwell law): The line integral of the magnetic field vec H around a closed path equals the total current, conduction plus... 1
- An optical fiber has an acceptance angle of 30° and core refractive index of [value not printed in the source paper] calculate refractive index of cladding? Note: The core index is missing from the printed paper. As the paper allows, we assume n₁=1.5 (a typical glass core) and take the fibre to b... 1
- Derive an expression for numerical aperture of a step index optical fiber. Set-up: A step-index fibre has a core of index n₁ and a cladding of index n₂ ( n₁>n₂ ). Light enters the core from a medium of index n₀ (air... 1
- Compare Step Index fiber and Graded Index Fiber. PointStep-index fibreGraded-index fibre Index profileCore index n₁ is constant and drops abruptly (in a step) to n₂ at the claddingCore inde... 1
- Explain why an excessively thin film appears black in reflected light? For a thin film of index μ and thickness t , the effective path difference between the two reflected rays is Δ=2μ tcos r-λ/2 The extra λ/2 c... 1
- In a Newton’s ring arrangement, a drop of liquid of refractive index 1.33 is placed in between the lens and plate. The diameter of 9th dark ring is found to be 0.58 cm. Obtain the radius of curvature, if the wavelength of light is 6000 A°. Given: μ=1.33 , n=9 , D₉=0.58 cm =5.8×10⁻³ m, λ=6000 Å =6×10⁻⁷ m. Formula: For dark rings in reflected light with a liquid film, 2μ t=nλ and... 1
- With the experimental set up of Newton’s ring, explain the determination of refractive index of a liquid. Experimental set-up: travelling microscope | v S (sodium lamp)--> L (lens) --> G (glass plate at 45°) | reflects light down v ( plano-convex... 1
- What is metastable state? What is the role of metastable state in LASER? Metastable state: An excited energy level of an atom in which the electron stays for an unusually long time, about 10⁻³ s, compared with an... 1
- With the neat energy level diagram, explain the construction and working of He Ne Gas laser. He–Ne laser is a four-level gas laser that gives a continuous output at 6328 Å (red). Construction: discharge tube (He:Ne ≈ 10:1, ~1 mm Hg)... 1
- What is Lidar technology and how does it work? LiDAR (Light Detection And Ranging) is a remote-sensing technique. It uses pulsed laser light to measure distances and build accurate 3-D ma... 1
- State the properties of matter waves. Properties of matter waves (de Broglie waves): Wavelength λ=h/p=h/mv : lighter particles have larger wavelengths, and so do slower ones. If... 1
- An electron is bound in an 1D potential well of width 2 A° but of infinite height. Find its energy values in the ground state and in first excited state. (Given: Planck constant is 6.634x10-34 J-Sec) Given: L=2 Å =2×10⁻¹⁰ m, h=6.634×10⁻³⁴ J s, m=9.1×10⁻³¹ kg. Formula: E_n=n²h²/8mL² Ground state (n = 1): E₁=frac(6.634×10⁻³⁴)²8×9.1×10⁻³¹×(2... 1
- Prove that electron cannot reside inside the nucleus of an atom. Using Heisenberg's uncertainty principle: Δ x Δ p≥hbar/2=h/4π The diameter of a nucleus is about 10⁻¹⁴ m. If an electron were confined insid... 1
- An electron and a photon each have a wavelength of 2A°. What are their momentum and energies? Given: λ=2 Å =2×10⁻¹⁰ m, h=6.63×10⁻³⁴ J s, m_e=9.1×10⁻³¹ kg, c=3×10⁸ m/s. Momentum (same for both, since p=h/λ ): p=frac6.63×10⁻³⁴2×10⁻¹⁰=3.... 1
- Show that the minimum value of \(u=xy+a^3\left(\frac1x+\frac1y\right)\) is \(3a^2\). u_x=y-a³/x², u_y=x-a³/y² . Stationary points: x²y=a³,\ xy²=a³ . Dividing, x=y ; then x³=a³⇒ x=y=a . Point (a,a) . r=u_xx=2a³/x³,\ s=u_xy=1,\... 1
- If \(y=a\cos(\log x)+b\sin(\log x)\), prove that \(x^2y_{n+2}+(2n+1)xy_{n+1}+(n^2+1)y_n=0\). y₁=-a/xsin(log x)+b/xcos(log x)⇒ xy₁=-asin(log x)+bcos(log x) . Differentiate: xy₂+y₁=-1/x[acos(log x)+bsin(log x)]=-y/x⇒ x²y₂+xy₁+y=0 . Lei... 1