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B Q.1. A cylindrical capacitor of length L is formed by two coaxial cylindrical shells of radii a and b, (b>a) each carrying a charge of magnitude q. Assume L>>b, to neglect the fringing of electric field. The Capacitance is given as a) C= 4 0 ab L b a b) C= 4 0 ab L c) C= qL b ln 2 0 a d ) C= 2 0 L ln b a qL a-b e) C= 4 0 ab 2 Q.2. An isolated spherical conductor of radius R carries a surface charge density σ. The potential energy of the conductor would be equal to (A) 4𝜋𝑅 2 𝜎⁄ 𝜀𝑜 (B) 𝑅 2 𝜎⁄ 4𝜋𝜀𝑜 (C) 2𝜋𝑅 2 𝜎 2⁄ 𝜀𝑜 (D) 4𝜋𝑅𝜎⁄ 𝜀𝑜 (E) 4𝜋𝑅 2 𝜎 2⁄ 𝜀𝑜 Q.3. A circular loop lying in the xy plane has a radius of 4. The loop experiences a magnetic field 𝑩 = 𝐵𝑜 𝑐𝑜𝑠(𝜔𝑡)𝑧̂ + 𝐵𝑜 𝑠𝑖𝑛(𝜔𝑡)𝑦̂ . The total induced electromotive force, (EMF), would be equal to (A) 4𝜋𝐵𝑜 𝑐𝑜𝑠(𝜔𝑡)𝜔 (B) 4𝜋𝐵𝑜 𝑠𝑖𝑛(𝜔𝑡)𝜔 (C) 16𝜋𝐵𝑜 𝑠𝑖𝑛(𝜔𝑡)𝜔 (D) 16𝜋𝐵𝑜 𝜔 (−𝑐𝑜𝑠(𝜔𝑡) + 𝑠𝑖𝑛(𝜔𝑡)) (𝐸) 16𝜋𝐵𝑜 𝑐𝑜𝑠(𝜔𝑡)𝜔 1 B Q.4. Two long straight parallel wires, separated by a distance, carry currents in opposite directions, along ±𝑧 . Identifying the wires as left wire and right wire, the two wires experience forces such that (A) the force on both wires is to the right. (B) the force on both wires is to the left. (C) the left wire moves up while the right wire moves down. (D) the force on the left wire is to the right while on the right wire is to the left. (E) the force on the left wire is to the left while on the right wire is to the right. Q.5. A long cylindrical solid rod of radius R carries a uniform current I along its length parallel to the zaxis. The magnetic field B at a perpendicular distance r from the axis of the rod will be equal to (A) 𝜇𝑜 𝐼𝑟⁄ and parallel to 𝑧̂ z direction 2𝜋𝑅 2 (B) 𝜇𝑜 𝐼𝑅 ⁄ and parallel to 𝜑̂ direction 2𝜋𝑟 2 (C) 𝜇𝑜 𝐼𝑟 2⁄ and parallel to 𝑧̂ direction 2𝜋𝑅 2 (D) 𝜇𝑜 𝐼𝑅 ⁄ and parallel to 𝑧̂ direction 2𝜋𝑟 2 (E) 𝜇𝑜 𝐼𝑟⁄ and parallel to 𝜑̂ direction 2𝜋𝑅 2 Q .6. A satellite orbiting the planet Mars has a mass m and travels in an approximately circular orbit of radius R and period T. The mass of Mars as calculated from this information will be: (A) 4π2R3/T2G (B) 4πR2/T3G (C) mR2/G (D) 4πmR2/G (E) 4πR2/mT3G 2 B Q .7. A projectile weighing one Newton is fired vertically upward. It could reach a height of one kilometer if there was no air drag. Due to air drag the projectile dissipates 600 Joules during its ascent. What is the height attained by the same projectile in the presence of the air drag? (A) 940 m (B) 600 m (C) 400 m (D) 60 m (E) 450 m Q.8. A boy of mass 40 kg stands on a wooden log of mass 60 kg. The log is lying on the frictionless floor of a room. The boy walks along the log at 2 m/s in the direction away from the door of the room. How fast does the log move with respect to the door? (A) at 2 m/s towards the door (B) at 1.63 m/s towards the door (C) at 1.63 m/s away from the door (D) at 1.33 m/s towards the door (E) at 1 m/s towards the door Q.9. A mass undergoes Simple Harmonic Motion with amplitude A. What fraction of the total energy is kinetic, when the displacement is x =A/2? (A) 1/2 (B) 3/4 (C) 2/3 (B) 1/4 (C) none of the above 3 B Q .10. A disk of radius R and mass M is mounted with its axis vertical. It is initially at rest. A bullet of mass m and velocity v is fired horizontally and tangential to the disk. It lodges in the perimeter of the disk. What angular velocity will the disk acquire? (A) mv/[(M+2m)R] (B) mv/[(2M+m)R] (C) Mv/mR (D) mv/MR (E) (2M+m)v/MR Q.11. n moles of an ideal gas at an initial temperature To and volume Vo are compressed isothermally to a volume Vo/2 . Which of the following statements is true? (A) An amount of heat equal to 2nkBTo is absorbed from the atmosphere. (B) An amount of heat equal to 2nkBTo is released to the atmosphere. (C ) An amount of heat equal to 2nkBTo ln.2 is absorbed from the atmosphere. (D) An amount of heat equal to 2nkBTo ln.2 is released to the atmosphere. (E) There is no exchange of heat with the atmosphere. Q.12. A reversible heat engine takes in heat at the temperatures of 227oC and releases heat to a heat sink at 27oC. The efficiency of the heat engine is equal to (A) 0.12 (B) 0.88 (C ) 0.4 (D) ln.4. (E) 0.6 4 B Q.13. Which one of the following statements explains why interference patterns are not usually observed for light from ordinary sources? (A) Diffraction effects predominate. (B) The two sources are out of phase. (C ) The interference pattern is too small to observe. (D) Light from ordinary light bulbs is not polarized. (E) The two sources are not coherent. Q.14. A ball hanging from a vertical spring oscillates in a simple harmonic motion with an angular frequency of 2.6 radians/sec and an amplitude of 0.075 m. What is the maximum acceleration of the ball? (A) 0.13m/s2 (B) 0.20 m/s2 (C ) 0.51 m/s2 (D) 2.6m/s2 (E) 35 m/s2 Q.15. A transparent film (n=1.4) is deposited on a glass lens (n=1.5) to form a non-reflective coating. What thickness of the film would prevent reflection of light with wavelength 500nm in air? (A) 89 nm (B) 125nm (C) 170nm. (D) 250nm (E) 357nm 5 B Q.16. For x 1, which of the following series expansion of ln 1 x is correct: (A) 1 x 2 x3 x 4 ... 2! 3! 4! (B) x x 2 x3 x 4 ... 2 3 4 x3 x5 x 7 (C) x ... 3! 5! 7! (D) x x3 x5 x7 ... 3 5 7 (E) 1 2x2 4x4 6x6 ... 2! 4! 6! Q.17. The solution of differential equation y 2 xy2 0 with y0 1 , on {-1,1} is given by (A) y x x 1 3 (B) y 2e x x 2 (C) y sinh (D) y x x 1 2 (E) y ln x Q.18. Find the solution of differential equation y 4 y 0 ; with the boundary conditions y (0) 1 and y 2 2 , if the general solution to the differential equation is given by y C1 cos(2x) C2 sin( 2x), (A) C1 C2 1 (B) C1 1 and C2 2 (C) C1 1 and C2 3 2 (D) C1 and C2 0 (E) The solution does not exist. 6 B Q.19. If a vector V is defined as 𝑽 = (2𝑥𝑧 + 3𝑦 2 )𝑗̂ + (4𝑦𝑧 2 )𝑘̂, then 𝛁 × 𝑽 would be equal to (A) (4𝑧 2 − 2𝑥)𝑖̂ + 2𝑧𝑘̂ (B) (2𝑥 + 6𝑦)𝑖̂ + 8𝑦𝑧 𝑘̂ (C) 𝑥 𝑖̂ + 𝑦𝑗̂ + 𝑧 𝑘̂ (D) 6𝑦 𝑖̂ + 𝑦 𝑘̂ (E) None of the above If a complex number Z is defined as Z= 2+ 2i then it can be expressed in the polar form as Z= r eiθ where r and θ are equal to Q.20. (A) r =4, θ= 30o (B) r =4, θ= 45o (C ) r = r =2√2, θ= 45o (D) r = 2, θ= 30o (E) r = 2, θ= 60o Q.21. The ratio of the energies of the n=5 and n=3 levels, (E5/E3) of the electron in the H atom in the Bohr model is equal to (A) 9/25 (B) 3/5 (C) 5/3 (D) 25/9 (E) 27/125 Q. 22. How many times does an electron in an H atom go around the first orbit (n=1) in a second? (A) (B) (C) (D) (E) 4𝜋 2 𝑚𝑟 2 ℎ ℎ 4𝜋 2 𝑚𝑟 2 ℎ 2𝜋 𝑚𝑟 2𝜋 𝑚𝑟 ℎ ℎ 2𝜋 2 𝑚𝑟 2 7 B Q.23. The potential difference applied on an X-ray tube is increased. As a result, in the emitted radiation (A) The intensity increases (B) The minimum wavelength increases (C) The intensity decreases (D) The minimum wavelength decreases (E) The minimum wavelength remains unchanged Q.24. A particle of charge q and mass m has been accelerated from rest through the potential difference V . Find an expression of its de-Broglie wavelength (A) (B) ℎ 2𝑚𝑞 ∆𝑉 ℎ √2𝑚𝑞 ∆𝑉 ℎ (C) √2𝑚𝑞 ∆𝑉 (D) 2𝑚𝑞 ∆𝑉 ℎ (E) √2ℎ𝑚𝑞 ∆𝑉 Q.25. A rigid bar has a length of 1m when at rest with respect to an observer. When the bar is moving at a speed 𝑣 = 𝑐⁄ parallel to its length, its length with respect to the stationery observer will √2 be equal to, (A) √2 (B) 1 (C) 1⁄ √2 (D) 1 2 (E) 1- √2 8