NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF PHYSICS ADVANCED PLACEMENT TEST (SAMPLE)
NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF PHYSICS ADVANCED PLACEMENT TEST (SAMPLE) PC1143 PHYSICS III MMM-YYYY----Time Allowed: 2 Hours INSTRUCTIONS TO CANDIDATES 1. This paper contains 5 short questions in Part I and 3 long questions in Part II. It comprises 9 printed pages. 2. The total marks for Part I is 40 and that for Part II is 60. 3. This is a CLOSED BOOK test. 4. Non-Programmable calculator is allowed to be used. 5. Answer ALL the questions. 6. Answers to the questions are to be written in the answer booklets. 7. A table of constants and mathematical formulae is attached. 2 PART I This part of the examination paper contains five short-answer questions on pages 2 to 4. Answer ALL questions. Question 1 Consider the box-like Gaussian surface G as shown in Figure 1. y y2 x2 x1 z1 x z1 z2 z2 z x1 x2 Figure 1 The bottom face is in the xz-plane, while the top face is in the horizontal plane passing through y2 2.00 m . Suppose x1 1.00 m , x2 4.00 m , z1 1.00 m , and z2 3.00 m , and G lies in an electric field given by E [(6.00 x 10.0)iˆ 3.00 ˆj zkˆ] N C , with x and z in metres and is a constant. If G encloses a net charge of 48.00 C , find . [8] Question 2 (a) Consider a particle P of mass m and positive charge q moving with velocity v in a uniform static magnetic field B . (i) Describe the magnitude and direction of the magnetic force that B exerts on P. [3] (ii) Can B do work on P? Explain briefly. [2] 6 (b) A source injects an electron of speed v 1.5 10 m s into a uniform magnetic field of magnitude B 1.0 102 T . The velocity of the electron makes an angle 30 with the direction of the magnetic field. Find the distance d from the point of injection at which the electron next crosses the field line that passes through the injection point. [3] 3 Question 3 Figures 2 and 3 show two circuits, each with R 14.0 , C 6.20 F , L 56.0 mH , and a battery having emf E 34.0 V . E R C R E L a a b b C L Figure 2 Figure 3 The switch in Figure 2 is kept in position a for 2C s , while the switch in Figure 3 is kept in position a for 2 L s . The switches are then separately thrown to their respective position b’s. Here, C and L are the capacitive and inductive time constants respectively. Find (a) C and L . [2] (b) the angular frequency of the resulting oscillations in each circuit. [1] (c) the current in each circuit 1.20 ms after the switches are being thrown to their respective position b’s. [5] Question 4 A series R-L-C circuit (Figure 4) consists of a resistor R 100 , an inductor L 0.15 H , and a capacitor C 30 F connected in series to an 120 V (rms) emf source. The frequency of the source is 60 Hz. Figure 4 Calculate (a) the current amplitude. (b) the phase angle . (c) the average power loss. [4] [2] [2] 4 Question 5 A plane electromagnetic wave travelling in the positive direction of an x-axis in vacuum has electric field components Ex E y 0 and Ez Emax cost kx , with Emax 2.0 V m and 1015 s 1 . (a) Find k. [1] (b) What is the magnetic field associated with the wave? [3] (c) Calculate the wave intensity. [2] 2 (d) The wave uniformly illuminates a surface of area 2.0 m . If the surface totally absorbs the wave, calculate the radiation pressure and the magnitude of the corresponding force on the surface. [2] END OF PART I 5 PART II This part of the examination paper contains three long questions on pages 5 to 9. Answer ALL questions. Question 6(a) Positive electric charge Q is uniformly distributed along a non-conducting rod R with length L, lying along the x-axis between x 0 and x L , as shown in Figure 5. z P z y L L2 L2 x Figure 5 Consider a point P that lies on a line, which goes through a point on the xy-plane with coordinates x L 2 and y L 2 , and is parallel to the z-axis. P is at a height z above the xy-plane. Show that the electric potential V at P is given by 2 L2 4 z 2 L Q . V z ln 2 2 4 0 L 2 L 4 z L [7] 6 Question 6(b) Now, we have four such rods as R, that are arranged into a square as shown in Figure 6. z P z y L L2 L2 x Figure 6 Find the electric field E at point P. Explain clearly how you obtain your answer. [9] Question 6(c) If an electron of mass m is placed at the centre of the square and is then displaced a small distance z along the vertical dashed line as shown in Figure 6 z L , determine its oscillating frequency. You may ignore gravity. [4] 7 Question 7(a) Consider a positive point charge q moving with velocity v as shown in Figure 7. P q v Figure 7 Describe the magnitude and direction of the magnetic field at point P due to q. [6] Question 7(b) Hence, derive the infinitesimal magnetic field dB at point P due to a current element Idl as shown in Figure 8. [6] P Idl Figure 8 Question 7(c) Derive an expression for the magnetic field strength at distance d from the centre of a straight wire of finite length L that carries current I. You have to provide a diagram of the straight wire with clear labels of all the quantities you use in your derivation. [8] Question 8(a) (i) A wire with resistivity carries current iC . The current is increasing at the rate diC dt . Show that there is a displacement current iD in the wire equal to 0 diC dt . [4] (ii) Consider a long, straight silver wire with resistivity 1.62 108 m . The current in the wire is uniform and changing at the rate of 2000 A s when the current is 100 A. What is the ratio of the magnitude of the magnetic field due to the displacement current to that due to the current at a small distance r from the wire? [2] 8 Question 8(b) The square loop in Figure 9 is made of rigid conducting rods, each 3.00 m in length, with a total series resistance of 10.0 . It is placed in a uniform 0.100 T magnetic field directed perpendicularly into the plane of the paper. 0.100 T b F a c d F Figure 9 The loop, which is hinged at each corner, is pulled until the separation between points b and d is 3.00 m as shown in Figure 10. 9 0.100 T b a F d c F Figure 10 This process takes 0.100 s to be completed, and is executed in such a way that the rate of change in area bounded by abcd is a constant. (i) What is the direction of the induced current in the loop? Justify your answer. (ii) What is the power delivered to the resistors? (iii)Using [2] [4] Eind show that the induced emf Eind in the loop at any instant during the 0.100 s is given by d Eind 1.80 cos 2 , dt where is the angle that bc makes with the horizontal through c. [8] YY END OF PART II END OF PAPER A. Fundamental Physical Constants Speed of light, c ≈ 2.998 × 108 m/s Magnitude of charge of electron, e ≈ 1.602 × 10−19 C Mass of electron, me ≈ 9.109 × 10−31 kg Mass of proton, mp ≈ 1.673 × 10−27 kg Permitivity of free space, ²0 ≈ 8.854 × 10−12 C2 · N−1 · m−2 Permeability of free space, µ0 = 4π × 10−7 Wb · A−1 · m−1 Acceleration due to gravity, g ≈ 9.807 m/s−2 B. Solutions to a Quadratic Equation ax2 + bx + c = 0, x= C. −b ± √ b2 − 4ac . 2a Derivatives d n x = nxn−1 dx d sin ax dx d cos ax dx d ax e dx d ln ax dx d ln f (x) dx d P (x) dx Q(x) = a cos ax = −a sin ax = aeax = a x 1 d f (x) f (x) dx · ¸ 1 d d = Q(x) P (x) − P (x) Q(x) [Q(x)]2 dx dx = 1 D. Power series n(n − 1) 2 n(n − 1)(n − 2) 3 x + x + ··· 2! 3! x3 x5 x7 x− + − + ··· 3! 5! 7! x2 x4 x6 1− + − + ··· 2! 4! 6! x3 2x5 17x7 x+ + + + ··· 3 15 315 x2 x3 1+x+ + + ··· 2! 3! x2 x3 x4 x− + − + ··· 2 3 4 (1 + x)n = 1 + nx + sin x = cos x = tan x = ex = ln(1 + x) = E. Integrals Z xn dx = Z xn+1 (n 6= −1) n+1 dx = ln x x Z 1 sin axdx = − cos ax a Z 1 cos axdx = sin ax a Z 1 eax dx = eax a Z dx x √ = arcsin 2 2 a a −x Z p xdx √ = a2 + x2 a2 + x2 Z p dx √ = ln(x + a2 + x2 ) a2 + x2 Z dx x 1 = arctan 2 2 a +x a a Z dx 1 x = 2√ a a2 + x2 (a2 + x2 )3/2 Z xdx 1 = −√ 3/2 2 2 2 (a + x ) a + x2 Z dx 1 p p = ln (x − a)2 + b2 (a − x) + (a − x)2 + b2 Z (x − a)dx 1 = −p 3/2 2 2 [(x − a) + b ] (x − a)2 + b2 Z dx x−a = 2p [(x − a)2 + b2 ]3/2 b (x − a)2 + b2 2
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