# PHY114 S11 Term Exam 3

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2 Figure :. A coil has.0 Ω resistance and 400 mh inductance. If the current is.89 A and is increasing at a rate of.62 As, what is the potential drop across the coil at this moment?(8 points) An alternating current of peak value of.5a and frequency 46Hz is passed through a coil with inductance of 92mH. Whatisthepeakvoltage? Ignore the resistance.(8 points) You are given two inductors L = 200mH, L 2 = 00mH and two capacitors C =.00µF, C 2 =2.2µF.Howwouldyouconnectthemuptogeta circuit with the largest possible natural frequency of oscillation? What is this frequency? Ignore resistance. (9 points) 2. Determine the total energy stored per unit length in the magnetic field between the conductors of a coaxial cable carrying a current I in each direction (see Fig.). (5 points.)what is the inductance per unit length of this coaxial cable?(0 points.) Ignore the magnetic field inside the conductors. Also, recall that the energy density of a magnetic field is B 2.. Suppose a 9kW radio station emits EM waves uniformly in all directions.how much energy per second crosses a 0.6 m 2 area km from the transmitting antenna?(8 points) What is the range of wavelengths for FM radio (88MHz to 08MHz )? What is the range of wavelengths for AM radio (55 khz to 700 khz)? (9 points) Alightbeamcomingfromanunderwaterspotlightexitsthewateratan angle of 56.6 to the normal. At what angle of incidence did it hit the 2

3 Figure 2: 4. air-water interface from below the surface? The refractive index of water is.. (8 points) Asmallcandleis5cmfromaconvexlensoffocallength2cm. Where is the image located? Is it real or virtual? Is it larger or smaller? Is it inverted or upright?(8 points) The magnification of the image in a convex mirror is m, thedistanceto the object is d. Whatistheradiusofcurvatureofthemirror?(8points) Suppose we place two convex mirrors of equal radius of curvature 2 meters,facingeachotheratadistanceof2metersalso. Anobjectisplacedatthe midpoint in between the mirrors. What are the positions of the images formed by multiple reflections?what are the limit points to which the sequence of images tend?(hint It is useful to choose a co-ordinate system whose origin is at the object.see Fig 2)(9 points)

4 Solutions. The potential difference is the sum of the those caused by the resistance and the inductance of the coil. Both cause a drop in the potential (the resistor always causes a drop and in our case the current is increasing, so the induced emf on the coil is negative as well). Thus the potential drop across the coil is RI + L di dt = = 4.V Again, the induced emf is L di dt. The current as a function of time is I(t) =I 0 sin ωt.herei 0 is the peak current and ω =2πf where f is the frequency. Thus L di dt = LI 0ω cos ωt The peak voltage (also called emf) is LI 0 ω =2πLI 0 f =2π = 69.9V. The natural frequency of an LC circuit is 2π.Thus to have a large LC frequency we need both a small inductance and a small capacitance. Two inductances in parallel have smaller inductance L = LL2 L +L 2 than each of them separately or in series (L +L 2 ),wemustconnecttheminparallel. Since capacitances connected in series have smaller capacitance C = C C 2 C +C 2 than either of them in parallel (C + C 2 ) we must connect them in series. Thus.2. L = = 20mH C = =.688µF f = 2π =0.55Hz 2. The energy per unit volume of a magnetic field is B 2. The magnetic field has a magnitude that only depend on the radial distance. Thus the energy over some some length l of the cable will be given by integrating this density over the cross-section of the cable: l ˆ r2 r B 2 (r)2πrdr Thus the energy per unit length is ˆ B 2 (r)2πrdr. 4

5 By Ampere s law Thus 2πrB = µ 0 I ˆ r2 B = µ 0I 2πr B 2 2πr = µ 2 0I 2 2πr = µ 0I 2 4πr B 2 (r)2πrdr = µ 0I 2 r 4π ˆ r2 r dr r = µ 0I 2 4π ln r 2 r. The energy stored in an inductor is 2 LI2. The energy per unit length is L 2 l I2 = µ 0I 2 4π ln r 2 r We see that the inductance per unit length of the coaxial cable is L l = µ 0 2π ln r 2 r. Another Method You can use Farday s law to calculate the inductance per unit length. energy of the magnetic field per unit length is then L 2 l I2.. The The power that crosses a sphere of radius r that surrounds the antenna must be equal to the power emitted by conservation of energy. Thus the P power crossing a unit area is 4πr.The power crossing an area A at a 2 distance r is P 4πr 2 A = π[000] 2 =.86mW Just a matter of using λf = c where λ is the wavelength, f the frequency and c is the velocity of light. λ = c f so that at 88MhZ, λ =.88m =.4m; at08mhz, λ =.08 m =2.78m; at.55mhz λ =.55 0 m = 560m ;at.7mhz λ = = 76m. 2 By Snell s law n sin θ = n 2 sin θ 2. Thus sin θ 2 = n n 2 sin θ = Thus the angle of incidence is θ = 8.9. sin = 4. 5

6 The distance to the image i is given by i + d = f, = i = df d f = = 8.cm This is a real image located on the opposite side of the lens from the candle. The magnification 8. 5 < so the image is smaller. Also, it is inverted. (No need to draw diagrams if the answer is clearly explained) The magnification is related to the distance of the image i and the distance to the object d by m = i d.foraconvexmirrortheimageisalwaysvirtual and i<0. Thus i = md. The lens formula gives i + d = f = f = id d + i = md2 d md f = m m d Since f < 0, the magnification is always less than one. curvature is R =2 f = 2md m. The radius of It is useful to choose a co-ordinate system whose origin is at the object. Let x be the position of the first image formed by the right mirror. By symmetry, the image formed by the left mirror will be at x. Now the distance to the object from the right mirror is d =. The distance to the image is (x ) Since the image is virtual i = (x ).The focal length is f =. (Halftheradiusofcurvatureandnegativebecausethemirrors are convex).thus, x = = x = 2 Similarly the left mirror forms an image at x = 2. For the second reflection, the object is at the point x. So the distance to the object from the mirror is d =+x.again f = ;the distance to the image from the mirror is i = (x 2 ) Similarly x + x 2 = x n + x n+ = As n becomes large x n x n+ so that the limiting position of the images satisfies x + x =, = x = ± ±.7 Thus the images accumulate at two points 0.7 behind each mirror. Special thanks to Emily Redman. 6

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