PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT OF PHYSICS AND ASTRONOMY ADVANCED QUANTUM MECHANICS 2 hours Spring 2017 Electricity nd Mgnetism 2 hours Answer question ONE (Compulsory) nd TWO other questions, one ech from section A nd section B. Instructions: All questions Answer questions re mrked 1 ndout 4 (Compulsory), of ten. The ONE brekdown other question on from the right-hnd Section A ndside ONEof other the question pper from is ment Sections B. guide to the mrks tht cn be obtined from ech prt. Use seprte books for Sections A nd B! All questions re mrked out of twenty. The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. Plese clerly indicte the question numbers on which you would like to be exmined on the front cover of your nswer book. Cross through ny work tht you do not wish to be exmined. PHY102-A TURN OVER 1
1. COMPULSORY SECTION A Electricity () A point chrge of 3.0 10 6 C is t the origin. Find the mgnitude nd direction of the electric field t (x, y, z) = (2 m, 1 m, 0 m). [2] (b) A bttery hs n EMF of 12.0 V nd n internl resistnce of 0.050 Ω. Its terminls re connected to lod resistnce of 3.00 Ω. Wht is the current in the circuit nd the terminl voltge of the bttery? [2] (c) Clculte the combined cpcitnce of the network shown below. [3] 1.0 µf 3.3 µf 1.0 µf 4.7 µf (d) A proton is relesed from rest in uniform electric field of mgnitude 8.0 10 4 V m 1. The proton trvels distnce of 0.55 m in the direction of the field. Wht speed does it ttin? (e) Explin wht is ment by the term electric dipole moment. [2] (f) Two point chrges q nd q re seprted by smll distnce d long the x xis, being situted t x = d/2 nd x = d/2 respectively. Clculte the electric field t the point x = long the x xis from the pir in the limit tht d, nd show tht in this limit it depends only on the product of q nd d, nd not on either of these prmeters individully. [3] (g) A prticle with chrge +9 µc is plced t the origin, nd nother prticle of chrge 4 µc is t x = L. At wht point is the net electrosttic field equl to zero? [3] [3] 9 µc 4 µc x 0 L (h) Explin wht is ment by the term drift velocity when referring to chrge crriers in conductor. On wht properties of the conductor does the drift velocity depend? [2] PHY102-A CONTINUED 2
2. () Stte Guss s lw of electrosttics in words. Also present it s n eqution, defining ll the symbols tht you use. [3] (b) An infinitely long insulting solid cylinder of rdius R contins non-uniform chrge distribution, whose density is given by ρ(r) = ρ 0 r/r for r R nd zero otherwise. Use Guss s lw to clculte the electric field distnce r from the xis of the cylinder, justifying your choice of Gussin surfce: i. when r > R; ii. when r < R. (c) How cn the electric field be determined from the electrosttic potentil? [2] (d) The potentil in region of spce is given by V (x, y, z) = A(x 2 + 2x y + 3 y 2 ) where A = 150 V m 2 with x nd y mesured in metres. Find the mgnitude nd direction of the electric field t the point x = 0.57 m, y = 0.75 m, z = 0.95 m. [6] [6] [3] 3. () A cpcitor of cpcitnce C is chrged to voltge V 0. i. How much energy is stored on the cpcitor? [1] ii. At time t = 0, resistor of resistnce R is connected cross the cpcitor. Give n expression for the power dissipted in the resistor t subsequent time t. [3] iii. By integrting this expression between pproprite limits, show tht the totl energy dissipted in the resistor is equl to the energy originlly stored on the cpcitor. iv. If the vlue of the cpcitnce is 50 µf, the resistor is 1.0 MΩ nd V 0 = 25 V, find the time when the power dissipted by the resistor is 1% of its vlue t t = 0. (b) Three chrges re plced t the vertices of n equilterl tringle of side 12 cm. Two of the chrges re +25 µc while the third is 25 µc. i. Clculte the mgnitude nd direction of the electric field t the centre of the tringle (the point which is equidistnt from ll three chrges). [3] ii. Find the mgnitude nd direction of the forces cting on ech of the three chrges. (You my indicte the directions with sketch, if you wish.) [4] iii. Determine the totl electrosttic energy of the system of the three chrges by clculting the work done when we move the three chrges from infinity to their positions in the equilterl tringle. [3] [3] [3] PHY102-A TURN OVER 3
SECTION B Mgnetism 4. COMPULSORY () The stndrd units of mgnetic field nd mgnetic flux re Tesl nd Weber respectively. Find equivlent units in terms of combintions of Newtons, Amperes nd metres. (b) An electron moving with velocity v = 2.0 10 6 î + 3.2 10 6 ĵ moves through uniform mgnetic field with B = 0.75 ˆk T. Wht is the force on the electron? [3] (c) A neutron with mgnetic dipole moment of 9.66 10 27 A m 2 pointing long the z direction enters region with mgnetic field grdient lso in the z direction, with db/dz = 5.0 T m 1. Wht is the ccelertion of the neutron? [3] (d) Explin the purpose of the commuttor in DC electric motor. [2] (e) Clculte the mgnetic field generted by current segment of length 5.0 mm crrying current of 60 A t point 10 cm to the side of the wire. In your nswer you should clerly specify the direction of the field. Sketch digrm of the current segment nd the mgnetic field, clerly indicting the coordinte system tht you use. (f) Two prllel wires re seprted by distnce of 5.0 cm. One of them crries current of 2.0 A. Find the current in the other wire tht would give repulsive force of 4.0 10 6 N m 1, stting crefully the direction of the current reltive to the other one. (g) Use Ampère s lw to find the mgnetic field t the core of long solenoid with n turns per unit length tht crries current I. [3] (h) The field inside prmgnetic mteril is found to be 1.5023 T when uniform externl mgnetic field of 1.5000 T is pplied. Wht is the mgnetic susceptibility of the prmgnetic mteril? [2] [2] [3] [2] PHY102-A CONTINUED 4
5. () A chrged prticle enters velocity selector in which it experiences uniform electric nd mgnetic fields of mgnitude E nd B respectively. The fields re perpendiculr to the direction of the prticle s direction nd to ech other. i. Mke sketch of the velocity selector nd find the condition on the velocity v for the prticle to be undeflected. [2] ii. A He + ion (mss 4m p, chrge +e) enters velocity selector in which E = 2.0 10 6 V m 1 nd B = 0.85 T. It then emerges into region of uniform mgnetic field B nd zero electric field. The ion is detected t distnce of 10.0 cm to the side of the exit slit of the velocity selector. Wht is the vlue of B? [3] iii. Helium hs nother isotope, nmely 3 He, with mss 3m p. The detector cn move in sidewys direction reltive to the exit slit of the velocity selector. By how fr would it hve to be moved to detect the 3 He + ion? [3] (b) A circulr current loop of rdius 1.5 cm crries current of 6.0 A. The loop lies in the x y plne. A uniform mgnetic field of strength 0.80 T is pplied long the (0, 1, 1) direction. Find the torque on the loop. [5] (c) i. Explin how the Hll effect cn be used to determine the sign nd density of the chrge crriers in conducting medium. [2] ii. A copper wire with squre cross-section crries current of 10 A in the x direction. The width of the wire is 3.0 mm, nd it is orientted such tht the edges re prllel to the y nd z directions. A mgnetic field of strength 5.0 T is pplied in the y direction. Clculte the Hll voltge generted cross the wire, drwing digrm tht clerly shows where the voltge ppers, nd which side hs the positive voltge. (The free electron density of copper is 1.2 10 29 m 3.) [5] PHY102-A TURN OVER 5
6. () At prticulr instnt in time, two electrons move through vcuum in opposite, prllel directions t speed v c. Their sptil seprtion is r. i. Clculte the mgnetic force between them when they re t their minimum seprtion. ii. Clculte the rtio of the mgnetic nd electric forces between the prticles, mking use of the reltionship ε 0 µ 0 c 2 = 1 to simplify your nswer. Which force is dominnt? (b) An infinite stright wire with circulr cross section of rdius R crries current I. Use Ampère s lw to find the mgnetic field strength B s function of r, the rdil distnce from the centre of the wire. You my ssume tht the current is uniformly distributed throughout the wire. Your nswer should include cler sttement bout the direction of the field, nd sketch of the vrition of B with r from 0 to. (c) Clculte the current required to generte mximum field strength of 1.0 mt using wire of rdius 2 mm. (d) Specify the sign nd reltive mgnitude of the mgnetic susceptibility for (i) ferromgnetic, (ii) prmgnetic, nd (iii) dimgnetic mterils. In the cse of the ferromgnetic mteril, explin how two different smples of the sme mteril could either be mgnetised or non-mgnetised t zero externl field. [6] [4] [2] [6] [2] END OF EXAMINATION PAPER 6
PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = 1.60 10 19 C electron mss m e = 9.11 10 31 kg = 0.511 MeV c 2 proton mss m p = 1.673 10 27 kg = 938.3 MeV c 2 neutron mss m n = 1.675 10 27 kg = 939.6 MeV c 2 Plnck s constnt h = 6.63 10 34 J s Dirc s constnt ( = h/2π) = 1.05 10 34 J s Boltzmnn s constnt k B = 1.38 10 23 J K 1 = 8.62 10 5 ev K 1 speed of light in free spce c = 299 792 458 m s 1 3.00 10 8 m s 1 permittivity of free spce ε 0 = 8.85 10 12 F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = 6.02 10 23 mol 1 gs constnt R = 8.314 J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = 6.67 10 11 N m 2 kg 2 Rydberg constnt R = 1.10 10 7 m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = 0.529 10 10 m Bohr mgneton µ B = 9.27 10 24 J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = 2.898 10 3 m K Stefn s constnt σ = 5.67 10 8 W m 2 K 4 rdition density constnt = 7.55 10 16 J m 3 K 4 mss of the Sun M = 1.99 10 30 kg rdius of the Sun R = 6.96 10 8 m luminosity of the Sun L = 3.85 10 26 W mss of the Erth M = 6.0 10 24 kg rdius of the Erth R = 6.4 10 6 m Conversion Fctors 1 u (tomic mss unit) = 1.66 10 27 kg = 931.5 MeV c 2 1 Å (ngstrom) = 10 10 m 1 stronomicl unit = 1.50 10 11 m 1 g (grvity) = 9.81 m s 2 1 ev = 1.60 10 19 J 1 prsec = 3.08 10 16 m 1 tmosphere = 1.01 10 5 P 1 yer = 3.16 10 7 s
Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ 2 = 1 ( r ) + 1r 2 r r r 2 θ 2 Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r 2 sin θ dr dθ dφ 2 = 1 ( r 2 ) + 1 r 2 r r r 2 sin θ ( sin θ ) + θ θ 1 r 2 sin 2 θ 2 φ 2 f(x) f (x) f(x) f (x) x n nx n 1 tn x sec 2 x e x e x sin ( ) 1 x ln x = log e x 1 x cos 1 ( x sin x cos x tn ( 1 x cos x sin x sinh ( ) 1 x cosh x sinh x cosh ( ) 1 x sinh x cosh x tnh ( ) 1 x ) ) 1 2 x 2 1 2 x 2 2 +x 2 1 x 2 + 2 1 x 2 2 2 x 2 cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v 2 Definite Integrls 0 + + x n e x dx = n! (n 0 nd > 0) n+1 π e x2 dx = π x 2 e x2 dx = 1 2 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx
Series Expnsions (x ) Tylor series: f(x) = f() + f () + 1! n Binomil expnsion: (x + y) n = (1 + x) n = 1 + nx + k=0 ( ) n x n k y k k n(n 1) x 2 + ( x < 1) 2! (x )2 f () + 2! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x2 2! + x3 x3 +, sin x = x 3! 3! + x5 x2 nd cos x = 1 5! 2! + x4 4! ln(1 + x) = log e (1 + x) = x x2 2 + x3 3 n Geometric series: r k = 1 rn+1 1 r k=0 ( x < 1) Stirling s formul: log e N! = N log e N N or ln N! = N ln N N Trigonometry sin( ± b) = sin cos b ± cos sin b cos( ± b) = cos cos b sin sin b tn ± tn b tn( ± b) = 1 tn tn b sin 2 = 2 sin cos cos 2 = cos 2 sin 2 = 2 cos 2 1 = 1 2 sin 2 sin + sin b = 2 sin 1( + b) cos 1 ( b) 2 2 sin sin b = 2 cos 1( + b) sin 1 ( b) 2 2 cos + cos b = 2 cos 1( + b) cos 1 ( b) 2 2 cos cos b = 2 sin 1( + b) sin 1 ( b) 2 2 e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) 2 nd sin θ = 1 ( e iθ e iθ) 2i cosh θ = 1 ( e θ + e θ) 2 nd sinh θ = 1 ( e θ e θ) 2 Sphericl geometry: sin sin A = sin b sin B = sin c sin C nd cos = cos b cos c+sin b sin c cos A
Vector Clculus A B = A x B x + A y B y + A z B z = A j B j A B = (A y B z A z B y ) î + (A zb x A x B z ) ĵ + (A xb y A y B x ) ˆk = ɛ ijk A j B k A (B C) = (A C)B (A B)C A (B C) = B (C A) = C (A B) grd φ = φ = j φ = φ x î + φ y ĵ + φ z ˆk div A = A = j A j = A x x + A y y + A z z ) curl A = A = ɛ ijk j A k = ( Az y A y z φ = 2 φ = 2 φ x + 2 φ 2 y + 2 φ 2 z 2 ( φ) = 0 nd ( A) = 0 ( A) = ( A) 2 A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk