UNIVERSITY OF EAST ANGLIA School of Mthemtics My/June UG Exmintion 2007 2008 ELECTRICITY AND MAGNETISM Time llowed: 3 hours Attempt FIVE questions. Cndidtes must show on ech nswer book the type of clcultor used. Do not turn over until you re told to do so by the Invigiltor. Copyright of the University of Est Angli
- 2-1 A sptilly bounded chrge distribution occupying volume V hs chrge density ρ nd electric sclr potentil φ. The electrosttic energy of the chrge distribution W is given by W = 1 ρφ dv. 2 Show, using the equtions of electrosttics, tht this my be written s W = ɛ 0 E 2 dv, 2 where E is the electric field, nd the integrl is tken over ll spce. V Totl chrge Q is distributed with density ρ = ρ 0 r/ throughout the sphere r <. Explin why the electric field tkes the form E = (E(r), 0, 0) in sphericl polr coordintes. Show tht Q = πρ 0 3 nd use Guss s lw to clculte E(r) in terms of Q for r < nd r >. Hence clculte the potentil. [8 mrks] Using either of the two expressions for electrosttic energy bove, show tht the electrosttic energy of this chrged sphere s W = Q2 7πɛ 0.
- 3-2 Stte clerly Guss s lw in integrl form nd derive from it the lw in differentil form. [3 mrks] Consider n infinite stright line chrge, with uniform chrge density q per unit length, lying long the z-xis. In wht direction is the electric field? Find the electric field E nd show tht the electric potentil of this chrge distribution is, in cylindricl polr coordintes, where r represents the distnce from the z-xis. φ = q 2πɛ 0 ln r Now suppose tht n infinite line chrge, uniform density +q per unit length, lies long the line x = 0, y = 0, < z < nd tht prllel line chrge, of density q per unit length, lies long the line x = d, y = 0, < z <. Using the bove result for single line chrge write down the potentil for this system of two line chrges. [2 mrks] Denote the vector seprtion of the two line chrges by d. A line dipole is formed by letting q, d 0 in such wy tht p = qd remins finite. Show tht, in cylindricl polrs, the electric potentil is given by φ = p r 2πɛ 0 r 2. Show tht the equipotentils re cylindricl surfces nd clculte the electric field. [3 mrks] PLEASE TURN OVER
- 4-3 Stte Mxwell s equtions governing electromgnetic fields in free spce, in the bsence of ny chrge or current densities. From these show tht the electric field E stisfies the wve eqution, nmely 2 E 1 c 2 2 E t 2 = 0, where c is the speed of light. Show tht the sme eqution governs the mgnetic field B. [8 mrks] Consider rectngulr wveguide which consists of hollow cvity enclosed by perfectly conducting wlls t x = 0, x =, y = 0, y = b. Wht boundry conditions must E nd B stisfy on the wlls? Consider the electric field where E = (E x, E y, E z )e i(kz ωt) E x = α cos( mπx ) sin(nπy b ), E y = β sin( mπx ) cos(nπy b ), E z = γ sin( mπx ) sin(nπy b ), where m nd n re non-zero integers. Show tht E stisfies the pproprite boundry conditions. [3 mrks] By using Mxwell s equtions for E find n expression for γ in terms of α nd β. Using the eqution for E show tht ( ( iω)b x = ikβ nπγ b ) sin( mπx ) cos(nπy b )ei(kz ωt) nd find the other two components of B = (B x, B y, B z ). Show tht B stisfies B = 0 identiclly. By using one component of the remining Mxwell eqution obtin the dispersion reltion ω 2 c 2 = m2 π 2 2 + n2 π 2 b 2 + k 2. [3 mrks]
- 5-4 Strting from Guss Lw derive the boundry condition the norml component electric field E must stisfy cross n interfce crrying surfce chrge of density σ. Wht is this boundry condition t the surfce of perfect conductor? Consider point chrge q is situted t the origin. Either write down or clculte the electric field E nd the potentil φ due to this chrge. A point chrge q is plced t the point (x 0, y 0 ), x 0, y 0 > 0. Intersecting plnes x = 0, y = 0 re perfectly conducting surfces which re held t zero potentil φ = 0. Use the method of imges to find n expression for the potentil φ. Wht chrge density is induced on the plnes x = 0, y = 0? Show tht, if x 0 = y 0 =, the force experienced by the point chrge is given by ( F = q2 4 ) 2 (, ). 4πɛ 0 3 16 [4 mrks] [4 mrks] PLEASE TURN OVER
- 6-5 Wht is the physicl significnce of the eqution where B is the mgnetic field? B = 0, potentil A. Explin why A is not uniquely defined. Explin how this eqution implies the existence of vector Write down the Mxwell eqution governing the mgnetic field B nd the current density J for the cse of stedy flow. Show tht, under certin condition on A, which should be stted nd explined, A nd J re relted by the eqution 2 A = µ 0 J. [10 mrks] Stte the Biot-Svrt lw giving B in terms of n integrl of the current density J nd obtin from it the lw in terms of current I flowing in thin wire. A stedy current flows round n ellipticl wire loop locted in the plne z = 0 nd described by the eqution x 2 / 2 + y 2 /b 2 = 1. Using the Biot-Svrt lw, show tht the mgnetic field t the point with coordintes (0, 0, z) cn be expressed s B = µ 0Ib 4π 2π 0 dθ ( 2 cos 2 θ + b 2 sin 2 k. θ + z 2 ) 3/2 [10 mrks]
- 7-6 () The eqution of continuity of chrge is given by J + ρ t = 0. Show tht it cn be derived from Mxwell s equtions nd explin physiclly why it must hold. At time t = 0, chrge density ρ 0 (r) is distributed throughout conducting substnce, which hs constnt conductivity σ. Stte Ohm s lw relting J nd E. Show tht the chrge density ρ stisfies differentil eqution with solution of the form ρ(r, t) = ρ 0 e t τ, where τ should be found. Wht impliction does this hve for good conductors? Wht bout for perfect conductnce? (b) Define the Poynting vector S in terms of the electric nd mgnetic fields. [5 mrks] Strting from Mxwell s eqution for B nd tking the sclr product with E show tht S cn be written in terms of the electric energy, the mgnetic energy nd the Ohmic het loss s S + E J = [ ɛ0 t 2 E2 + 1 ] B 2. 2µ 0 Explin wht ech term mens nd the significnce of this eqution. [You my ssume the vector identity ( b) = b b, where nd b re vector fields.] [9 mrks] END OF PAPER