University of Illinois at Chicago Department of Physics. Electricity & Magnetism Qualifying Examination

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Edwin Harrington
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2 Univesity of Illinois at Chicago Depatment of Physics Electicity & Magnetism Qualifying Examination Januay 3, 6 9. am : pm Full cedit can be achieved fom completely coect answes to 4 questions. If the student attempts all 5 questions, all of the answes will be gaded, and the top 4 scoes will be counted towad the exam s total scoe.

3 Miscellaneous Equations: n nn ( ) ( + x) = + nx+ x ln( + x) = x x + L x e = + x+ x + L 3 sin( x) = x x + L cos( x) = x + L () D 3! E = ε ρ B E = t B = B= + D = ρ µ J µ ε D H= J f + t ( ) B B = // // () 3 E E = // // ( 4) H H = K n$ ( n$ : ) f D = σ f E t E = E e B= k E e + L ik ( ωt) ik ( t), ω ε = 885. µ = 4π 7 f C /Nm N/A ω D= ε E+ P H= B M µ D= εe H= B µ σ b = Pn $, ρb = P Jb = M, Kb = M n$ z ρ = ( ) $ E ( ) 4πε τ d z µ J ( = ) B ( ) $ 4π dτ p $ Vdip( ) = 4πε 3( p $)$ p Edip( ) = 3 4πε v E = V A, t B= A z ρ(, t ) V( t, ) = dτ 4πε z µ J (, t) At (, ) = 4 π dτ = f d l = ( E+ v B) d l ε ε z dφ dt ε di =, = L dt v E = V A, t B= A z C Q, U = CV, L Φ B / I, U = LI V uem = ( E D+ B H) ρ J = t

. A unifomly chaged, solid, non-conducting sphee of adius a (chage density ρ ) has its cente located a distance d fom a unifomly chaged, non-conducting, sheet (chage density σ ) as shown in the figue

4 . A unifomly chaged, solid, non-conducting sphee of adius a (chage density ρ ) has its cente located a distance d fom a unifomly chaged, non-conducting, sheet (chage density σ ) as shown in the figue below. (a) (b) Detemine the potential diffeence between the cente of the sphee and the neaest point on the chaged sheet. Detemine the net electostatic foce on the chaged sphee.. A thin disc of adius R and thickness d has a unifom, fozen-in polaization along the +x axis. It lies in the xy-plane and is centeed at the oigin as shown in the figue below. In you calculations below, assume that d << R. a) Compute the electic field fo a point along the z-axis, E(z). (Do not assume fo this pat that z is lage) b) Wite the electic field at the oigin in tems of P. Explain the micoscopic intepetation of this esult (fo example, is this an accuate desciption of the field nea an atomic nucleus at the cente of the disc? why?) c) Wite an appoximate expession fo the electic field on the z-axis fo z>>r and show that you esult fom pat (a) has the expected leading ode dependence on z fo z >> R.

5 3. Conside the cicuits shown below. Cicuit consists of a esisto (R) and capacito (C) in seies with a voltage souce (V ). A long, staight wie in cicuit goes, along the axis, though an N-tun, tooidal coil of ectangula cosssection (inne adius = a, oute adius = b, height = h). Cicuit consists of the coil, having negligible esistance, in seies with anothe esisto (R'). The capacito is initially unchaged. At t = the switch is closed. (a) (b) (c) What is the cuent I (t) that flows though cicuit as a function of time? Assume that this esult does not depend on any inteaction with cicuit. [It is acceptable eithe to just wite the esult, o to deive it fom scatch.] What is the EMF induced in the coil as a function of time? In this situation, the cuent in cicuit is not accuately given by the EMF divided by R'. Why is this? Discuss and make qualitative gaphs to indicate how the actual cuent vs. time will diffe fom EMF/R'. 4. Conside a thin disc of adius R and thickness d, having linea, magnetic susceptibility χ. It lies in the xy-plane and is centeed at the oigin as shown in the figue below. In you calculations below, assume that d << R. The disc is placed in an othewise unifom, oscillatoy magnetic field: B ext (t) = B cos(ωt) diected along the z-axis. Assume the susceptibility is sufficiently small that the magnetization is appoximately unifom, with magnitude: M = (χ/µ )B ext (a) Fo this pat, assume that the fequency is low, so that the field due to the disc can be calculated in the magnetostatic limit (i.e. no appeciable etadation effects). What ae the constaints on ω fo this magnetostatic limit? Calculate the magnetic field due to the disc at a point along the z axis.

6 4(b) No longe assuming that the fequency is low, wite an integal expession fo the vecto potential A(x,y,z,t) due to the disc, defining any symbols you use (it is not necessay to do the integation, but do wok it into the fom of a definite integal such that the esult will be a function of x,y,z,t). 5. Conside an electomagnetic, sinusoidal plane wave, nomally incident on a laye of dilute plasma fom vacuum. Assume that the plasma fills the egion z>. The dispesion elation fo the plasma is appoximately: p ω = ω + ck whee ω p is a constant called the "plasma fequency" and c is the speed of light. You can assume the pemeability of the plasma is the same as vacuum. (a) (b) (c) Descibe the diffeence in the behavio of the wave in the fequency anges ω < ω p and ω > ω p. Daw a sketch of the wave in each case (ty to get the pope elationship between wavelengths on eithe side of the bounday). Apply the bounday conditions (as with a nomal dielectic, you can assume thee is no fee chage o fee cuent anywhee), and solve fo the amplitudes of the eflected and tansmitted waves in tems of the incident electic field amplitude. Evaluate the amplitude of the magnetic field of the tansmitted wave fo the fequency: ω = 3 ω p You answe should be a complex numbe, witten in pola fom, times the incident amplitude of the electic field.

Solutions. V in = ρ 0. r 2 + a r 2 + b, where a and b are constants. The potential at the center of the atom has to be finite, so a = 0. r 2 + b.

Solutions. V in = ρ 0. r 2 + a r 2 + b, where a and b are constants. The potential at the center of the atom has to be finite, so a = 0. r 2 + b. Solutions. Plum Pudding Model (a) Find the coesponding electostatic potential inside and outside the atom. Fo R The solution can be found by integating twice, 2 V in = ρ 0 ε 0. V in = ρ 0 6ε 0 2 + a 2