MAGNETIC FIELDS & UNIFORM PLANE WAVES

Transcription

1 MAGNETIC FIELDS & UNIFORM PLANE WAVES Nme Sectin Multiple Chice 1. (8 Pts). (8 Pts) 3. (8 Pts) 4. (8 Pts) 5. (8 Pts) Ntes: 1. In the multiple chice questins, ech questin my hve me thn ne cect nswe; cicle ll f them.. F multiple chice questins, yu my dd sme cmments t justify yu nswe. 3. Mke sue yu clcult is set t pefm tignmetic functins in dins & nt degees. Regul Questins 6. (0 Pts) 7. (0 Pts) 8. (0 Pts) Ttl (100 Pts) Sme Cmments nd Helpful Inf: In this test, we use tw types f nttin f unit vects. Keep in mind tht $ x$ x $ y$ $ z$ y z $ $ $ $ φ φ $ $ θ θ Be sue t shw yu wk f the multiple chice questins. Dw pictues f ech pblem t be sue tht yu undestnd the pblem sttement. 1 K. A. Cnn Renssele Plytechnic Institute 4 Apil 006

2 MULTIPLE CHOICE QUESTIONS Except f pblem 4, thee is nly ne nswe t ny f these questins. 1. Fce (8 pints) A ectngul cuent lp is fmed using tw pllel cnducting ils, cuent suce with time-invint cuent I, nd sliding cntct between the ils. The fce n the sliding cntct will be diected t the ight f which f the fllwing cses? I Fce?. when the cuent is s shwn but nt when the cuent is in the ppsite (dwnwd) diectin. 4 pints f this nswe b. nt when the cuent is s shwn but when the cuent is in the ppsite (dwnwd) diectin. 4 pints f this nswe c. when the cuent is s shwn nd when the cuent is in the ppsite (dwnwd) diectin.. 8 pints f this nswe d. neve, thee is n fce n the sliding cntct.. Shielding (8 pints) In de t pevent the electic nd mgnetic fields fm enteing leving m, the wlls f the m e shielded with 1-mm thick luminum fil. The best ptectin is chieved t. 1 Hz b. 1 khz c. 1 MHz d. 1 GHz e. N diffeence Shielding wks bette t highe fequencies. K. A. Cnn Renssele Plytechnic Institute 4 Apil 006

3 3. Mutul Inductnce (8 Pints) Of the fu mutul psitins f the tw lps shwn, the mgnitude f the mutul inductnce between the lps is lgest f the psitin in i. Figue () ii. Figue (c) iii. Figue (d) iv. Figue (b) Cupling wks bette when the cils e s simil s pssible. Als, the e is mximized whee the field is lgest. v. Cnnt tell 4. Fields nd Wves Hees (8 Pints) Identify which nme ges with ech equtin. Equtin (c) des nt hve nme nd ne nme ges with ll f the equtins. ) b) c) d) d E dl B ds dt d H dl J ds + D ds dt Fdy B ds 0 Guss D ds ρ dv Ampee v Mxwell 3 K. A. Cnn Renssele Plytechnic Institute 4 Apil 006

4 5. Ampee s Lw (8 pints) Tw cylindicl cnducts f cicul css sectin (dius ) cy time-invint cuent I (I > 0) diected int the pge t the left ut f the pge t the ight. The line integl f the mgnetic flux density vect, B, lng clsed cicul cntu C psitined s shwn is ) μ I b) μ I c) gete thn μ I d) less thn μ I e) less thn μ I nd psitive f) gete thn μ I nd negtive g) ze the cntu suunds bth cnducts s tht the ttl cuent enclsed is ze. This uses Ampee s Lw, which is equtin b in the pevius pblem. Becuse the cuent is time-invint, we cn dp the lst tem in the equtin. C 4 K. A. Cnn Renssele Plytechnic Institute 4 Apil 006

5 REGULAR QUESTIONS 6. Ampee s Lw (0 pints) y x σ σ A lng, stight, slid cylindicl cnduct with dius f is shwn bve. The suunding medium is fee spce. Thee is ttl cuent I cied by this cnduct diected int the pge.. Wht is the cuent density vect? (5) The cuent flws int the pge, s it flws in the negtive z diectin. The e f the I cnduct is π. Thus, the cuent density is given by J z $ π b. Wht is the mgnetic field intensity vect H inside the cnduct (<)? (5) Inside the cnduct, t sme dius, the enclsed cuent will be given by the fctin f the cnduct e within tht dius. Using the ed dshed cicul cntu shwn bve, Ampee s lw lks like H dl Hφ π Ienclsed I The mgnetic field intensity H is then given by H $ φ I π c. Hw much enegy is sted pe unit length in the mgnetic field f the egin inside the cnduct (<)? (5) The sted enegy in egin is given by I I I I I Wm B Hdv dz d π μ d μ 3 μ μ φ π π π 4π 4 8π nte tht thee is missing d in the bve expessin. It shuld be lcted befe the thid equl sign. d. Wht is the intenl inductnce f the cnduct? Tht is, wht is the inductnce sscited with the egin inside the cnduct (<)? (5) Since Wm 1 LI we hve tht L μ 8π 5 K. A. Cnn Renssele Plytechnic Institute 4 Apil 006

6 6 K. A. Cnn Renssele Plytechnic Institute 4 Apil 006

7 7. Mgnetic Cicuit nd Fdy s Lw (0 Pints) A thin tidl ce, mde f femgnetic mteil f pemebility μ, hs n i gp, s shwn in the figue. Thee is time-invint cuent thugh the winding. The cuent-cying wies e pefect cnducts. The depth f the ce is w.. Which f the fllwing is cect f the mgnitude f the mgnetic field H g in the gp? (6) b z g x H H g H g 0 H μ H g c c Hg μ Hc μ H μ H g since B μ c g μhg Bc μ H c b. Nw detemine the mgnetic field in the gp in tems f V, R,, b, g, w nd μ. Any esnble ppximtin will be ccepted. (8) This cn be dne eithe by the mgnetic cicuit methd diectly fm Ampee s Lw. Since mst will pbbly use the mgnetic cicuit methd, we will fllw tht ppch. g Fist evlute the eluctnce f the ce nd gp. These e Rgp nd μ w πb g πb Rce. The MMF f this cicuit is NI. The flux is the given by μw μw NI ψ m. The flux is diected clckwise becuse f the wy the cuent πb g g + μw μw cying wies e wpped und the ce. The mgnetic field is then given by ψ m NI NI μ NI B Finlly, t find I, we nte w πb g g w w w πb g g g + + μ μ μ μ 7 K. A. Cnn Renssele Plytechnic Institute 4 Apil 006

8 V NV tht I nd B μ. Eithe fm f the slutin is fine. The mgnetic flux density R gr is diected dwnwd thugh the gp in the negtive z diectin. B NV μ gr z $ c. Nw ssume tht ne tun sque cnducting lp psses thugh the gp s shwn t velcity v. Detemine the vltge induced und the lp s functin f time. Nte tht the e f the lp is exctly equl t the e f the gp ( times w). (6) z v x Side View v w Tp View The time when the cil eches the mgnetic field is nt specified s the exct timing is nt equied f the nswe. F simplicity, we will ssume tht the induced vltge stts t t 0. F u nswe, we will ssume n finging. Thus, the flux tht psses thugh the cil will ise linely fm ze t the full flux nd then immeditely fll linely bck t ze. The ise time nd the fll time e given by the ce width divided by the velcity τ. Thus the flux linked is given by ψ v t Bw t m() f τ 0 t τ nd ψ t Bw t m () 1 f τ t τ. The induced vltge is given by the τ negtive time deivtive (the sign is nt imptnt hee becuse we hve nt pecisely d 1 specified the cnnectins t the cil) f the flux. V ψ m () t Bw f dt τ 0 t τ nd V d 1 ψ m () t Bw f τ t τ. dt τ 8 K. A. Cnn Renssele Plytechnic Institute 4 Apil 006

9 8. Unifm Plne Wves in Lssless nd Mteils (0 Pints) A unifm plne wve is ppgting in i nd incident nmlly n lssless dielectic medium. The fequency f the wve is 300MHz. The vege pwe density f the wve is 53 Wtts pe sque mete. Yu might find it helpful t dw the vect digm f ech wve t be sue yu hve the diectins cect. Yu d nt hve t simplify ny expessin (yu cn leve it in tems f pmetes), except f pt f, which equies numbe. η η 1 x η 043. η (Ai) Regin 1 z 0 (Lssless Dielectic) Regin. Detemine the ngul fequencyω, the ppgtin cnstnt β nd the wvelength λ f this wve. (3) c π f x 0 λ 1 β π f λ ω π π( ) b. Detemine the mgnitude f the electic field E nd bth the electic nd mgnetic fields in phs vect nttin. (4) E E η ( 53) E η ( 53) 00 jβz E xe $ e H y e jβz $ η c. Find the eflectin cefficient Γ f the electic field. (3) η η Γ η + η K. A. Cnn Renssele Plytechnic Institute 4 Apil 006

10 d. Wite the phs vect fm f the eflected nd tnsmitted electic fields. (4) E x$γ E e β $ ( 1+ Γ) + j z E x E e t jβz π π whee β 465. π λ 043. e. Find the phs vect fm f the tnsmitted wve mgnetic field. () H t ( ) E y$ 1 Γ e η + jβz f. Find the tnsmitted pwe. (4) (Be sue yu pvide numbe f this pt.) P 1 ve ( E H ) Re z$ (. ) ( ) * z$ (. 0 43)( 10π ) 10 K. A. Cnn Renssele Plytechnic Institute 4 Apil 006