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# Note for ECE 2317 with Professor Jackson at UH

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This 8 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 32 views.

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Date Created: 02/06/15

Electromagnetics Workshop Solutions Boundary Conditions Problem 1 The uniform elds shown in the gure below are near a dielectricdielectric boundarybut on opposite sides of it Which con gurations are correct Assume that the boundary is charge ee and that 52 gt 1 G D G D or m m m d I m H l Solution a Normal D is not continuous a this con guration is not correct b Normal D is not continuous a this con guration is not correct c Normal D is continuous D1r gt D2 gt 51E gt 52E E1r E2r gt 51 gt 52 4 this con guration is not correct d Tangential E is not continuous 4 this con guration is not correct e This con guration is correct 0 Tangential E is not continuous a this con guration is not correct Electromagnetics Workshop Solutions Boundary Conditions Problem 2 A silver coated sphere of radius 5 cm carries a total charge of 12 nC uniformly distributed on its surface in free space Calculate a IDI on the surface of the sphere b D external to the sphere Solution Q 3nC 3nC 2 D 38197 C a ps 4an 5cm2 n25x10quot n m Q A3nCAN0955A bD W1 7rr Zr nCmz Electromagnetics Workshop Solutions Boundary Conditions Problem 3 Match the following descriptions with the gures shown below Fields are near the interface but on opposite sides of the boundary j l H N y l y l N W a Medium 1 and medium 2 are dielectrics with 81 gt 82 b Medium 1 and medium 2 are dielectrics with 81 lt 82 c Impossible d Impossible e There is a positive surface charge on the boundary between two dielectrics f Medium 2 is a perfect conductor Solution i c iif iii b iV a V d Vi e Electromagnetics Workshop Solutions Boundary Conditions Problem 4 Match the following descriptions with the gures shown The elds in the gures are very near the interface but on opposite sides of the boundary Descriptions can be used once more than once or not at all a b c d e f g h Medium 1 and medium 2 are dielectrics with 81 gt 82 Medium 1 and medium 2 are dielectrics with 81 lt 82 Impossible Possible but description is not provided There is a positive surface charge on the boundary between two different dielectrics There is a negative surface charge on the boundary between two different dielectrics Medium 2 is a perfect conductor There is a surface current on the boundary between the dielectrics E1 2 1 p 1 1 o 7 l 2 2 D2 1 2 n in D1 1 1 DZ 2 2 iv V E1 E1 1 E2 1 2 39l 2 39 E2 0 vii viii D2 D 1 1 1 D2 1 D l J o D 2 2 D1 2 2 X x1 xii Electromagnetics Workshop Solutions Boundary Conditions Solution i b ii g iii 0 iV a V d Vi e Vii Viii ix 0 X d Xi f Xii c Electromagnetics Workshop Solutions Boundary Conditions Problem 5 Given the potential d3 20 cos for p lt 2 and d3 50 cos l20pcos for p gt 2 a Find E in both regions b Show that the boundary conditions are satis ed at the surface p 2 for all if g so for pgt2 and ggrgo for plt2 Find 5 S olution For cylindrical coordinates E9 V qgt a 2 1 ad A ad A 6p 2 E Ep 20cos Afpi20psin Aga 20cosA20sin A plt2 p E9 cos 50l20p2fsin 50 l20p2A pgt2 At p 2 the tangential components of the electric eld should be equal 20sin sin 50 120p2 p2 The tangential electric eld components are equal for all at the surface p 2 At p 2 the normal components of the electric ux density should be equal 2 20cos sr cos 50l20p p2 The normal electric ux density components are equal for all at the surface p 2 if g 4 Electromagnetics Workshop Solutions Boundary Conditions Problem 7 A cylindrical capacitor as shown carries a total charge of Q and 7Q on the inner and the outer conductors respectively The capacitor is not connected to the ground or to any voltage source A dielectric washer with inner and outer diameters matching the dimensions of the cylindrical capacitor is placed x meters inside the capacitor as shown in the gure below a Find the electric eld in the capacitor in terms of Q h Find the voltage between the two conductors 6 VII39llIIlllA eo Solution aQplxp0hxEp2ii amp lp 27r 0p 1 0 7 EOQ 51g p0 1X 0hixpl 1x 0hix E A F 27rp 1x 0 hex 7 Q Q a b Gabi 2 p 1x 0hixdp 2 1x 0hixnb Electromagnetics Workshop Solutions Boundary Conditions Problem 8 A parallelplate structure with a plate separation d is lled With a material that contains a volume charge density pV az Cm3 and has permittivity g Both plates are held at ground potential 0 V a Using Poisson s equation determine the potential between plates b Determine the surface charge density on the top plate c Determine the surface charge density on the bottom plate Solution a Poisson s equation 2 Z V2q3amp3 a q 36 0 Z C1 g 62 g 62 25 3 3 z ClzC2 00C20 a adz z 23 dzz V q d 03C 1 68 b Fortop plate pm D i s V i 3zz d2 24 adz Z 3 Cm c Forbottom plate 20p 5 00m A A a pf Dzs V zg3zZ dz 20 pfm39 2 Cmz

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