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# Note for ECE 2317 with Professor Jackson at UH divergence

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KARMA
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This 5 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 41 views.

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Date Created: 02/06/15
Electromagnetics Workshop Divergence and Divergence Theorem Problem 1 Vector eld E is characterized by the following properties a E points along 1 b The magnitude of E is a function of only the distance from the origin c E vanishes at the origin d VE 6 everywhere Find an expression for E that satis es these properties Solution E frf 1 6 1 6 VEr ZE 2Err Zgrzfr6 3fr2r 2 r Since E vanishes at the origin 3 C 0 3 fr 2r 39 E 2rf Vm Solutions Electromagnetics Workshop Solutions Divergence and Divergence Theorem Problem 2 a The electric eld in a free space region is given by E 3yz 2 4yzz y 23y i Vm What is the charge density in the region if any b Use a volume integral to nd the total charge in the unit cube with one corner at the origin one at 100 another at 010 and another at 001 c Check your result in part b by using a flux integral to compute the charge in the cube Solution a VDpV 80 8yz3zzypy pV 50 8yz3zzy Cm3 b Q oIp oIp e 8yz3zzy dxdydz sJ 4z zz dz 0 Q c c QmD dSgo 1 I o 0 580 T oIp oIp 11 5 4dd dd C 2 xz y xy 250 Electromagnetics Workshop Solutions Divergence and Divergence Theorem Problem 3 a Find the net ux out ofthe surface shown if D cos 3 sin z i i By examining D explain why the ux through each of the surfaces and is zero ii Evaluate the ux through surfaces and and hence determine the net ux out of the closed surface shown b Determine the net charge enclosed in the surface shown Solution a i D D D A 2 D 0 L0 Z 20 It ID dS licos pd dz 00 ah pm ha ii JD dSIIsin dszl g ah 7r za jD dSHzpdpd 00 71761211 Ira2 h 2 Hence P 2ah C 20201 b an 13 dS Qmm 71761211 Qencloszd zah Electromagnetics Workshop Divergence and Divergence Theorem Problem 4 Solutions a Show that the volume charge density corresponding to D 10sin2 3 p i10 Zcos2 2 is p independent of and z b Verify the divergence theorem for D with the cylindrical volume shown below Solution 1 6 1 6D 6D a pV V D pr 6 p By 62 1 1 6 6 102 10 s1n2 co 2 py p p pa p 61 vi 39 Z V 10s1n 10cosz p p b 10 Cm3 py p hZ Z alo RHSJ VDdV j H pdpd dz20nah V rhZ 0 0 p LHS jD dSjD dS j D dS 5de top bottom hZ 27 27rd LHS j j 10sin2 pd dz 2 I j Zcoszgjpdpdgj 412 0 Fa 0 0 p FMZ LHS 107rah107rah 207239ah Thus LHS RHS Electromagnetics Workshop Divergence and Divergence Theorem Problem 5 A static electric eld in free space is given by E i2r2 rsin639 r2 sin6cos Find the volume charge density that produced this electric eld Solution pv VQ50VE In spherical coordinates we have vgi23r2E 1E9sin6 E r 6r rs1n666 rs1n639 6 where Er2r2 E9rsin6 2 Eq r s1n6cos Hence we have pvso8r2cos6 rsin Cm3 Solutions

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