3209-2009-Solutions

3209-2009-Solutions - ECE 3209 Electromagnetic Fields...

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Unformatted text preview: ECE 3209 Electromagnetic Fields University of Virginia Fall 2009 Homework # 2 Solutions 1. To verify Gausss Theorem, lets first consider the surface integral, which we can break into two parts: I S ~ F d~a = Z side ~ F d~a side + Z top ~ F ~a top where d~a side = r sin drd, and d~a top = rr 2 sin dd Thus, I S ~ F d~a = 4 Z r = R r =0 Z =2 =0 r 3 cos sin fl fl fl =30 drd + Z =30 =0 Z =2 =0 r 4 sin 2 fl fl fl r = R dd = 8 cos sin fl fl fl =30 Z r = R r =0 r 3 dr + 2 R 4 Z =30 =0 sin 2 fl fl fl r = R d = 2 3 h r 4 4 i R + R 4 h - sin 2 2 i / 6 = 3 R 4 2 + R 4 6- 3 4 I S ~ F d~a = R 4 3 4 + 6 Now, lets take the divergence of ~ F : ~ ~ F = 1 r 2 r r 2 F r + 1 r sin F sin + 1 r sin F = 1 r 2 r r 4 sin + 1 r sin 4 r 2 cos sin + 1 r sin ( r 2 tan ) ~ ~ F = 4 r sin + 4 r cos 2 - sin 2 sin = 4 r cos 2 sin Integrating this over the volume of the ice cream cone, we have Z V ~ ~ F dv = 4 Z R r 3 dr Z = / 6 =0 cos 2 d Z =2 =0 d = R 4 + sin 2 2 / 6 = R 4 6 + 3 4 Which is the same result we obtained for the surface integral. This shows that Gausss Theorem is satisfied. 2. Cheng, P. 2-32. Consider the vector field, ~ D = r cos 2 r 3 between two spherical shells of radius r = 1 and r = 2 . (a) Evaluating the surface integral, we have I S ~ D d~a = Z sin d 1 r Z 2 cos 2 d fl fl fl r =2- 1 r Z 2 cos 2 d fl fl fl r =1 = 1 4 h cos i + sin2 2 2 =- (b) Evaluating the volume integral of the divergence, we have ~ ~ D = 1 r 2 r r 2 cos 2 r 3 =- cos 2 r 4 Z V ~ ~ D dv =- Z R =2 R =1 dr r 2 Z = =0 sin d Z =2 =0 cos 2 d = 1 r fl fl fl R =2 R =1 h- cos i h 2 + sin2 4 i 2 =- 1 2 (2)( ) =- which is the same as part (a), as expected. 3. Cheng, P. 2-36. Given the vector function, ~ A = sin( / 2) we have (for the only non-zero components of the curl), ~ ~ A = r 1 r sin (sin( / 2)sin )- 1 r r ( r sin( / 2)) = sin( / 2) r r cot - Integrating this over the hemispherical surface, we get Z S ~ ~ A d~a = r Z = / 2 =0 cos d Z =2 =0 sin( / 2) d...
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This note was uploaded on 10/16/2010 for the course ECE 309 taught by Professor Weikle during the Fall '08 term at UVA.

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3209-2009-Solutions - ECE 3209 Electromagnetic Fields...

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