problemset2

in limit c 0 s s if c is 0 then c a d l

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Unformatted text preview: λ0 az λ0 az |2 Ez = ≈ = 4πε0Qz| − 2 + z2) 3 2ε0 |z |3 2 z<0 2ε0 (a 4πε |z |2 0 5 Problem Set 2 6.641, Spring 2009 a r dr Figure 3: A line charge ring of width dr in the disk (Image by MIT OpenCourseWare.) D From (C), Φ = λ0 r 1 2ε0 (r 2 +z 2 ) 2 for a ring of radius r. But now we have σ0 , not λ0 . How do we express λ0 in terms of σ0 ? Take a ring of width dr in the disk (see figure). Total charge in the ring = (r)(2π ) (dr)σ0 . � �� � circumference Line charge density = λ0 = So: λ0 = σ0 dr total charge length = σ0 dr σ0 rdr dΦ = 1 2ε0 (r2 + z 2 ) 2 �a �a σ0 rdr σ0 rdr Φtotal = = 2 + z2) 1 2ε0 0 (r2 + z 2 ) 1 2 2 0 2ε0 (r �� ��r=a �� � σ0 σ0 � = = r2 + z 2 � a2 + z 2 − |z | 2ε0 2ε0 r =0 � � σ0 1 1 → − √ −√ E = −�Φtotal = z iz 2ε0 a2 + z 2 z2 E As z → ∞, 1 (a2 + z 2 ) 2 → |z | + Φtotal → a2 ; 2|z | 1 (a2 + z 2 )− 2 → 1 |z | � 1− π a2 σ0 4�0 π |z | πa2 σ0 ¯ E→ iz 4πε0 z 2 just like a point charge of σ0 πa2 . F As a → ∞, z in the Φtotal → √ a2 + z 2 can be neglected, so σ0 [a − |z |] 2ε0 6 a2 2z 2 � Problem Set 2 6.641, Spring 2009 � � � σ0 σ0 z 1 0 − 0 = 2εσ0 Ez → − 2ε0 |z | 2ε z>0 z<0 0 just like a sheet charge. Problem 2.3 A By the divergence theorem: i � V − → � · (� × A )dV = � S − → → (� × A ) · d− a where S encloses V . ii By Stokes’ Theorem: � � − → −→ →− → −= (� × A ) · d a A ·d l S� C Suppose S is as in figure 4 S Figure 4: Closed surface S (Image by MIT OpenCourseWare.) and S � is as in figure 5 S C Figure 5: Open surface S � bounded by contour C (Image by MIT OpenCourseWare.) i.e., S � is the same as S , except for the contour curve C , which makes S � slightly unclosed. Now consider limit as C → 0 (Figure 6) 7 Problem Set 2 6.641, Spring 2009 Figure 6: Limit as C → 0 (Image by MIT OpenCourseWare.) � → → � − − → − → In limit C → 0, S � → S . If C is 0, then C A · d l = 0. By equation (ii), S (� × A ) · d− = 0. By a � − → equation (i), V � · (� × A )dV = 0. Since V can be any volume, argument of integral must be identically 0. − → � · (� × A...
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This document was uploaded on 03/19/2014 for the course EECS 6.641 at MIT.

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