problemset1

1c opposite polarity voltages between holes and

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Unformatted text preview: E Force caused by E (which is caused by charge on electrodes) Force caused by B - Force caused by B Figure 1: Figure for 1.1C. Opposite polarity voltages between holes and electrons (Image by MIT OpenCourseWare.) A By Gauss’ Law: � �a �0 E · d� = � ρdV ; Sr = sphere with radius r SR VR � � As shown in class, symmetry ensures E has only radial compoent: E = Erˆr i LHS of Gauss’ Law: � �a �0 E · d� = SR 2π � 0 π � 0 � �� � �0 Erˆr · r2 sin θdθdφˆr i i � �� � d� in spherical coord. a = 2 4πr� Er �0 � �� surface area of sphere of radius r RHS of Gauss’ Law: For r < b: � � r � 2π π � ρdV = VR 0 0 0 ρr 2 r sin θdθdφdr �� � b� diff. vol. element 2 Problem Set 1 = 6.641, Spring 2009 4 π r4 π r 4 ρb ρb = 4b b � �� � vol of sphere For r > b & r < a (b < r < a): � � b 2π � π � ρdV = VR 0 0 0 ρb r 2 r sin θdθdφdr + b r � 2π � b 0 � π ρa r2 sin θdθdφdr 0 4πρb b3 4πρa (r3 − b3 ) =� + 3 4 � 4 = π ρb b3 + π ρa (r3 − b3 ) b < r < a 3 B Equating LHS and RHS ⎧ 4 ⎪ πr ⎨ ρb ; b 4πr2 Er �0 = 3 3 ⎪ ⎩ πρb b3 + 4πρa (a − b ) ; 3 ⎧2 ⎪ r ρb ; ⎨ 4�0 b Er = ρ...
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This document was uploaded on 03/19/2014 for the course EECS 6.641 at MIT.

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