problemset1

# 1c opposite polarity voltages between holes and

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/19/2014 for the course EECS 6.641 at MIT.

Ask a homework question - tutors are online