problemset1

Sinh 2y0 d y a cosh a20 2 2 for plotting purposes it

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: � � sinh ad + sinh a(y − d) − sinh ay 1 1 ad −1 cosh 2 (d − y ) 2 tanh + ln − sinh a(d − 2y ) dy = cosh cosh a(y − d) − cosh ay a cosh a2y 2 2 � � � �� � � � cosh a (d − y ) a ad ln(cos ax) 1 2 2 tanh−1 + ln sinh (2y − d) + cosh = C (x0 , y0 ) + ay a a cosh 2 2 2 � �� � � � �� cosh a (d − y0 ) 1 ad a 2 C (x0 , y0 ) = cosh ln(cos ax0 ) + 2 tanh−1 + ln sinh (2y0 − d) y a cosh a20 2 2 For plotting purposes, it is best to use non-dimensional variables: ˜ Ex = Ex �a/ρ0 , ˜ Ey = Ey �a/ρ0 , x = a x, ˜ y = ay , ˜ ˜ d = ad, ˜˜ ˜ C (x0 , y0 ) = aC (x0 , y0 ) ˜ ˜ ˜ Ex = − cos x[sinh d + sinh(y − d) − sinh y ]/ sinh d ˜ ˜˜ ˜ ˜ ˜ Ey = − sin x[cosh(y − d) − cosh y ]/ sinh d ˜ ˜˜ ˜ Field lines: −1 ln(cos x) + 2 tanh ˜ ⎡ ⎣ cosh �˜ d−y ˜ 2 cosh ⎡ �⎤ � � ⎦ cosh y ˜ 2 cosh �˜ d−y0 ˜ 2 �� ˜ d 2 � � �� ˜ d + ln sinh y − ˜ 2 �⎤ � � �� ˜ d � � ⎦ cosh + ln sinh y0 − ˜ = ln(cos x0 ) + 2 tanh−1 ⎣ ˜ ˜ 2 cosh y0 2 �� ˜ d 2 e) 2.0 ay 1.5 1.0 0.5 0.0 0 1 2 3 4 5 ax Figure 1: Electric field vector arrow plot. � � ˜ Ey cosh(y − d) − cosh y ˜˜ ˜ sin x ˜ dy ˜ = = ˜ ˜ dx ˜ cos x sinh d + sinh(y − d) − sinh y ˜ Ex ˜˜ ˜ 2 6 2.0 ay 1.5 1.0 0.5 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �...
View Full Document

This document was uploaded on 03/19/2014 for the course EECS 6.642 at MIT.

Ask a homework question - tutors are online