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Unformatted text preview: −Hx (y = 0) = + · (1 − cosh ad) a sinh ad J0 sin ax (cosh ad − 1) Kz (y = d) = Hx (y = d) = − · a sinh ad =− 19 � Problem Set 1 6.642, Fall 2008 d) Equation of ﬁeld lines: Az (x, y ) = Az (x0 , y0 ). � � µJ0 Az (x, y ) = sin ax + C sinh ay + D cosh ay a2 � � sinh ay µJ0 (1 − cosh ad) − cosh ay = 2 sin ax 1 − a sinh ad � � µJ0 sinh ad − sinh ay + sinh a(y − d) = 2 sin ax a sinh ad x = a x, ˜ x = sin ˜ −1 A z a2 ˜ Az = µJ0 y = ay , ˜ � � ˜ ˜ sin x0 [sinh d − sinh y0 + sinh(y0 − d)] ˜ ˜ ˜ ˜ sinh d − sinh y + sinh(y − d) ˜ ˜˜ e) 2.0 ay 1.5 1.0 0.5 0.0 0 1 2 3 4 5 6 ax Figure 3: Magnetic ﬁeld vector arrow plot. � � � � � � 0.5 0.0 0 1 � � � � � � � � � � � � � � � � � � � � � 1.0 � � � � � ay 1.5 � � � � � � � � � � � � � � 2.0 � � � � � � 2 3 4 5 6 ax Figure 4: Magnetic ﬁeld line plot. f) In Problem 1e, electric ﬁeld lines terminate perpendicularly to the electrodes on surface charge. In Problem 2e, the magnetic ﬁeld lines are parallel to the conducting walls resulting in surface currents. 20 Problem Set 1 6.642, Fall 2008 Mathematica File for Computations Necessary to Generate Magneti...
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