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8.022 Lecture Notes Class 26 - 10/31/2006 N Turns, Length L B ﬁeld lines �| = N |B • Single Loop ? � µ0 I 2πR � · d�l = µ0 I B Can’t do this b/c of ”edge” eﬀect If we have enough loops, no edge eﬀects! each gives µ0 IN • Use inﬁnite loop for Ampere’s - location inside � in does not matter, so max B solenoid � | = µ0 IN = µ0 In (n = N , ”turn density” ) |B L L 2 � ×B � = µ0 J� � � ·B � =0 � � ×E � =0 � � ·E = ρ � �0 � Φ = V = − E · dl � =− � E �V So: � ×E =� � × (�V � ) = 0 , b/c gradient of scalar is 0 � So we could write similarly, � ×B � =� � × (� � )=0 � Use : � ·B � =0 � � · (� � ×A �) = 0 � � = (� � × A) � B � · A, � Multiple A’s have curls equal to B, so can choose easiest � � ·A �=0 Coulomb gauge � � = B0 ẑ Find A � B � = B0 × ŷ or − B0 yx̂ A 3 � c = B0 (xŷ − yx̂) A 2 � ×B � = � � × (� � ×A �) = � µ0 J µ0 J� � ·A � � � ) − �2 A = �( � = −µ0 J� (Poisson) �2 A � = 0 (Laplace) �2 A � x ) = µ0 A(� 4π � � �� 3 � J (x )d x |�x − x�� | A is the magnetic vector potential