# Document 13604271 ```8.022 Lecture Notes Class 26 - 10/31/2006
N Turns, Length L
B ﬁeld lines
�| = N
|B
• Single Loop ?
�
&micro;0 I
2πR
� &middot; d�l = &micro;0 I
B
Can’t do this b/c of ”edge” eﬀect
If we have enough loops, no edge
eﬀects!
each gives &micro;0 IN
• Use inﬁnite
loop for Ampere’s - location inside
� in
does not matter, so max B
solenoid
� | = &micro;0 IN = &micro;0 In (n = N , ”turn density” )
|B
L
L
2
� &times;B
� = &micro;0 J�
�
� &middot;B
� =0
�
� &times;E
� =0
�
� &middot;E = ρ
�
�0
�
Φ = V = − E &middot; dl
� =−
�
E
�V
So:
� &times;E =�
� &times; (�V
� ) = 0 , b/c gradient of scalar is 0
�
So we could write similarly,
� &times;B
� =�
� &times; (�
� )=0
�
Use :
� &middot;B
� =0
�
� &middot; (�
� &times;A
�) = 0
�
� = (�
� &times; A)
�
B
� &middot; A,
�
Multiple A’s have curls equal to B, so can choose easiest �
� &middot;A
�=0
Coulomb gauge �
� = B0 ẑ Find A
�
B
� = B0 &times; ŷ or − B0 yx̂
A
3
� c = B0 (xŷ − yx̂)
A
2
� &times;B
� =
�
� &times; (�
� &times;A
�) =
�
&micro;0 J
&micro;0 J�
� &middot;A
� �
� ) − �2 A
= �(
� = −&micro;0 J� (Poisson)
�2 A
� = 0 (Laplace)
�2 A
� x ) = &micro;0
A(�
4π
� � �� 3 �
J (x )d x
|�x − x�� |
A is the magnetic vector potential
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