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