Calculating B for circuits of various shapes
Basic idea:
I
}
I
Close up, any element of a
circuit basically looks like a
short straight wire
Intuition guided by
long straight wire :
. . . Near wire, B field lines will be approx. concentric circles
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Field lines of a circular
current
loop
zˆ
B
!
I
x
Field lines near wires are concentric circles similar to long straight wire field.
Field lines for two circular current loops
(field lines for each loop shown separately)
x
current
out
current
in
fields
cancel
x
3-D view
Arrow size indicates field strength. Contributions from red and
blue loops reinforce each other along central axis (purple line).
Fields cancel where arrows are equal and opposite.
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Field lines for two circular current loops
(combined field pattern)
x
x
3-D view
Near central axis, the contributions from the two loops give a roughly
uniform field, with a weaker field outside.
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Field lines for 5 circular current loops
.
.
.
.
.
x
x
x
x
x
The field strength
is increasing inside
and weakening
outside the loops.
3-D view
5 /23
Field lines for 21 circular current loops
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
As the number of current loops increases, the field near the axis becomes
stronger and more uniform, while the field outside becomes weaker.
In the limit of an infinite number of closely spaced loops, the field is uniform
inside and vanishes completely outside. This arrangement is called an
Ideal Solenoid
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Real
solenoids
are of finite
length and
typically
comprise
multiple
turns of a
single wire
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Magnetic field of Ideal Solenoid
B =0
!
uniform B
!
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
B =0
!
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