i i. magnetic field similar to a bar magnet for a very long solenoid, the magnetic field can be...
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I
B
I
I
B
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Magnetic field similar to a bar magnet
For a very long solenoid, the magnetic field can be considered to be confined to the region inside the coils.
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Magnetic field from current loop
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2a
z
x
2a
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x
z
a
zr
dB
dB
cos sinz xdB dB dB dB
x components cancel
2 2 2r a z
z >> a
cosz
r
a
r
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i
B
m
At a point along the axis z >> a
032zB i A
z
current loop in xy-plane
m i A
magnetic dipole momentright hand screw rule
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H
B
Diamagnetic material
m < 0 (small)
B = o (1+ m ) H
Permeability
= o (1+ m ) =slope of B-H line
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Ideal magnetic material or paramagnetic material
m > 0 (small)
B = o r H = H
= constant = slope of B-H curve
B
H
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L1L111.5 .5 : : MagnetizationMagnetizationIf H is large or substance strongly magnetic (e.g. ferromagnetic), as H increases, the magnetization M (and hence B) may increase nonlinearly:
Measure from thegraph
So r varies with H. Could also use “differential permeability”
High field region where slope decreases is called "saturation" region.
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L1L111.6 .6 : : MagnetizationMagnetizationHysteresis
Ferromagnetic materials also show a “hysteresis” effect, where decreasing the applied magnetic field, or H, doesn’t produce the reverse effect of increasing the field:
Br = “remanence” or“residual magnetism”
Hc = “coercivity”
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L1L111.7 Magnetization.7 Magnetization“hard” magnetic materials: Hc is high, area of the loop is
large, used for permanent magnets.“soft” magnetic materials: Hc is small, area of loop is
small, used for transformer cores & electromagnets.
Material can be demagnetized by strikingor heating it, or go round the hysteresis loop,gradually reducing its size. "Degaussing"
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LL99.1 .1 : : Magnetic fields due to Magnetic fields due to currents currents
Magnetic fields are produced by currents.
Biot-Savart law
Ampere’s law
Example:
so
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LL99.2 .2 : : Magnetic fields due to Magnetic fields due to currents currents
A solenoid:
(n is number of turns/length)
Therefore (inside)
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LL99.3 .3 : : Magnetic fields due to Magnetic fields due to currents currents
Use the Biot-Savart law to derive the magnetic field on the axis of a current loop:
and
Therefore
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LL99.4 .4 : : Magnetic fields due to Magnetic fields due to currents currents
Magnetic field of the Earth
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LL99.5 Magnetic fields due to .5 Magnetic fields due to currents currents
The magnetic field of a magnetic dipole:
This magnetic field has the same shape as the electric field of anelectric dipole: do the exercise in the Exercise Set.
(I, A0)
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w
tI
B
-
+ + + + + + +
- - - - - - - - -
charge carriers are electrons for copperRight hand rule electrons are deflected down bottom of probe is negative
Z
X
Y
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w
tI
B
Z
X
Y
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Hysteresis Curve for an Iron sample
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
-100 -50 0 50 100
H (A.m-1)
B
(T
)
M H
Saturation of M
coercivity
retentivity (remanence)
retentivity (remanence)
Saturation of M
Area enclosed = energy dissipated in a cycle in reversing the magnetic domains
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A B
CD
F
J
E
K
H
I
LI
G