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I
I
B
B
I
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.
Magnetic field from current loop
0.9
2a
x
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
z
2a
dBz  dB cos 
dBx  dB sin 
z
x components cancel
dB

dB
r2  a2  z2
cos  
r
r
z
z >> a

a
z
r
a
x
B
right hand
screw rule
magnetic dipole moment
m
miA
At a point along the axis
i
current loop in xy-plane
Bz 
z >> a
0
iA
2 z 3
B
Diamagnetic material
m < 0 (small)
B = o (1+ m ) H
H
Permeability
 = o (1+ m ) =slope of B-H line
B
Ideal magnetic material
or paramagnetic material
m > 0 (small)
B = o r H =  H
 = constant = slope of B-H curve
H
L11.5 : Magnetization
If H is large or substance strongly magnetic (e.g. ferromagnetic), as H increases, the
magnetization M (and hence B) may increase nonlinearly:
Measure
from the
graph
So r varies with H.
Could also use “differential permeability”
High field region where slope decreases is called
"saturation" region.
L11.6 : Magnetization
Hysteresis
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”
L11.7 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 striking
or heating it, or go round the hysteresis loop,
gradually reducing its size. "Degaussing"
L9.1 : Magnetic fields due to
currents
Magnetic fields are produced by
currents.
Biot-Savart law
Example:
Ampere’s law
so
L9.2 : Magnetic fields due to
currents
A solenoid:
(n is number of turns/length)
Therefore
(inside)
L9.3 : Magnetic fields due to
currents
Use the Biot-Savart law to derive the magnetic field
on the axis of a current loop:
and
Therefore
L9.4 : Magnetic fields due to
currents
Magnetic field of the Earth
L9.5 Magnetic fields due to
currents
The magnetic field of a magnetic dipole:
(I,
A0)
This magnetic field has the same shape as the electric field of an
electric dipole: do the exercise in the Exercise Set.
Y
X
+
Z
+
+
+
+
+
+
-
-
-
w
I
-
-
-
B
-
-
-
-
t
charge carriers are electrons for copper
Right hand rule  electrons are deflected down
 bottom of probe is negative
Y
X
Z
w
I
t
B
Hysteresis Curve for an Iron sample
Saturation of M
2.0
1.5
retentivity (remanence)
1.0
MH
B (T)
0.5
0.0
coercivity
-0.5
-1.0
retentivity (remanence)
-1.5
-2.0 of M
Saturation
-100
-50
0
H
50
100
-1
(A.m )
Area enclosed = energy dissipated in a cycle in reversing the magnetic domains
H
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