Download Magnetic field

29-30 Magnetism - content
Magnetic Loadspeaker
•Magnetic Force – Parallel conductors
•Magnetic Field
•Current elements and the general magnetic
force and field law
•Lorentz Force
•Origin of magnetic force
Recording and playback unit
•Application of magnetic field formula
•”Amperes” circuital law
•Application of the circuital law
•Magnetic dipoles
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29-30 Magnetism
is the interactions between charge in motion, i.e. currents.
Ampere 1820-1825 measured interactions between currents in
closed conductors
2
Starting point is the parallel currents
I1
I2
L
r
F12
 0 I1 I 2

Lrˆ
2r
Sign of current according to
direction =>
•anti-parallel currents repell
•parallel currents attract
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Magnetic field
F12
 0 I1 I 2

Lrˆ
2r
F12  I 2 L  B12
B12
L in current direction
 0 I1

ˆ
2r
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Current element
A current element is a vector defined
as
Magnified
current
element
dt
da
J
dl
Idl  J  da dl  nev dt  qv
dI = J . da
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Magnetic field from a current element
We will show that contribution from a current element is
0 I dy sin 
dB 
4
r2
r  r 2  y2
Total field in point B is then
r
0 I
B
4
0 I

4
For
a

a
a

dy
a
dy sin  0 I

2
r
4
r
( r 2  y 2 )3 / 2
a

a
dy
r
r
3

0 I
2a

4 r r 2  a 2
0 I
a  , B 
2r
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General magnetic force law
dF12  I 2 dL2  dB12
0 I1 dy sin  0 I1 dL1  rˆ
dB12 

2
4
r
4
r2
dF12
Since
Law of Biot-Savart 1820
0 I1I 2
dL1  rˆ

dL2 
4
r2
Idl  J  da dl  nev dt  qv
v2
F12
 0 q1q2
v1  rˆ

v2  2
4
r
v1
q2
r
q1
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Field Theory
8
The Hall effect
A current carrying conductor in a magnetic field
DV = V2-V1 = EHL = vdBL.
A Hall probe can be used to
”measure” the magnetic field.
L
9
(Interaction between moving free charges)
Consider two electron beams:
From this we conclude:
R
e-
fe
fm
fm
e-
fe
e-
fe
fm
e
-
V
v
fe
V’
v’
1
R
fe 
fm
 3.0 108 m / s  c
Use
 0 0
q2
v2q 2 ˆ
fm  
R
2
2
40c R
40 R 2
fTOT
Rˆ
1
q2  v2  ˆ
1  2  R

2 
40 R  c 
1
10
(Relative motion)
Electromagnetism
Observer at
rest
V
V
Electric Force
V
Observer in
motion
Relative rest
q1 q 2 Rˆ
fe 
40 R 2
1 q1q2 v 2 ˆ
Magnetic Force f m  
R
40 R 2 c 2
0 
1
 0c 2
R
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(Origin of magnetic effect – interactions take time)
Assume
• Interaction speed c
R*=ct
• Invariance of interaction speed
vt
v
v
R=ct0
R
R* 2
R *  R  (vt)  R  (v
)
c
v2
2
R * (1  2 )  R 2
c
R2
2
R* 
v2
1 2
c
2
2
2
2
In motion, interaction occurs over a larger
distance, R*, and the strength decreases.
Coulombs law
changes to
f em
q1 q 2 Rˆ
fe 
40 R 2
1 q1q2 ˆ
1 q1q2  v 2  ˆ
1  2  R

R
2
2 
*
40 R
40 R  c 
which is electric plus magnetic force
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Calculations of the magnetic field
1. Field on axis from a circular current loop
0 I dL  rˆ
dB 
4
r2
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2. Field from an ”infinite” current plane
Consider plane to consist of parallel threads of infinitesimal thickness
y

K is
current line
density
(A/m)
x
r
Q
Q
y
From one thread
0 dI
dB 
ˆ
2r

dB  0
2
0

2
Kdy
y r
2
K
y r
2
2
(
2

0
0 K
1
ˆ
ˆ
By
Kr  2
dy  y
2
2
y r
2

(sin xˆ  cos yˆ ) 
y
y r
2
2
xˆ 
r
y r
2
2
yˆ )dy
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3. The solenoid field
A solenoid is an infinitely long coil. It is built up by parallel loops:
On the axis
Sum all contributions from the
loops ( see example 30.4 in
Benson) to get
N
B I
L
where N is number of turns
and L is length of solenoid
equivalent to two parallel planes15
”Ampere’s” circuital law for the magnetic field
If C is a circle with radius r
I
C
0 I
0 I
0 I
C B  dl  2r C ˆ  dl  2r C dl  2r 2r  0 I
I
r
dl
For an arbitrary integration curve
C
I
I
r
dl
0 I
0 I
ˆ
C B  dl  C 2r  dl  C 2r rd  0 I
 B  dl   I
Current enclosed
by curve C
0 encl
C
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Verification of Amperes circuital law
1. Current carrying plate
Integration
path C
 B  dl   I
B
0 encl
C
B 2 L  0 KL
I = KL
B
L
2. Solenoid
0 K
2
since solenoid approximation means
neglecting all field outside coil
L
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Application of Circuit Law
Coaxial cable with
homogenous current over
cross sectional area:
a. Identify symmetry: cylindrical, i.e. circles around axis.
b. Choose integration path as circles around axis
 B  dl   I
0 encl
I
 0  J  da
C
S
where S is the surface bounded by C
I
1.
r  r1
Current density
Integration path
I
J  2 nˆ
r1
I
B 2r   0 2 r 2
r1
I
B  0
r
2
2r1
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 B  dl   I
Coaxial cable with
homogenous current over
cross sectional area:
0 encl
C
S
2.
I
I
 0  J  da
r2  r  r1
B 2r   0 I
B  0
I
2r
r
Integration path
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 B  dl   I
0 encl
Coaxial cable with
homogenous current over
cross sectional area:
C
3.
S
r3  r  r2
Current density
I
I
r
Integration path
 0  J  da
I
J
nˆ
2
2
 (r3  r2 )
I
2
2
B 2r   0 I   0
(

r


r
2 )
2
2
 (r3  r2 )
 0 I  (r32  r22 )  (r 2  r22 )
B

2
2
2r
 (r3  r2 )
 0 I r32  r 2

2r (r32  r22 )
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 B  dl   I
0 encl
Coaxial cable with
homogenous current over
cross sectional area:
C
4.
I
I
 0  J  da
S
r  r3
I encl  I  I  0  B  0
Integration path
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Magnetic dipoles
Compare a solenoid with a permanent bar magnet
A current loop is the infinitesimal
magnetic dipole.
What is its dipole moment?
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Torque and energy for interacting magnetic dipole
Torque is
Magnetic dipole moment is defined
so that
and vectorially
t   B
Energy
Work to rotate from
aligned to anti-aligned is
So that magnetic energy is
U m    B
Equivialent with electric dipole
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formulas. (Minus sign is
conventional, but not correct)
Earth Magnetism
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Magnetism in Biology
Magnetite found in animals
Solomon fish
Bacteria
Pigeon bird
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