11
29-30 Magnetism - content
•Magnetic Force – Parallel conductors
•Magnetic Field
•Current elements and the general magnetic force and field law
•Lorentz Force
•Origin of magnetic force
•Application of magnetic field formula
•”Amperes” circuital law
•Application of the circuital law
•Magnetic dipoles
Magnetic Loadspeaker
Recording and playback unit
2
Ampere 1820-1825 measured interactions between currents in closed conductors
29-30 Magnetism
is the interactions between charge in motion, i.e. currents.
3
L
I2I1
Starting point is the parallel currents
ˆ
2210
21 LII
F Sign of current according to direction =>
•anti-parallel currents repell
•parallel currents attract
4
Magnetic field
ˆ
2210
21 LII
F
21221 BLIF
ˆ
210
21
IB
L in current direction
5
da dl
d
dI = J . da
Magnified current element
J
Current element
A current element is a vector defined as
vqdvnedladJlId
6
Magnetic field from a current element
20 sin
4 r
dyIdB
We will show that contribution from a current element is
Total field in point B is then
22
02/322
0
30
20
2
4)(4
4
sin
4
a
aI
ydy
I
rdy
I
r
dyIB
a
a
a
a
a
a
For 2
, 0IBa
22 yr
7
2110
210
21
ˆ
4
sin
4 r
rLdI
r
dyIdB
General magnetic force law
212221 BdLdIFd
21
2210
21
ˆ
4 r
rLdLd
IIFd
Since vqdvnedladJlId
21
2210
21
ˆ
4 r
rvv
qqF
q1
q2
r
v2
v1
Law of Biot-Savart 1820
8
Field Theory
9
The Hall effect
A current carrying conductor in a magnetic field
V = V2-V1 = EHL = vdBL.
L
A Hall probe can be used to ”measure” the magnetic field.
10
(Interaction between moving free charges)
vv’
fm fm
R
fefe
e-e-
Consider two electron beams:
fm fm
V
V’
fe
fe
e- e-
From this we conclude:
RR
qv
cfm ˆ
4
12
22
20
Rc
v
R
qfTOT ˆ1
4
12
2
2
2
0
RR
qfe ˆ
4 20
2
csm /100.31 8
00
R
Use
11
V V
Observer at rest
Observer in motion
V
Relative rest
(Relative motion)
Electromagnetism
Electric Force
Magnetic Force
20
21
4
ˆ
R
Rqqf e
Rc
v
R
qqfm ˆ
4
12
2
221
0
R
20
0
1
c
12
(Origin of magnetic effect – interactions take time)
vR=ct0
R
vt
v
R*=ct
2
2
22
22
22
22222
1*
)1(*
)*
()(*
cv
RR
Rc
vR
c
RvRvtRR
Assume
• Interaction speed c
• Invariance of interaction speed
In motion, interaction occurs over a larger distance, R*, and the strength decreases.
Coulombs law
changes to2
0
21
4
ˆ
R
Rqqf e
Rc
v
R
qqR
R
qqfem ˆ1
4
1ˆ4
12
2
221
02*
21
0
which is electric plus magnetic force
13
2
0 ˆ
4 r
rLdIBd
1. Field on axis from a circular current loop
Calculations of the magnetic field
14
2. Field from an ”infinite” current plane
x
r
y
y
Q
K is current line density (A/m)
Consider plane to consist of parallel threads of infinitesimal thickness
dyyry
rx
ry
y
ry
K
yxry
KdyBd
)ˆˆ(2
)ˆcosˆ(sin2
222222
0
22
0
ˆ
20dIBd
From one thread
2ˆ
1
2ˆ 0
220 K
ydyry
KryB
15
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
IL
NB
where N is number of turns and L is length of solenoid
equivalent to two parallel planes
16
”Ampere’s” circuital law for the magnetic field
dlrII
C
Irr
Idl
r
Ild
r
IldB
CCC
0000 2
22ˆ
2
If C is a circle with radius r
II
C dlr
For an arbitrary integration curve
Irdr
Ild
r
IldB
CCC
000
2ˆ
2
encl
C
IldB 0Current enclosed by curve C
17
Verification of Amperes circuital law
1. Current carrying plate
I = KL
L
B
Integration path C
encl
C
IldB 0
2
2
0
0
KB
KLLB
2. Solenoid
since solenoid approximation means neglecting all field outside coil
L
18
Application of Circuit Law
Coaxial cable with homogenous current over cross sectional area:
a. Identify symmetry: cylindrical, i.e. circles around axis.
1. 1rr Current density n
r
IJ ˆ
21
b. Choose integration path as circles around axis
S
encl
C
adJIldB 00 I
I
Integration path
where S is the surface bounded by C
rr
IB
rr
IrB
21
0
22
10
2
2
19
I
I
Integration path
2. 12 rrr
r
IB
IrB
2
2
0
0
S
encl
C
adJIldB 00 Coaxial cable with homogenous current over cross sectional area:
r
20
I
I
Integration path
3. 23 rrr
Current density nrr
IJ ˆ
)( 22
23
S
encl
C
adJIldB 00
)(2
)(
)()(
2
)()(
2
22
23
2230
22
23
22
222
230
22
22
22
300
rr
rr
r
I
rr
rrrr
r
IB
rrrr
IIrB
Coaxial cable with homogenous current over cross sectional area:
r
21
I
I
Integration path
S
encl
C
adJIldB 00
4. 3rr
00 BIIIencl
Coaxial cable with homogenous current over cross sectional area:
22
Magnetic dipoles
Compare a solenoid with a permanent bar magnet
A current loop is the infinitesimal magnetic dipole.
What is its dipole moment?
23
Torque and energy for interacting magnetic dipole
Torque is
Magnetic dipole moment is defined
so that
and vectorially BEnergy
Work to rotate from aligned to anti-aligned is
So that magnetic energy is BUm Equivialent with electric dipole formulas. (Minus sign is conventional, but not correct)
24
Earth Magnetism
25
Magnetism in Biology
Magnetite found in animals
Bacteria
Pigeon bird
Solomon fish
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