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Unit 7: Sources of magnetic field
� Oersted’s experiment.
� Biot and Savart’s law.� Magnetic field created by a circular loop
� Ampère’s law (A.L.).
� Applications of A.L. Magnetic field created by a:
� Straight current-carrying wire
� Coil
� Magnetic flux trough a surface.
� Maxwell’s equations for Magnetostatics.
� Magnetism in matter. Ferromagnetism.
Jean-Baptiste Biot Félix Savart
André Marie Ampère
Oersted’s experiment. 1820
Tipler, chapter 27,2
� 1. If switch is off, there isn’t
current and compass needle is
aligned along north-south axis
� 2. If switch is on, current
aligns compass needle
perpendicular to current.
� 3. If current flows in opposite
direction, compass needle is
aligned in opposite direction.
An electric current creates a magnetic field
F
F
Biot and Savart’s law
� Magnetic field created by acurrent is perpendicular tocurrent, and depends on theintensity of current and distancefrom current.
Tipler, chapter 27-2
� Magnetic field created by acurrent element (Idl ) at a pointP is:
Br
d
i
P
2
r0
3
0
r
di
4r
di
4d
ulrlB
rrrrr ×
=×
=π
µ
π
µ
dl direction is the same as iµ0 (vacuum magnetic permittivity)=4π10-7 Tm/A
rr
ru
rr
=
rr
l
rd
∫×
=
B
A
3
0
r
d
4
i rlB
rrr
π
µ
� Magnetic field created by a finite piece of wire is the sum (integral) of each current element at P:
l
rd
Br
i
A
B
Biot and Savart’s law
P
This equation can be applied to different conductor shapes, straight
conductors, circular conductors,…….
rr
i
Tipler, chapter 27.2
� Magnetic field lines created by a straight current-carrying wire are circular in shape aroundconductor:
Magnetic field lines
Direction of magnetic field comes from right-hand or screw rule
Br
Br
Br
� Magnetic field lines created by a circular loop:
i
Magnetic field lines
https://www.youtube.com/watch?v=V-M07N4a6-Y
Br
Br
Br
� As magnetic poles cannot exist isolated (north pole orsouth pole), any field line exiting from a north pole must goto a south pole, and all magnetic field lines are closed lines.
Magnetic field lines
� Ampère’s law relates the integral of magnetic fieldalong a closed line and the intensity passing througha surface enclosed by this line. Closed line C must bechosen by us (if possible, should be a magnetic fieldline):
∫ ∑=⋅c i
i0 Id µlBrr
I1I2
Ii...
c
Tipler, chapter 27.4
Ampère’s law.
Ampère’s law
� Each intensity has it own sign, according to the r¡ght-hand or screw rule.
� Ampère’s law is equivalent (in Electromagnetism), to Gauss’s law in Electrostatics.
I1>0I2>0
Ii<0
c
I>0 ...
Ampère’s law.
� It’s used to compute magnetic fields where symmetryexists.
� In order to easily compute the integral of line, thechosen closed line C should have two features:
� a) Modulus of magnetic field should be equal at everypoints on closed line C.
� b) Magnetic field vector (B) should be parallel to closedline C at every points along C.
� In this way: ∫ ∫∫ ===⋅c cc
BLdlBBdld lB
rr
Ampère’s law.
i
� Let’s take a straight current-carrying wire. Fieldlines of this conductor are circumferences.Choosing one of such lines of radius R, surfaceenclosed by such line and applying A.L:
R
B
B
B
BL
IR2BdlBd 0
LL
µπ =⋅==⋅ ∫∫ lBrr
R2
IB 0
π
µ=
Tipler, chapter 27.4
Application of A.L: straight current-carrying wire.
� A conductor creates a magnetic field on second conductorand a force appears on this conductor. The same happenson first conductor.
Tipler, chapter 27.4
Force between two straight current-carrying wires
d
I2
I1 r
B2
r
B1
r
F21 r
F12r
l
http://www.youtube.com/watch?v=43AeuDvWc0k
SBrr
dd ⋅=φ
Sdr
Br
�Unit: Weber
�Wb = T m2
� Given a surface element dS, magnetic flux through
such surface element is defined as (inner product):
�Tipler, chapter 28.1
Magnetic flux
� If surface is finite (surface S): ∫ ⋅=S
SdBrr
φ
� On a closed surface, as
magnetic field lines are
closed lines, an entering
line must always exit
from volume, and
magnetic flux through a
closed surface is always
zero:
entering flux (-) must be equal
to exiting flux (+).
Magnetic flux
0SdB
SurfaceClosed
=⋅∫rr
� On 1865, J.C. Maxwell stated his four famous Maxwell’s
equations, a summary of electromagnetic field. For steady
magnetic fields (magnetic fields not changing on time),
these equations can be written as:
�Maxwell’s equations for Magnetostatics
0ldE =⋅∫rr
0
i
SurfaceClosed
QSdE
ε
∑∫ =⋅
rr
0SdB
SurfaceClosed
=⋅∫rr
∑∫ =⋅ i0 IldB µrr
�E is conservative �Gauss’s law
�Ampere’s law �Monopoles don’t exist
For Magnetodynamics, it’s necessary modify these equations, and a no conservative electric field
appears and a new term must be added on Ampere’s law.
R2
NiB
0
π
µ=
� Applying A.L. to middle line of toroid and to circleenclosed by this line:
Tipler, chapter 27.4
i
i
R
N turns
B
NIR2BdlBd 0
LL
µπ =⋅==⋅ ∫∫ lBrr
Application of A.L: toroid (circular solenoid).
By applying A.L. at points outsideof toroid, result is that magneticfield is zero at any point outside
toroid.
• B=0
• B=0
B
B
B
niL
NiB 0
0µ
µ==
� On a solenoid, if L>>>r, the magnetic field can be taken asuniform inside solenoid and null outside solenoid. From toroid(L=2πR):
N turns
B
L
Application of A.L: solenoid.
r
L
Nn =
Number of turns
by unit of lenght
If we put a ferromagnetic material inside solenoid , magnetic field is multiplied by thousands (with the same intensity of current flowing along solenoid).
SNirr
=µMagnetic moment of a solenoid is:
Magnetic moment by unit of volume inside a solenoid
is called magnetization:
0
B
SL
SNi
VM
µ
µrrr
r===
B
B
Magnetism in matter. Ferromagnetism.
� Magnetic properties of ferromagneticmaterials can be explained by theiratomic structure.
� An electron in its atomic orbit can beconsidered as an electric current flowingthrough a loop. So, the electronproduces a magnetic field, and themagnetic moment (µ) of the electroncan be computed.
e-
Sirr
=µ
� In the atoms of many materials, such magnetic moments arecancelled, but in ferromagnetic materials, a resultingmagnetic moment is not zero:
0=µr 0≠µ
r
Atom ofNon ferromagnetic
material
Atom ofFerromagnetic
material
Tipler, chapter 27.5
Magnetism in matter. Ferromagnetism.� In ferromagnetic materials, there are regions (magneticdomains) with their magnetic moments all pointing in thesame direction. Magnetic moment by unit of volume is called
Magnetization (it is a vector):dV
dµr
r=M
� In a domain: but in all domains0domainone
≠∑Mr
0domainsall
=∑Mr
� Directions on wich domains are oriented, are called easymagnetization directions, and are related to the crystallinestructure of the material.
� Magnetization is not zero and a magnetic field Bm appears (due to
magnetization) reinforcing the applied magnetic field. Bm depends on Bappthrough a characteristic of material called magnetic susceptibility (χm):
Magnetism in matter. Ferromagnetism.� When we apply an external magnetic field (Bapp), the magnetic moments
at edge of domains change their direction according to Bapp.
appmm BBrr
χ=
appmmapp )1( BBBBrrrr
χ+=+=
BappBm
B=Bapp+Bm > Bapp
1
� So, the resulting magnetic field will be:
Magnetism in matter. Ferromagnetism.
� By increasing the applied magnetic field Bapp, some domainsare pointing in the direction of easy magnetization(Barkhausen effect). Such magnetization increases Bm andtotal field still more……
Bm
B=Bapp+Bm > Bapp
Bapp
2
Magnetism in matter. Ferromagnetism.
� Until all domains are pointing in easy magnetization directionscloser to the applied magnetic field Bapp.
Bm
B=Bapp+Bm > Bapp
Bapp
3
Magnetism in matter. Ferromagnetism.� The last step occurs when all domains are pointing in thedirection of the external applied field. We have got the highermagnetization (saturation magnetization).
Bm
B=Bapp+Bm > BappBapp
� Resulting field (B) is thousand times the applied field:
appmmapp )1( BBBBrrrr
χ+=+=
Permalloy: χm = 25000
4
Ferromagnetism. First magnetization curve.
� Magnetization process is non linear (but only in one region (3)of curve), and drawing B vs Bapp we get first magnetizationcurve of a ferromagnetic material:
M
First imantation curve of a ferromagnetic material
Saturation magnetization
Bapp
4
3
2
1
Ferromagnetism. Removing magnetization.� Removing the external applied field, the magnetic momentsreturn to their easy magnetization directions, but not thosethey had initially, and some magnetization remains (remnantmagnetization):
0≠M
� To cancel remnant magnetization, an opposite magnetic fieldmust be applied (coercive field, Bc):
0=M
Bc
Ferromagnetism. Hysteresis curve.� If the coercive field grows, saturation magnetization can bereached in opposite direction to the first. In an alternatingfield, magnetization and demagnetization lose energy byfriction, and this process can be represented by a hysteresiscurve (cycle):
Bapp
First magnetization curve
Saturation magnetization
Coercivefield
Remnant magnetization
M
Saturation magnetization
� Area enclosed by hysteresis curve is related to quantity ofenergy lost by friction. Hard materials (high remnantmagnetization) are suitable to make magnets or data storagedevices . Soft materials are suitable to make electromagnets
Ferromagnetism. Hard and soft magnetically materials.
“Soft” material“Hard” material
1 0 1 1 0 0 1 0 0 1 0
I I
Writing and reading magnetic devices� A solenoid wound around a soft magnetic material can be usedto “organize” (write as 0 or 1) a hard magnetic material.Stored information can be read by electromagnetic induction.
x 2400