Slides Unit 7. Sources of magnetic field - UPV 7/Slides Unit 7. Sources of magnetic... · Unit 7: Sources of magnetic field ... 3 0 r d 4 i l r B r r r π µ Magnetic field created

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


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.


� 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


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=


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.


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:


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


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:


� 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


� 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


Coercivefield

Remnant magnetization

M


� 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

Documents

Slides Unit 7. Sources of magnetic field - UPV 7/Slides Unit 7. Sources of magnetic... · Unit 7: Sources of magnetic field ... 3 0 r d 4 i l r B r r r π µ Magnetic field created