Lecture 8MAGNETOSTATICS

Magnetic Fields

Fundamental Postulates of Magnetostatics in Free Space

Prof. Viviana Vladutescu

Magnetic Fields

Magnetism and electricity have not been considered distinct phenomena until Hans Christian Oersted conducted an experiment that showed a compass deflecting in proximity to a current carrying wire

Produced by-time varying electric fields

-permanent magnet (arises from quantum mechanical electron spin/ can be considered charge in motion=current )

-steady electric currents

m

AH SI

If we place a wire with current I in the presence of a magnetic field, the charges in the conductor experience another force Fm

Fm ~ q, u, B

q –chargeu –velocity vectorB -strength of the field (magnetic flux density)μr –relative permeabilityμ –absolute permeabilityμ0-permeability of the free spaceM=χmH- is the magnetization for linear andhomogeneous medium

BH1

)(104 70 H

Relative permeabilities for a variety of materials

Material μ/(H m-1) μr Application

Ferrite U 60 1.00E-05 8 UHF chokes

Ferrite M33 9.42E-04 750 Resonant circuit RM cores

Nickel (99% pure) 7.54E-04 600 -

Ferrite N41 3.77E-03 3000 Power circuits

Iron (99.8% pure) 6.28E-03 5000 -

Ferrite T38 1.26E-02 10000 Broadband transformers

Silicon GO steel 5.03E-02 40000 Dynamos, mains transformers

supermalloy 1.26 1000000 Recording heads

BuqF

EqF

m

e

FFF me

)( BuEqF

Lorentz’s Force equation

Note: Magnetic force is zero for q moving in the direction of the magnetic field (sin0=0)

When electric current is passed through a magnetic field a force is exerted on the wire normal to both the magnetic field and the current direction. This force is actually acting on the individual charges moving in the conductor.

nmm aFrFr sin

The magnetic force is exerting a torque on the current carrying coil

Cross product

Fundamental Postulates of Magnetostatics in

Free Space

JB

B

0

0

0 J

0s

sdB

Law of conservation of magnetic flux

There are no magnetic flow sources, and the magnetic flux lines always close upon themselves

Ampere’s circuital law

c

ss

IldB

sdJsdB

0

0

The circulation of the magnetic flux density in free space around any closed path is equal to μ0 times the total current flowing through the surface bounded by the path

Stoke’s Theorem

cs

ldHsdH

Note: For a closed surface there will be no surface bounding external contour

s

ldH

H c

s

0

lim

Proof: Sum over

where

nj asH

Note: The net contribution of all the common parts in the interior to the total line integral is 0 and only the contribution from the external contour C remains after summation

c

N

j cs

ldHldHj 1

0lim

cs

ldHdsH

c

ldHJds

The maximum circulation of H per unit area as the areashrinks to zero is equal to the current density through that area

Two possible Amperian paths around an infinite length line of current.

c

encIldH

Differential Form Integral Form

Postulates of Magnetostatics

in Free Space

JB

B

0

0

c

s

IldB

sdB

0

0

Given a 3.0 mm radius solid wire centered on the z-axis with an evenly distributed 2.0 amps of current in the +az direction,

plot the magnetic field intensity H versus radial distance from the z-axis over the range 0 ≤ ≤ 9 mm.

Example

The field from a particular line of current making up the distributed current

The field from a second line of current results in the cancellation of ar

component

, ; 2enc encd I where H and d d H I H L H a L a

This will be true for each Amperian path.AP1:

2 2

2 2 20 0

I I I, = ,

a a aenc z encI d I d d

J S J a

2 for

2

Ia

a

H a

AP2: Ienc = I,

for 2

Ia

H a