Sources of the Magnetic FieldMoving charges – currentsAmpere’s Law Gauss’ Law in magnetismMagnetic materials

Biot-Savart

Law

sr

dr̂

r̂

outdBr

indBr

I

P′

P

θ

rB

sBˆ⊥

⊥r

rr

d

dd

θsin

12

∝∝∝

∝

dBdsdBIdBr

dB

20 ˆ

4 rIdd rsB ×

=π

μr AmT /.104 70

−×= πμpermeability of free space∫

×= 2

0 ˆ4 r

dI rsBπ

μr

Magnetic Field Surrounding a Thin, Straight Conductor

θcosˆˆˆˆ dxrdsd kkrs =×=×r

( ) kkB ˆcos4

ˆ2

0

rdxIdBd θ

πμ

==r

20 cos

4 rdxIdB θ

πμ

=

θθθθ

θθ

22

cossec

tancos

addadx

ax

ar

−=−=

−=

=θθ

πμ

θθθθ

πμ d

aI

adaIdB cos

4coscoscos

40

22

20 ==

( )2100 sinsin

4cos

4

2

1

θθπ

μθθπ

μ θ

θ

−== ∫ aId

aIB

If the wire is very long,°≈°−≈

9090

2

1

θθ then

aIB

πμ2

0=

Field Due to a Circular Arc of Wire

20 90sin

4 RidsdB °

=π

μ

∫∫ ′==φ

φπμ

0

0

4d

RidBB

φ′= Rdds

RiB

πφμ

40=

RiB

20μ

= Full Circle (φ = 2π)

Magnetic Force Between Two Parallel Conductors

laII

aIlIlBIF

πμ

πμ

2212020

1211 =⎟⎠⎞

⎜⎝⎛==

BFFF == 21

aII

lFB

πμ2

120=

Concept QuestionOn a computer chip, two conducting strips carry charge from P to Q and from R to S. If the current direction is reversed in both wires, the net magnetic force of strip 1 on strip 2

1. remains the same.2. reverses.3. changes in magnitude, but not in direction.4. changes to some other direction.5. other

Ampère’s

Law

( ) IrrIdsBd 0

0 22

. μππ

μ=== ∫∫ sB

Id 0. μ=∫ sB rr

A line integral of B.ds around a closed path equals μ0 I, where I is the total continuous current passing through any surface bounded by the closed path.

Long Current Carrying Wire

( ) IrBdsBd 02. μπ === ∫∫ sBrr

rIB

πμ2

0=

For r >= R

IRrI

Rr

II

2

2

2

2

=′

=′

ππ

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=′==∫ I

RrIrBd 2

2

002. μμπsBrr

For r < R

rRIB ⎟

⎠⎞

⎜⎝⎛= 2

0

2πμ

The ToroidBy symmetry, B is constant over loop 1 and tangent to it

Bdsd =sB.

( ) NIrBdsBd 02. μπ === ∫∫ sB

rNIBπ

μ20=

Outside the toroid: 0≈B (Consider loop 2 whose plane is perpendicular to the screen)

The Solenoid

The Magnetic Field of a Solenoid

BldsBddpathpath

=== ∫∫∫11

.. sBsB

NId 0. μ=∫ sB

nIB 0μ= where lNn =

Along path 2 and 4, B is perpendicular to ds

Along path 3, B=0

Consider loop 2

Magnetic Flux

∫=Φ AB dB .

θcosBAB =Φ

For a uniform field making an angle θ with the surface normal:

(Weber=Wb=T.m2)

0=ΦB ABBB =Φ=Φ max,

Gauss’

Law in Magnetism

Unlike electrical fields, magnetic field lines end on themselves, forming loops. (no magnetic monopoles)

Magnetic Field LinesElectric Field Lines

0. =∫ AB d

Magnetic Moments of Atoms

Aμrr I=

RIB

20μ

=

Current carrying loop Electron in orbit

ωπ2

=T

reve

TeI

ππω

22===

rv

=ω

revrr

evIA22

2 === ππ

μ

vrmL e=

h⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

ee menL

me

22

2μ

Electron spin TJme

espin /1027.9

224−×=⎟⎟

⎠

⎞⎜⎜⎝

⎛= hμ

Magnetic Moments of AtomsThe net orbital magnetic moment in most substances is zero or very small since the moments of electrons in orbit cancel each other out.Similarly, the electron spin does not contribute a large magnetic moment in substances with an even number of electrons because electrons pair up with ones of opposite spin.

FerromagnetismIn certain crystals, neighboring groups of atoms called domains have their magnetic moments aligned. A ferromagnetic crystal can be given a permanent magnetization by applying an external field.Examples include Fe, Co, Ni.

Unmagnetized crystal has domains of aligned

magnetic moments.

An external field increases the sizes

of the domains aligned with it.

As the external field gets larger, the

unaligned domains become smaller.

ParamagnetismParamagnetic substances have a small but positive magnetism that comes from atoms with permanent magnetic moments.An external field can align these moments for a net magnetization but the effect is weak and not permanent.Certain organic compounds such as myoglobin are paramagnetic.

DiamagnetismA very small effect caused by the induction of an opposing field in the atoms by an external field.Superconductors exhibit perfect diamagnetism (Meissner effect).

SummaryBiot-Savart Law gives the magnetic field due to a current carrying wire.Ampere’s Law can simplify these calculations for cases of high symmetry.The magnetic flux through a closed surface is zero.Solenoids and toroids can confine magnetic fields.Orbital and spin magnetic moments of electrons give rise to magnetism in matter.

For Next ClassReading Assignment

Chapter 31 – Faraday’s Law

WebAssign: Assignment 8