Moving charges – currents Ampere’s Law Gauss’ Law in magnetism Magnetic materials

s

dr̂

r̂

outdB

indB

I

P

P

rB

sB

ˆ

d

dd

sin

12

dB

dsdB

IdB

rdB

2

0ˆ

4 r

Idd

rsB

AmT /.104 7

0

permeability of free space

2

0ˆ

4 r

dI rsB

cosˆˆˆˆ dxrdsd kkrs

kkB ˆcos

4ˆ

2

0

r

dxIdBd

2

0 cos

4 r

dxIdB

2

2

cossec

tan

cos

addadx

ax

ar

d

a

I

a

daIdB cos

4cos

coscos

4

0

22

2

0

2100 sinsin

4cos

4

2

1

a

Id

a

IB

If the wire is very long,

90

90

2

1

then

a

IB

2

0

2

0 90sin

4 R

idsdB

0

0

4d

R

idBB

Rdds

R

iB

4

0

R

iB

2

0 Full Circle ( 2)

la

II

a

IlIlBIF

22

120201211

BFFF 21

a

II

l

FB

2

120

On 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

Irr

IdsBd 0

0 22

.

sB

Id 0. sB

A line integral of B.ds

around a closed path

equals 0I, where I is

the total continuous

current passing

through any surface

bounded by the

closed path.

IrBdsBd 02. sB

r

IB

2

0

For r >= R

IR

rI

R

r

I

I

2

2

2

2

I

R

rIrBd

2

2

002. sB

For r < R

rR

IB

2

0

2

By symmetry, B is constant over

loop 1 and tangent to it

Bdsd sB.

NIrBdsBd 02. sB

r

NIB

2

0

Outside the toroid: 0B(Consider loop 2 whose plane is

perpendicular to the screen)

BldsBddpathpath

11

.. sBsB

NId 0. sB

nIB 0 where lNn

Along path 2 and 4, B is perpendicular to ds

Along path 3, B=0

Consider loop 2

AB dB .

cosBAB

For a uniform field making

an angle with the surface

normal:

(Weber=Wb=T.m2)

0B ABBB max,

Unlike electrical fields,

magnetic field lines end

on themselves, forming

loops. (no magnetic

monopoles)

Magnetic Field Lines Electric Field Lines

0. AB d

Aμ

I

R

IB

2

0

Current carrying loop Electron in orbit

2T

r

eve

T

eI

22

r

v

rev

rr

evIA

22

2

vrmL e

ee m

enL

m

e

22

2

Electron spin TJm

e

e

spin /1027.92

24

The 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.

In 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.

Paramagnetic 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.

A very small effect caused by the induction of an opposing field in the atoms by an external field.

Superconductors exhibit perfect diamagnetism (Meissner effect).

Biot-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.

Reading Assignment

Chapter 31 – Faraday’s Law

WebAssign: Assignment 8