Wed. Fri.

(C 21.6-7,.9) 1.3.4-1.3.5, 1.5.2-1.5.3, 5.3.1-.3.2 Div & Curl B (C 21.6-7,.9) 5.3.3-.3.4 Applications of Ampere’s Law

Mon. Wed. Thurs. Fri.

1.6, 5.4.1-.4.2 Magnetic Vector Potential 5.4.3 Multipole Expansion of the Vector Potential Review

HW8

q1

1r

1E

q2

2r

2E

netE

Memory Lane: Electrostatics Field of charge distributions

rr 2

ˆ1

1

41

1

qrE

o

drdqd

dqr )()(

drrEo

24

1ˆ

r

r

ech i

dqqi i

inet dq

qrE

oo

arg

41

lim1

41

ˆˆ

2i2 r

rr

r

Found Electric fields due to variety of charge distributions

-orr

or

q

Biot-Savart Law

24

ˆ

r

r

ldIrBd o

I

r

r

Source

Observation location

dL

r

2

0 ˆ4 r

r

IdrB

I

r

r

dl

r

y

x

z

x y

z

r

f f

r

l

r

r

a

aa

dr

2sin

top

bottom

o dIn

a

a

aasin

2

r

z

𝑥

𝑦

𝑧 𝜌 𝑟 ′ = 𝜌𝑜𝑐𝑜𝑠 𝜃′

r’dq’

r’sin(q’)df’ r q’

f’

dr’

Magneto-statics Fields of current distributions

Del Operator

zz

yy

xx

ˆˆˆ

Gradient – vector representing the local slope of a scalar field. Example: temperature across a sheet of metal heated at one corner.

T T

xˆ x

T

yˆ y

T

zˆ z

Divergence – scalar representing in/out flow from a point in a vector field. Example: velocities of water filling a bath tub.

v vx

x

vy

y

vz

z

Curl – vector representing circulation of a vector field. Example: velocities of water swirling down a drain.

v

ˆ x ˆ y ˆ z

x y z

vx vy vz

ˆ x vz

y

vy

z

q3

q2

Memory Lane: Electrostatics Flux from Charge Sources

q1

11 EadE

441

1

o

q

E

o

q

EadE

1

11

||daEE 11

2

2

2

1

2

0

2

41sin

f

q

qqf ddo

q

E rr

Ditto for q2, q3, …

o

enclosednetQ

net adE.

(integral form) Gauss’s Law

Vol

QE encl

VolVol o1

0

E

0lim

Vollimdiv

x

y

z

x ,y,z

Ex(x)

Ex(x x)

o

E 1

oVol

adEE

Vol

1

0limdiv

(derivative form)

0 ldE

0 E

Motivated scalar potential

Simplified symmetric problems

ldEV

EV

Pause & Put together Gauss’s Th’m

Gauss’s Theorem – explicitly putting it together

but o

enclosedQadE

oE 1

Though we were thinking specifically about electric field while we did the math that got us to this relation, it’s quite general and true for any vector field. So, as expressed in Ch. 1, for generic function F,

denclQ And we’d gone off and proven

Putting these together:

dadE

o

dEadE

dFadF

24

ˆ

r

r

ldIrBd o

I

r

r

Source

Observation location

dL

r

Magneto-Statics Flux from Current Sources

ad

ldIdadrBd o

B

24

ˆ

r

r

Not so easy; start with special case to suggest / understand general result infinite line charge

f

ˆ2s

IrB o

line

ad

IadrB o

Bline

f

ˆ2s

Surface of any radius would work

ss

sˆˆ

2dzd

Io ff

sˆ2

ff

dzIdo 0

Similarly,

zsline Bz

Bs

sBss

rB

f

f

11

dzsfds ss ˆdzdad f

01

01

2z

I

ss

sso

s

f0

24

ˆ

r

r

ldIrBd o

I

r

r

Source

dL

r

Magneto-Statics Circulation from Current Sources

ld

ldIldrBd o

24

ˆ

r

r

Not so easy; start with special case to suggest / understand general result infinite line charge

f

ˆ2s

IrB o

line

0 adrB line

fff

ˆˆ2

dI

o ss

fff

ˆˆ2

Ido

What of ?

0 rBline

s

rBCurl line

ld

IldrB o

line

f

ˆ2s

ff ˆdld s

path of any radius would work

2

2Io Io

Takes a little work to get right answer and understand.

Goes Differential: Curl

Ampere’s Law

JA

I

A

I

A

I

A

dB

A

dB

A

dB

JA

I

A

dB

z

zpiercing

y

ypiercing

x

xpiercing

zyx

I

piercing

I

0

,,,

0

00

,,,,

patchz

patchz

Acomponentz

A

dBB

patchz.

.

0.

.

limcurl

Curl=circulation density (per area encircled) Zoom in to differential scale:

Break closed path into paths around differential patches

Break area into differential area ‘patches’

(ultimately all internal path legs cancel with each other)

I

B 𝐵 ∙ 𝑑𝑙 = 𝜇𝑜𝐼𝑝𝑖𝑒𝑟𝑐𝑖𝑛𝑔

Project onto coordinate planes

...curl . componentxB ...curl . componentyB

Goes Differential: Curl

Ampere’s Law

...curl zB

Curl=circulation density (per area encircled)

I

B 𝐵 ∙ 𝑑𝑙 = 𝜇𝑜𝐼𝑝𝑖𝑒𝑟𝑐𝑖𝑛𝑔

patchx

patchxo

Apatchx

patchx

Ax

A

I

A

dBB

patchxpatchx.

.

0.

.

0 ..

limlimcurl

...curl yB

z

y

By(z+z)

By(z)

Bz(y)

Bz(y+y) Ix

zy

zyByzzBzyyByzB zyzy

zy 0lim

zy

yzByzzB

zy

zyBzyyB yyzz

zy 0lim

z

B

y

B yz

xB

curl

patchx

xo

dA

dI

.

xo J

Similarly, yozx

y Jx

B

z

BB

curl and zo

xy

z Jy

B

x

BB

curl

zyxoxyzxyz JJJ

y

B

x

B

x

B

z

B

z

B

y

BB ,,,,curl

or JB

0

Magneto-Statics Divergence and Circulation from Current Sources

f

ˆ2s

IrB o

line

0 adrB line

0 rBline

s

IldrB oline

JB

0

Derivation specific to constant, Infinite Line-Current

Gauss’s Theorem Stokes’ Theorem

Note: both follow from applying Biot-Savart, which holds only for steady currents

Now for more general (more mathematical / less intuitive) proof

Pause and put together Stoke’s:

Stoke’s Theorem – explicitly putting it together

but

Though we were thinking specifically about magnetic field while we did the math that got us to this relation, it’s quite general and true for any vector field. So, as expressed in Ch. 1, for generic function F,

daJI

And we’d gone off and proven

Putting these together:

adFldF

IldrB oline

JB

0

daJldrB oline

daBldrB line

Derive Divergence of B 𝛻𝑟 ∙ 𝐵 𝑟 = 𝛻𝑟 ∙ 𝜇𝑜

4𝜋 𝐽 ×r r2 𝑑𝜏′

observation location

𝐵 𝑟

𝑟

𝑟 ′

r = 𝑟 -𝑟 ′

source currents

𝛻𝑟 ∙ 𝐵 𝑟 =𝜇𝑜

4𝜋 𝛻𝑟 ∙ 𝐽 ×

r r2 𝑑𝜏′

Can slip del inside integral since not taking derivative with respect to integration variable

Use Product Rule (6): BAABBA

𝛻𝑟 × 𝐽 ×r r2 =

r r2 ∙ 𝛻𝑟 × 𝐽 - 𝐽 ∙ 𝛻𝑟 ×

r r2

taking derivative to see how field varies from one observation location to another, not for changes in source locations

using

dJdaldda

IldI

r r2 has no curl (can write out in Cartesian to convince, or see in spherical)

𝛻𝑟 ×r r2 =

1

𝑟

1

𝑠𝑖𝑛𝜃

𝜕

𝜕𝜑

1

r2 𝜃 - 1

𝑟

𝜕

𝜕𝜑

1

r2 𝜑 = 0

𝛻𝑟 ∙ 𝐵 𝑟 =𝜇𝑜

4𝜋 0𝑑𝜏′ = 0

So by Gauss’ Theorem

𝛻𝑟 ∙ 𝐵 𝑟 𝑑𝜏 = 𝐵 𝑟 ∙ 𝑑𝑎 = 0

Derive Ampere’s Differential form 𝛻𝑟 × 𝐵 𝑟 = 𝛻𝑟 × 𝜇𝑜

4𝜋 𝐽 ×r r2 𝑑𝜏′

observation location

𝐵 𝑟

𝑟

𝑟 ′

r = 𝑟 -𝑟 ′

source currents

𝛻𝑟 × 𝐵 𝑟 =𝜇𝑜

4𝜋 𝛻𝑟 × 𝐽 ×

r r2 𝑑𝜏′

Can slip del inside integral since not taking derivative with respect to integration variable

Use Product Rule (8): ABBABAABBA

𝛻𝑟 × 𝐽 ×r r2 =

r r2 ∙ 𝛻𝑟 𝐽 - 𝐽 ∙ 𝛻𝑟

r r2+𝐽 𝛻𝑟 ∙

r r2 -

r r2 𝛻𝑟 ∙ 𝐽

taking derivative to see how field varies from one observation location to another, not for changes in source locations

= - 𝐽 ∙ 𝛻𝑟r r2+𝐽 𝛻𝑟 ∙

r r2

using

dJdaldda

IldI

𝛻𝑟 ∙r r2=4𝜋𝛿3 𝑟

Motivation

add

22 r

r

r

r ˆˆ

by Gauss’s theorem

fqq ddsin

1 2

2r

r4

𝛻𝑟 × 𝐵 𝑟 =𝜇𝑜

4𝜋 − 𝐽 ∙ 𝛻𝑟

r

r2 𝑑𝜏′ + 𝐽 𝛻𝑟 ∙

r

r2 𝑑𝜏′

𝐽 4𝜋𝛿3 𝑟 𝑑𝜏′

4𝜋𝐽 𝑟

+ 𝐽 ∙ 𝛻𝑟′r

r2 𝑑𝜏′

Looking at just one component

𝐽 ∙ 𝛻𝑟′𝑥−𝑥′

r3 𝑥 = 𝛻𝑟′ ∙ 𝐽 𝑥−𝑥′

r3 −𝑥−𝑥′

r3 𝛻𝑟′ ∙ 𝐽 𝑥

Derive Ampere’s Differential form 𝛻𝑟 × 𝐵 𝑟 = 𝛻𝑟 × 𝜇𝑜

4𝜋 𝐽 ×r r2 𝑑𝜏′

observation location

𝐵 𝑟

𝑟

𝑟 ′

r = 𝑟 -𝑟 ′

source currents

using

dJdaldda

IldI

𝛻𝑟 × 𝐵 𝑟 =𝜇𝑜

4𝜋 𝐽 ∙ 𝛻𝑟′

r

r2 𝑑𝜏′ +4𝜋𝐽 𝑟

𝛻𝑟′ ∙ 𝐽 =-𝑑𝜌

𝑑𝑡

where

= 0

𝛻𝑟′ ∙ 𝐽 𝑥 − 𝑥′

r3𝑑𝜏′ = 𝐽

𝑥 − 𝑥′

r3∙ 𝑑𝑎 ′

𝐽 ∙ 𝛻𝑟′𝑥−𝑥′

r3 𝑥 = 𝛻𝑟′ ∙ 𝐽 𝑥−𝑥′

r3 −𝑥−𝑥′

r3 𝛻𝑟′ ∙ 𝐽 𝑥

Ditto for other two components

For now, with electro-magnetic statics

Gauss’s theorem

If area fully encloses current, then no current penetrates area

= 0

Ditto for other two components

𝛻𝑟 × 𝐵 𝑟 =𝜇𝑜

4𝜋0 +4𝜋𝐽 𝑟

𝛻𝑟 × 𝐵 𝑟 = 𝜇𝑜𝐽 𝑟

So by Stokes’ Theorem

𝐵 𝑟 ∙ 𝑑𝑙 = 𝜇𝑜 𝐽 𝑟 ∙ 𝑑𝑎 ′ = 𝜇𝑜𝐼

Limitations: holds when Biot-Savart holds - statics

Maxwell’s laws for electro-statics

𝛻𝑟 ∙ 𝐵 𝑟 = 0

𝐵 𝑟 ∙ 𝑑𝑎 = 0

𝛻𝑟 × 𝐵 𝑟 = 𝜇𝑜𝐽 𝑟

𝐵 𝑟 ∙ 𝑑𝑙 = 𝜇𝑜𝐼

𝛻𝑟 × 𝐸 𝑟 = 0

𝐸 𝑟 ∙ 𝑑𝑙 = 0

𝛻𝑟 ∙ 𝐸 𝑟 = 1𝜀0

𝜌 𝑟

𝐸 𝑟 ∙ 𝑑𝑎 = 1𝜀𝑜

𝑄

For arguably symmetric fields, useful for finding fields

Gauss’s

Ampere’s

Wed. Fri.

(C 21.6-7,.9) 1.3.4-1.3.5, 1.5.2-1.5.3, 5.3.1-.3.2 Div & Curl B (C 21.6-7,.9) 5.3.3-.3.4 Applications of Ampere’s Law

Mon. Wed. Thurs. Fri.

1.6, 5.4.1-.4.2 Magnetic Vector Potential 5.4.3 Multipole Expansion of the Vector Potential Review

HW8