Chapter 22

Patterns of Fields in Space

• Electric flux• Gauss’s law• Ampere’s law• Maxwell equations

What is in the box?

no charges? vertical charged plate?

Patterns of Fields in Space

Box versus open surface

Seem to be able to tellif there are charges inside

…no clue…

Gauss’s law: If we know the field distribution on closed surface we can tell what is inside.

Patterns of Fields in Space

3. Surface area

flux through small area:

AnEflux ˆ~

Definition of electric flux on a surface:

surface

AnE ˆ

Electric Flux: Surface Area

Symmetry: Field must be perpendicular to surfaceEleft=Eright

0

ˆ

inside

surface

qAnE

2EAbox Q / A Abox

0

E Q / A 20

The Electric Field of a Large Plate

Symmetry: 1. Field should be radial2. The same at every location

on spherical surface

0

ˆ

inside

surface

qAnE

A. Outer sphere:

0

24

QrE 2

04

1

r

QE

B. Inner sphere:

0

2 04

rE 0E

The Electric Field of a Uniform Spherical Shell of Charge

Can we have excess charge inside a metal that is in static equilibrium?

Proof by contradiction:

0

ˆ

inside

surface

qAnE

=0

00

insideq

Gauss’s Law: Properties of Metal

0

ˆ

inside

surface

qAnE

=0

00

insideq

What is electric field inside?

0 ACBV

0ldEVADB

=

1. No charges on the surface of an empty hole

2. E is zero inside a hole

Gauss’s Law: Hole in a Metal

+5nC

0

ˆ

inside

surface

qAnE

=0

00

insideq

0 insidesurface qq

nC 5 surfaceq

Gauss’s Law: Charges Inside a Hole

0

ˆ

inside

surface

qAnE

Gauss’s Law: Screening

Is the field zero inside the box because the metal blocks the field?

Can we have excess charge inside in steady state?

0

ˆ

inside

surface

qAnE

surfacerightsurfaceleft

AnEAnE__

ˆˆ

00

insideq

Gauss’s Law: Circuits

Gauss’s Law: Junction Between Two Different Metal Wires

i1=i2

n1Au1E1 = n2Au2E2

E2 n1u1

n2u2

E1 E1

0

ˆ

inside

surface

qAnE

There is negative charge along the interface!

qinside 0 (E1A E2A) 0

n2<n1u2<u1

Magnet Cut in Half & Pulled Apart

No magnetic monopole! Try to cut a magnet down to a single pole, just get smaller magnets

No magnetic Charge!

Dipoles:Electric field: ‘+’ and ‘–’ charges can be separatedMagnetic field: no monopoles

Suppose magnetic dipole consists of two magnetic monopoles, each producing a magnetic field similar to the electric field.One cannot separate them total magnetic ‘charge’ is zero.

0

ˆ

inside

surface

qAnE

Gauss’s law for magnetism

0ˆ surface

AnB

0ˆ AnBor

Gauss’s Law for Magnetism

Patterns of Magnetic Field in Space

Is there current passing through these regions?

There must be a relationship between the measurements of the magnetic field along a closed path and current flowing through the enclosed area.

Ampere’s law

Quantifying the Magnetic Field Pattern

r

IBwire

2

40

Curly character – introduce: ldB

dlr

IldB

2

40

rr

I

22

40

IldB 0

Similar to Gauss’s law (Q/0)

All the currents in the universe contribute to Bbut only ones inside the path result in nonzero path integral

Ampere’s law is almost equivalent to the Biot-Savart law:but Ampere’s law is relativistically correct

Ampère’s Law

pathinsideIldB _0

pathinsideIldB _0

Can B have an out of plane component?

Is it always parallel to the path?

rBldB 2

IrB 02

r

IB

2

40

for thick wire: (the same as for thin wire)

Would be hard to derive using Biot-Savart law

Ampere’s Law: A Long Thick Wire

pathinsideIldB _0

Number of wires inside: (N/L)

What is on sides? ldB

Uniform: Does not depend on distance from sheet. Opposite directions above and below sheet.

Ampere’s Law: An Infinite Sheet

Each wire has 𝐼𝑑

𝐵

𝐵𝑑+𝐵𝑑=𝜇0𝑑 ¿

𝐵

𝐵=𝜇0 𝑁 𝐼

2𝐿

pathinsideIldB _0

Symmetry: B || path

INrB 02

r

NIB

2

40

Is magnetic field constant acrossthe toroid?

Ampere’s Law: A Toroid

Three equations:

Gauss’s law for electricity

Gauss’s law for magnetism

Ampere’s law for magnetism pathinsideIldB _0

0

ˆ

insideqdAnE

Is anything missing?

‘Ampere’s law for electricity’ ldE

0 ldE

(incomplete)

Maxwell’s Equations

0ˆ AnB

0

ˆ

insideqdAnE

0 ldE

pathinsideIldB _0

Gauss’s law for electricity

Gauss’s law for magnetism

Incomplete version of Faraday’s law

Ampere’s law(Incomplete Ampere-Maxwell law)

First two: integrals over a surfaceSecond two: integrals along a path

Incomplete: no time dependence

Maxwell’s Equations (incomplete)

0ˆ AnB

Motional EMF Revisited

Ampere’s lawB

Heat Ring and it expands

What is the direction of the electric field on the ring?

𝑣

Curly electric fields!

What is changing inside the ring?

𝑒𝑚𝑓=−𝑑𝜙𝐵

𝑑𝑡

𝐸

𝐸