MIDTERM 3

UTC 4.132 Thu-Nov 15, 7:00PM - 9:00PM

Course Summaries Unit 1, 2, 3 ProvidedTA session Monday Homework Review

(attendance optional)

Bring pencils, calculators (memory cleared)

Chapter 24

Classical Theory of Electromagnetic Radiation

Maxwell’s Equations

0

ˆ

insideqdAnE

pathinsideIldB _0

Gauss’s law for electricity

Gauss’s law for magnetism

Complete Faraday’s law

Ampere’s law(Incomplete Ampere-Maxwell law)

0ˆ AnB

∮𝐸 ∙𝑑 𝑙=− 𝑑𝑑𝑡 [𝐵 ∙ �̂�𝑑 𝐴 ]

No current inside

0 ldB

Current pierces surface

IldB 0

rIB 2

40

IrrIldB 0

0 224

pathinsideIldB _0Ampere’s Law

Time varying magnetic field leads to curly electric field.

Time varying electric field leads to curly magnetic field?

dAnEelec ˆ

00

0cosQA

AQ

elec

dtdQ

dtd elec

0

1

I

0

1

I

dtdI elec

0 ‘equivalent’ current

pathinsideIldB _0 combine with current in Ampere’s law

Maxwell’s Approach

dtdIldB elec

pathinside 0_0

Works!

The Ampere-Maxwell Law

Four equations (integral form) :

Gauss’s law

Gauss’s law for magnetism

Faraday’s law

Ampere-Maxwell law

0

ˆ

insideqdAnE

dAnBdtdldE ˆ

dtdIldB elec

pathinside 0_0

+ Lorentz force BvqEqF

Maxwell’s Equations

0ˆ AnB

Time varying magnetic field makes electric field

Time varying electric field makes magnetic field

Do we need any charges around to sustain the fields?

Is it possible to create such a time varying field configuration which is consistent with Maxwell’s equation?

Solution plan: • Propose particular configuration• Check if it is consistent with Maxwell’s eqs• Show the way to produce such field• Identify the effects such field will have on matter• Analyze phenomena involving such fields

Fields Without Charges

Key idea: Fields travel in space at certain speedDisturbance moving in space – a wave?

1. Simplest case: a pulse (moving slab)

A Simple Configuration of Traveling Fields

0

ˆ

insideqdAnE

0ˆdAnE

Pulse is consistent with Gauss’s law

0ˆ AnB

Pulse is consistent with Gauss’s law for magnetism

A Pulse and Gauss’s Laws

dtd

emf mag

Since pulse is ‘moving’, B depends on time and thus causes E

Area doesnot move

tBhvmag

Bhvdt

dt

magmag

emf

EhldEemf

E=Bv

Is direction right?

A Pulse and Faraday’s Law

dtdIldB elec

pathinside 0_0

=0

tEhvelec

Ehvdt

dt

elecelec

BhldB

EvhBh 00

vEB 00

A Pulse and Ampere-Maxwell Law

vEB 00 E=Bv

vBvB 00

2001 v

m/s 8

00

1031

v

Based on Maxwell’s equations, pulse must propagate at speed of light

E=cB

A Pulse: Speed of Propagation

Question

At this instant, the magnetic flux mag through the entire rectangle is:

A) B; B) Bx; C) Bwh; D) Bxh; E) Bvh

Question

In a time t, what is mag?

A) 0; B) Bvt; C) Bhvt; D) Bxh; E) B(x+vt)h

Question

emf = mag/t = ?

A) 0; B) Bvh; C) Bv; D) Bxh; E) B(x+v)h

Question

What is around the full rectangular path?

A) Eh; B) Ew+Eh; C) 2Ew+2Eh; D) Eh+2Ex+2Evt; E)2Evt

Question

emf dmag

dtBvh

rEgd

rl Eh—

What is E?

A) Bvh; B) Bv; C) Bvh/(2h+2x); D) B; E) Bvh/x

Exercise

If the magnetic field in a particular pulse has a magnitude of 1x10-5 tesla (comparable to the Earth’s magnetic field), what is the magnitude of the associated electric field?

E cB

Force on charge q moving with velocity v perpendicular to B:

E 3x108 m / s 1x10 5 T 3000V / m

𝐹𝑚𝑎𝑔

𝐹𝑒𝑙=𝑣𝐵

𝐸¿𝑣𝐵𝑐𝐵=

𝑣𝑐

Direction of speed is given by vector product BE

Direction of Propagation

Electromagnetic pulse can propagate in spaceHow can we initiate such a pulse?

Short pulse of transverseelectric field

Accelerated Charges

1. Transverse pulse propagates at speed of light

2. Since E(t) there must be B

3. Direction of v is given by: BE

E

Bv

Accelerated Charges

We can qualitatively predict the direction.What is the magnitude?

Magnitude can be derived from Gauss’s law

Field ~ -qa

rcaqEradiative 2

041

1. The direction of the field is opposite to qa

2. The electric field falls off at a rate 1/r

Magnitude of the Transverse Electric Field

Field of an accelerated charge

1

2

3

4

vT

𝑎A B

Φ𝑆

𝛼

Φ𝐵

Φ𝐴

Accelerates for t, then coasts for T at v=at to reach B.

cT

ct

𝜃𝑣𝑇sin𝜃

r>cT ; outer shell

inner shell of acceleration zone

> since B is closer, but = since areas compensate

Φ𝐵+Φ𝐴=0 No chargeΦ𝑆=0

𝐸𝑆

𝐸𝑟𝑎𝑑

𝐸𝑡𝑎𝑛

tan (𝛼)=𝐸 𝑡𝑎𝑛

𝐸𝑟𝑎𝑑¿𝑣𝑇𝑠𝑖𝑛(𝜃)

𝑐𝑡

𝐸𝑡𝑎𝑛=𝐸𝑟𝑎𝑑𝑣𝑇𝑠𝑖𝑛 (𝜃)

𝑐𝑡

Field of an accelerated charge

1

2

3

4

vT

𝑎A B

Φ𝑆

𝛼

Φ𝐵

Φ𝐴

cT

ct

𝜃𝑣𝑇sin𝜃

𝐸𝑆

𝐸𝑟𝑎𝑑

𝐸𝑡𝑎𝑛

𝐸𝑡𝑎𝑛=𝐸𝑟𝑎𝑑𝑣𝑇𝑠𝑖𝑛 (𝜃)

𝑐𝑡

𝐸𝑟𝑎𝑑=1

4𝜋 𝜀0

𝑞𝑟2

𝐸𝑡𝑎𝑛= 14𝜋 𝜀0

𝑞𝑟2𝑣𝑇𝑠𝑖𝑛 (𝜃)

𝑐𝑡

𝑎=𝑣 /𝑡𝐸𝑡𝑎𝑛= 𝑞

4𝜋 𝜀0

𝑎𝑠𝑖𝑛(𝜃)𝑐2𝑟

c

𝑎𝑠𝑖𝑛 (𝜃 )=𝑎⊥𝐸𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑣𝑒=

𝑞4𝜋 𝜀0

−𝑎⊥

𝑐2𝑟