PDF Introduction to Electromagnetism with Practice - Seoul National University

Sunkyu Yu

Dept. of Electrical and Computer Engineering Seoul National University

Introduction

Introduction to Electromagnetism with Practice

Theory & Applications

Lecturer, TA & IWSL

Prof. Sunkyu Yu [email protected]

Disordered systems Non-Hermitian systems

Photonic AI

Hyungchul Park [email protected]

Ikbeom Lee [email protected]

Kyuho Kim [email protected]

TA Lecturer

Course Introduction

Course No. 430.202B 003

Textbook: David K. Cheng, Field and Wave Electromagnetics (2

Ed., Addison Wesley, 1989) Refs: A. Zangwill. Modern Electrodynamics, Cambridge University Press (2001).

기초전자기학 및 연습

Introduction to Electromagnetism with Practice

Task Project Midterms I + II Final Attendance Attitude Total

20 20 30 (15 + 15) 20 0 10 100

301-201, Mon. Wed. 14:00~15:15

Question: Q&A Board in eTL

Contact: 301-1103 or Zoom after prior appointment by email

All the lectures will be provided as recorded videos!

Lecture Schedule (Changed!)

01

week Introduction & Vector Analysis 02

week Vector Analysis

03

week Vector Analysis / Static Electric Fields (EXP 1) 04

week Static Electric Fields (EXP 2)

05

week Static Electric Fields (EXP 3)

06

week Midterm 1 (3-4 hours for online or 3-4 days for take-home) 07

week Laplace/Poisson Equations (EXP 4)

08

week Laplace/Poisson Equations (EXP 5) 09

week Steady Electric Currents (EXP 6)

10

week Midterm 2 (3-4 hours for online or 3-4 days for take-home) 11

week Static Magnetic Fields

12

week Static Magnetic Fields

13

week Time-Varying Fields and Maxwell's Equations (partly) 14

week Special Topics

15

week Final Exam (3-4 hours for online or 3-4 days for take-home)

Notices

✓ Online Lectures (Exams can be online or offline depending on COVID-19).

✓ Closed-book (only for offline), open-book & take-home exams will be combined.

✓ Term Project (numerical simulation of electromagnetic phenomena) : it can be replaced with homework

✓ Homework may include paper review & numerical calculations.

✓ The evaluation & schedule may change depending on preparation process or COVID-19.

4 Ways to get a point for “Attitude”

• I’ll (or TA’ll) upload some questions or problems in the eTL QnA board

➔ If you answer this question, you’ll get a point.

• If you upload a “very” good question, you’ll get a point

• If you point out a typo or incorrectness of lecture materials, you’ll get a point

• If you give a correct answer to the questions of other students, you’ll get a point

Grades

0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Chart Title

The students in the same cluster

will get the same grade...

Goal of This Lecture

• Understanding Theoretical Knowledges for static electromagnetics

• Learning a route to Maxwell’s Equations

• Learning Analytical & Numerical Tools to handle electromagnetic phenomena (with Experiment class for Matlab)

• Introducing Applications of electromagnetic phenomena

• Achieving Global Viewpoints to handle vector differential equations

• (If time allows) Studying Recent Achievements in Electromagnetics & Photonics

Electromagnetics ~ Optics (or Photonics)

Electromagnetics ~ Optics (or Photonics)

Ultimate Goal

: Understanding phenomena related to

(Quantum & Relativistic) Photons & Electrons

In many practical problems

: Understanding phenomena related to (Quantum) Photons & Electrons

In many large-scale problems

: Understanding phenomena related to (Classical) Photons & Electrons

In this course

: Understanding sufficiently slow phenomena related to

(Classical) Photons & Electrons

What is Electromagnetics? = Optics – Understanding Light

Optics (The Oxford English Dictionary)

= Branch of physics which deals with the properties and phenomena of light

= “What is Light & Light-Matter Interaction?”

Light-Matter Interaction

Light

What is Light?

Light

Particle

Wave

Newton: “Rays of light are very small bodies emitted from shining substances”

Light (unlike sound) does not turn corners:

“Why then may not light deflect from straight lines as well as sounds?”

Huygens: “...each point of a luminous body

…gives rise to its own waves, and is the

centre of these waves.”

What is Light?

https://physics.aps.org/story/v22/st8

Light shaping. Shown here are the

measured oscillations in a multi-photon light pulse less than 100 attoseconds long. Researchers have now shaped the

“pulse” of a single photon.

Energy

~ Frequency Momentum

~ Wavevector Spin

~ Polarization Orbital

~ Wavefront

Particle Theory of Light

Particle

Newton: “Rays of light are very small bodies emitted from shining substances”

“Why then may not light deflect from straight lines as well as sounds?”

Quantum Mechanics

Quantization

Einstein

Planck Bohr

Wave Theory of Light

Electromagnetics

Wave phenomena

Wave

Huygens: “...each point of a luminous body

…gives rise to its own waves, and is the centre of these waves.”

Maxwell

Faraday

Dual Nature of Light

Light

Particle

Wave

= Photons!

Quantum mechanics

Electromagnetics

Quantum Electrodynamics

(QED)

The Overview of Optics

Quantum Optics Electromagnetic Optics

Wave Optics Ray Optics

Rigorous & Difficult

Light = Straight Line Light = Scalar Wave

Phase Diffraction Direction

Reflection, Refraction, … Vectorial Nature

Polarization, Spin, Orbital, … Probabilistic Description

Nonlinearity, Lasing, …

Light = Vectorial Wave

Light = Quantized Photon

Lecture Plan I

01^st week Introduction / Vector Analysis 02^nd week Vector Analysis

03^rd week Vector Analysis / Static Electric Fields (EXP 1) 04^thweek Static Electric Fields (EXP 2)

05^thweek Static Electric Fields (EXP 3) 06^thweek Midterm 1 / Midterm 1 Sol.

Maybe tedious & monotonous, … but essential for later topics

Static Assumption

Lecture Plan II

05^thweek Laplace/Poisson Equations (EXP 4) 06^thweek Laplace/Poisson Equations (EXP 5) 07^thweek Steady Electric Currents (EXP 6) 08^thweek Midterm 2

Quasi-Static

Lecture Plan III

11^thweek Static Magnetic Fields 12^thweek Static Magnetic Fields

13^thweek Time-Varying Fields and Maxwell's Equations (partly) 14^thweek Special Topics

15^thweek Final Exam / Final Exam Sol.

Introducing Magnetic Fields

Maxwell’s Eqs

Sunkyu Yu

Dept. of Electrical and Computer Engineering Seoul National University

Mathematical Preliminaries

Introduction to Electromagnetism with Practice

Theory & Applications

Why should we learn a vector?

Physical Systems & Mathematical Spaces ~ Sets

Mathematical Space ~ Set

ψ 2

ψ 1

ψ 3

Element / State

Element / State Element / State

Physical Systems

A type of a certain space?

Determined by “operations” on the elements The operations should satisfy certain “rules”

= Axioms of the space Usually Dynamical & Complex…

Different masses & shapes, Various movement & interactions …

➔ Numerous “information”

with different physical aspects

How can one describe them?

Classifying the Spaces for Physics

Vector Space

Normed Vector Space

Inner-Product Space Banach Space

Hilbert Space

Vector Space

Preliminary: Mapping, Cartesian Product, and Field

• For the sets S and T,

ψ

ψ

ψ

ψ

ψ

ψ

ψ

S T

T

M

A Mapping (or Operator) M of S into T: A rule of assigning

ξ  S to M(ξ)  T - M(ξ) is the image of ξ under M

- S: domain of M, T: codomain of M, T

= {M(ξ):

ξ  S}  T: range of M

- When T

= T ➔ A mapping M of S onto T

Preliminary: Mapping, Cartesian Product, and Field

• For the sets S

, S

, S

, … , S

,

S

× S

× S

× … × S

: Cartesian Product of the sets S

, S

, S

, … , S

➔ The set of all “ordered” n-tuples (ξ

, ξ

, … , ξ

) where ξ

S

for k = 1, 2, … , n

Ex) Cartesian coordinate: X × Y × Z = {(x,y,z)|xX, yY, zZ}

• Field F: A set with two binary operations: addition and multiplication

These binary operations are the mappings F × F → F satisfying the field axioms

➔ S

× S

× S

× … × S

= {(ξ

, ξ

, ξ

, … , ξ

)|ξ

S

for k = 1, 2, … , n}

Associativity: a + (b + c) = (a + b) + c & a · (b · c) = (a · b) · c Commutativity: a + b = b + a & a · b = b · a

Distributivity: a · (b + c) = (a · b) + (a · c)

Identity: there exist 0 & 1  F such that a + 0 = a & a · 1 = a Inverses: there exists –a  F, such that a + (–a) = 0

& for every a ≠ 0 in F, there exists a

 F, such that a · a

= 1

Vector Spaces over Fields of Scalars

ψ 1 ψ 2 ψ 3

…

V

i + j

ψ ψ

aψ i

Vector Addition

Scalar Multiplication

Two ‘operations’ and their ‘rules’ are defined.

Vector Space V (~ Linear Space)

Real Vector Space V(R)

Complex Vector Space V(C)

Real Scalar F Complex Scalar F

Mapping of V × V → V

(f, g) → f + g, (f, g)  V × V, f + g  V

Mapping of F × V → V

(a, f) → af, (a, f)  F × V, af  V

Axioms of Vector Spaces

i + j

ψ ψ aψ i

Vector Addition Scalar Multiplication

Distributive

Associative

Commutative

Identity

i

+ =

ψ O ψ

Inverse Vector

( )

i

+ −

= O

ψ ψ

(

)

a ψ + ψ = a ψ + a ψ

( ψ

+ ψ

) + ψ

= ψ

+ ( ψ

+ ψ

)

i

+

=

+

ψ ψ ψ ψ

(

) ( )

a b ψ = ab ψ

( a b + ) ψ

= a ψ

+ b ψ

Compatible

F×(F×V)→V (F×F)×V →V

1 ψ

= ψ

There is no counterpart in one of the field axioms - inverse of multiplications for every a ≠ 0 in F,

there exists a^–1F, such that a · a^–1= 1

A field is a vector space, but the inverse is not true.

Vector Spaces: Features

( − ψ i ) = −  1 ψ i

0 ψ i = O O is unique

is unique

− ψ i

Null Vector

Inverse Vector

Vector Spaces: Linear Independence

1 n

i i

i

a O

=

 ψ =

If this equation is satisfied only when all a

= 0, The set of ψ

is said to be linearly independent.

If a

≠ 0,

1,

1 n

k i i

i i k k

a =  a

= − 

ψ ψ

0 0

+1 0 –1

Vector Spaces: Dimension

The dimension of a vector space

= The maximum number of linearly independent vectors that can be accommodated in the vector space

V

(R): n-dimensional Real Vector Space V

(C): n-dimensional Complex Vector Space A set of n linearly independent vectors = A basis

Each vector = Basis vector

1 n

i i

i

a

=

= 

Ψ ψ

Basis vector

Vector component for this basis vector

Vector Space: Finite-D Example

Consider n-dimensional vector space

1 n

i i

i

a

=

= 

Ψ ψ

Basis vector

Vector component for this basis vector

When we “know” the basis, we can express an arbitrary state with the N × 1 matrix representation

1 2 a

N

a a

a

   

=  

   

  Ψ

1 2 b

N

b b

b

   

=  

   

  Ψ

1 1

2 2

a b

N N

a b a b

a b

 + 

 + 

 

+ =

 

 + 

 

Ψ Ψ

1 2 a

N

ca c ca

ca

 

 

 

=  

 

  Ψ

Caution: The finite ‘dimension’ does not mean the finite ‘set’: Consider (x,y) plane

Classifying the Spaces

Vector Space: Vector Addition & Scalar Multiplication

Normed Vector Space: Norm as the distance

Inner-Product Space: An Orthonormal Basis Banach Space: Completeness

Hilbert Space

Euclidean Space R

 H

Euclidean (& Non-Euclidean) Geometry

Euclidean Geometry I

Euclid: Greek, living at around 300BC

Euclid’s “Elements”: one of the most important

mathematical set of books of all time

Euclidean Geometry II

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. [The parallel postulate]: If two lines are drawn which intersect a third in such a way that

the sum of the inner angles on one side is less than two right angles, then the two lines

inevitably must intersect each other on that side if extended far enough.

From Euclidean to Non-Euclidean Geometry

Janos Bolyai (1802-1860) Nikolai Lobachevsky

(1792-1856) Carl Friedrich Gauss

(1777-1855)

[The parallel postulate]: If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

19

century

[The revised Postulate]: Given a line and a point not on that line, there are (i) no parallel lines since any two lines must intersect: Elliptic Geometry

(ii) infinitely many lines parallel to the given line through the point: Hyperbolic Geometry

The Revised Form of the Fifth Euclidean Postulate

Hyperbolic Geometry Elliptic Geometry Euclidean Geometry

No parallel lines A single line Infinitely many lines

Tiling & Schläfli Symbol

Dividing 360 ° with integers

: 180 ° , 120 ° (Regular Hexagon), 90 ° (Square), 72 ° , 60 ° (Equilateral Triangle), 45 ° , …

➔ Only three possible regular polygons in 2D Euclidean Geometry…

Square Lattice {4,4}

= 4 squares at a vertex

Honeycomb Lattice {6,3}

= 3 regular hexagons at a vertex

Triangular Lattice {3,6}

= 6 equilateral triangle at a vertex

Schläfli symbol {p,q} = q contact of p-sided regular polygons at a vertex

Results of Modifying the Fifth Euclidean Postulate

{3,6}

{3,q<6}

The sum of internal angles = π

The sum of internal angles < π The sum of internal angles > π

{3,q>6}

Denser contact of polygons!

Sparser contact of polygons!

Hyperbolic Geometry Elliptic Geometry

Euclidean Geometry

We know Elliptic Geometry… But, Hyperbolic Geometry?

Lettuces

Coral

• We can find 2D hyperbolic planes in nature…

• But, it is extremely difficult to analyze/engineer

such shapes & platforms in a practical manner

Analysis: Poincaré Disk Model

https://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html https://twitter.com/mathemaniac/status/753728363563331584

Hyperbolic Geometry for Internet Mapping

More Efficient Internet Mapping in Hyperbolic Geometry

Nat. Commun. 1, 62 (2010)

Hyperbolic Lattices in Circuit QED & Photonics

Nature 571, 45 (2019) PRL 125, 053901 (2020)

Summary

• Vector Space

• Euclidean & Non-Euclidean Geometry