TFE4120 : crash course

Intensive course: 7-day lecture including exercises. Teacher: Anyuan Chen, Post-doctor in electrical , room E-421. e-post: [email protected]

Assistant: Hallvar Haugdal E-451. [email protected]. Exercises help: proposal time 13:00-15:00 place: E-451.

Paticipants: should have Bsc in electronic, electrical/ . Aim of the course: Give students a minimum of pre-requisities to follow a 2-year master program in or electrical /power engineering.

Webpage: https://www.ntnu.no/wiki/display/tfe4120/Crash+course+in+Electromagnetics+2017 All information is posted there . Lecture1: - and vector calulus

1) What does electro-magnetism mean? 2) Brief induction about 3) Electric : ’s law 4) Vector calulus (pure mathmatics) Electro-magnetism

Electro-magnetism: interaction between and magnetism. (1791-1867) • In 1831 Faraday observed that a moving could induce a current in a circuit. • He also observed that a changing current could, through its magnetic effects, induce a current to flow in another circuit.

James Clerk Maxwell: (1839-1879) • he established the foundations of electricity and magnetism as electromagnetism. Electromagnetism: Maxwell equations

• A distribution of charges produces an electric • Charges in (an electrical current) produce a • A changing magnetic field produces an , and a changing electric field produces a magnetic field.

Electric and Magnetic fields can produce on charges

Gauss’ law

Faraday’s law

Ampere’s law

Electricity and magnetism had been unified into electromagnetism! Coulomb’s law: force between electrostatic charges

풒ퟏ풒ퟐ 풒ퟏ풒ퟐ : 푭 = 풌 ퟐ = ퟐ 풓ퟏퟐ ퟒ흅휺ퟎ풓ퟏퟐ

풒ퟏ풒ퟐ Vector: 푭 = ퟐ 풓ෞퟏퟐ ퟒ흅휺ퟎ풓ퟏퟐ

풓ෞퟏퟐ is just for direction, its absolut value is 1.

The electrostatic force had the same functional form as Newton’s law of The magnitude of the electrostatic force between two point charges: 1) directly proportional to the product of the magnitudes of charges 2) inversely proportional to the square of the distance between them 3) The force is along the straight line joining them. Vector force:

풏 풒풒 푭 = ෍ i 풓ෝ 풕풐풕 4흅휺 풓2 풊 풊=1 0 풊 Integration and vector caculus

dL direction Vector: Effective part of A is the the component along L direction

Vector: Effective part of A the the component along S direction S direction is perpendicular to the tangent plane to that surface at S

Scalar: no directon Gradient:Greatest rate of increase

P Gradient: 3 derivative of a scalar function showing the direction and magnitude of the maximum spatial variation ( greatest rate of increase) of the scalar function V at a point space.

휕풇 휕풇 휕풇 훻풇 = 풙ෝ+ 풚ෝ+ 풛ො 휕푥 휕푦 휕푧 Divergence: Flux out of a point

Electric flux density: definition 푫 = 휀푬, independent of the material.


휕퐸푥 휕퐸푦 휕퐸푧 훻 ∙ 퐸 = + + 훻 ∙ 퐸 is a scalar. 휕푥 휕푦 휕푧 Divergence: Mathematical calculation Curl: how much does a field circulate around a point.

휕퐴 휕퐴 휕퐴 휕퐴 휕퐴 휕퐴 훻 × 퐴 = ( 푧- 푦)푥ො + ( 푥- 푧)푦ො + ( 푦- 푥)푧Ƹ 휕푦 휕푧 휕푧 휕푥 휕푥 휕푦 Curl

The curl around -axis, in yz plane

Similar to the curl around y and z-axis

휕퐴 휕퐴 휕퐴 휕퐴 휕퐴 휕퐴 훻 × 퐴 = ( 푧- 푦)푥ො + ( 푥- 푧)푦ො + ( 푦- 푥)푧Ƹ 휕푦 휕푧 휕푧 휕푥 휕푥 휕푦 Stokes’ Theorem Different coordinates

Spherial coordinate Cylindrical coordinate Cartesia Coordinate Examples: Probelm 3 Solutions: Solution for b Example:

휕퐴 휕퐴 휕퐴 휕퐴 휕퐴 휕퐴 훻 × 퐴 = ( 푧- 푦)푥ො + ( 푥- 푧)푦ො + ( 푦- 푥)푧Ƹ 휕푦 휕푧 휕푧 휕푥 휕푥 휕푦 Solution for i) and ii) Conservative vector: solution for iii)

휕퐴 휕퐴 휕퐴 휕퐴 휕퐴 휕퐴 훻 × 퐴 = ( 푧- 푦)푥ො + ( 푥- 푧)푦ො + ( 푦- 푥)푧Ƹ 휕푦 휕푧 휕푧 휕푥 휕푥 휕푦