TFE4120 Electromagnetism: crash course

Intensive course: 7-day lecture including exercises.

Teacher: Anyuan Chen, Post-doctor in electrical power engineering, room E-421. e-post:anyuan.chen@ntnu.no

Assistant: Hallvar Haugdal E-451. hallvar.haugdal@ntnu.no.

Exercises help: proposal time 13:00-15:00 place: E-451.

Paticipants: should have Bsc in electronic, electrical/ power engineering.

Aim of the course: Give students a minimum of pre-requisities to follow a 2-year master programin electronics or electrical /power engineering.

Webpage: https://www.ntnu.no/wiki/display/tfe4120/Crash+course+in+Electromagnetics+2017

All information is posted there .

Lecture1: electro-magnetism and vector calulus

1) What does electro-magnetism mean?

2) Brief induction about Maxwell equations

3) Electric force: Coulomb’s law

4) Vector calulus (pure mathmatics)

Electro-magnetism

Electro-magnetism: interaction between electricity and magnetism.

Michael Faraday (1791-1867)

• In 1831 Faraday observed that a moving magnet 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 static distribution of charges produces an electric field• Charges in motion (an electrical current) produce a magnetic

field

• A changing magnetic field produces an electric field, and a changing electric field produces a magnetic field.

Electric and Magnetic fields can produce forces on charges

Electricity and magnetism had been unified into electromagnetism!

Gauss’ law

Faraday’s law

Ampere’s law

Coulomb’s law: force between electrostatic charges

The electrostatic force had the same functional form as Newton’s law of gravity

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.

Scalar: 𝑭 = 𝒌𝒒𝟏𝒒𝟐

𝒓𝟏𝟐𝟐 =

𝒒𝟏𝒒𝟐

𝟒𝝅𝜺𝟎𝒓𝟏𝟐𝟐

Vector: 𝑭 =𝒒𝟏𝒒𝟐

𝟒𝝅𝜺𝟎𝒓𝟏𝟐𝟐 ෞ𝒓𝟏𝟐

ෞ𝒓𝟏𝟐 is just for direction, its absolut value is 1.

Vector force:

𝑭𝒕𝒐𝒕 =

𝒊=1

𝒏𝒒𝒒i

4𝝅𝜺0𝒓𝒊2 ෝ𝒓𝒊

Integration and vector caculus

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

dL direction

Gradient:Greatest rate of increase

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

𝛻𝒇 =𝜕𝒇

𝜕𝑥ෝ𝒙+

𝜕𝒇

𝜕𝑦ෝ𝒚+

𝜕𝒇

𝜕𝑧ො𝒛

P

Divergence: Flux out of a point

Electric flux density: definition 𝑫 = 𝜀𝑬, independent of the material.

𝛻 ∙ 𝐸 =𝜕𝐸𝑥𝜕𝑥

+𝜕𝐸𝑦

𝜕𝑦+𝜕𝐸𝑧𝜕𝑧

+Q

𝛻 ∙ 𝐸 is a scalar.

Divergence: Mathematical calculation

Divergence theorem

Curl: how much does a field circulate around a point.

𝛻 × 𝐴 = (𝜕𝐴𝑧

𝜕𝑦-𝜕𝐴𝑦

𝜕𝑧) ො𝑥 + (

𝜕𝐴𝑥

𝜕𝑧-𝜕𝐴𝑧

𝜕𝑥) ො𝑦 + (

𝜕𝐴𝑦

𝜕𝑥-𝜕𝐴𝑥

𝜕𝑦) Ƹ𝑧

Curl

𝛻 × 𝐴 = (𝜕𝐴𝑧

𝜕𝑦-𝜕𝐴𝑦

𝜕𝑧) ො𝑥 + (

𝜕𝐴𝑥

𝜕𝑧-𝜕𝐴𝑧

𝜕𝑥) ො𝑦 + (

𝜕𝐴𝑦

𝜕𝑥-𝜕𝐴𝑥

𝜕𝑦) Ƹ𝑧

The curl around x-axis, in yz plane

Similar to the curl around y and z-axis

Stokes’ Theorem

Different coordinates

Spherial coordinate Cylindrical coordinate Cartesia Coordinate

Examples: Probelm 3

Solutions:

Solution for b

Example:

𝛻 × 𝐴 = (𝜕𝐴𝑧

𝜕𝑦-𝜕𝐴𝑦

𝜕𝑧) ො𝑥 + (

𝜕𝐴𝑥

𝜕𝑧-𝜕𝐴𝑧

𝜕𝑥) ො𝑦 + (

𝜕𝐴𝑦

𝜕𝑥-𝜕𝐴𝑥

𝜕𝑦) Ƹ𝑧

Solution for i) and ii)

Conservative vector: solution for iii)

𝛻 × 𝐴 = (𝜕𝐴𝑧

𝜕𝑦-𝜕𝐴𝑦

𝜕𝑧) ො𝑥 + (

𝜕𝐴𝑥

𝜕𝑧-𝜕𝐴𝑧

𝜕𝑥) ො𝑦 + (

𝜕𝐴𝑦

𝜕𝑥-𝜕𝐴𝑥

𝜕𝑦) Ƹ𝑧