Chap1 Vector Analysis Stu

8/18/2019 Chap1 Vector Analysis Stu

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Electromagnetism in Energy Science 2015 1

Course Introduction

Electromagnetism in Energy Science (00184)

Class time: every Thu 14:00-16:40

References:1. David J. Griffiths, "Introduction to Electrodynamics 3rd Edition", Prentice-Hall Inc. (1999).

2. Walter Greiner, "Classical Electrodynamics", Springer (1998).

3. J. D. Jackson, “Classical Electrodynamics”, John Wiley & Sons Inc. (1999).

Grades:

1st exam (40%), final exam (40%),

homework for every 2 weeks (15%), attendance and enthusiasm (5%)

Office hours: every Thu 17:00-18:00, or

after an appointment by e-mail: [email protected]

Course material and homework will be uploaded in the e-class site.

Cheating in exam and copying homework of others will be graded to zero point.


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Electromagnetism in our everyday living: electricity, electronic goods, ...

in nature: atoms, molecules, light, ...

Phenomena related to electromagnetism:

electrostatic force, magnet, ...

H. C. Oersted found the connection btw. electricity and the magnetic field.

Most electromagnetic phenomena were explained by M. Faraday' experiments

and his ideas were mathematically formulated by J. C. Maxwell.

M. Faraday

dynamo by Faraday

J. C. Maxwell

Introduction: Electromagnetism


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E (elec. field), D (elec. displacement), (elec. charge density), H (mag. field),

B (mag. induction), and J (elec. current density) depend on t (time) and r (position).

In quasi-stationary states ( )

the elec. and the mag. fields are decoupled: electro- and magnetostatic regimes

Maxwell’s Equations


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Classification of physical observables into scalars and vectors.

While a scalar is given by a magnitude, a vector is defined by a direction

as well as a magnitude.Vector algebra:

1) addition, subtraction of two vectors

2) dot product of two vectors:

:projection of the vector A(B) on the vector B(A).

3) cross product of two vectors:

The direction of n is decided by the right-hand rule.

is area of the parallelogram given by A and B.

1. Vector Analysis

A

B

A

B

AB

An arbitrary vector A expanded in terms of basis vector:With three orthonormal basis vectors i , j , k


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Triple products

1) Scalar triple product:

is the volume of the parallelepiped spanned by A,B and C.

2) Vector triple product:

Differential vector calculus

Vectorial operator

When is


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3) Divergence of a vector

F is a measure of how much the vector F spreads out from the considered point.

4) Curl of a vector F

F is a measure of how much the vector F curls around the point considered.

2) Gradient of a scalar F(x,y,z):

It points in the direction of max. increase of the function F(x,y,z), and

its magnitude gives the slope along this max. direction.※ What does F(x,y,z)=0 mean?

p p

F(x,y,z)

F

Find the gradient of


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Product Rules: for a scalar f and vectors A and B

Second derivatives

Why is again a scalar and a vector?




Integral calculus

independent of path taken from i to f.

=0, if the path is closed, F is conservative.

Green's (divergence) theorem

Stokes' theorem

F

The result depends only on the boundary line, not on the particular surface.

The value is zero for any closed surface, since the boundary line can shrinks

down to a point.

※combination of Gauss's and Stokes' theorem:




Check the divergence theorem using the fnc. in unit volume.

Check the Stokes’ theorem using the fnc. in unit area.




Spherical and Cylindrical Coordinates

r

x

y

z

z

x

y

Relation btw. cart. and sph.:

Relation btw. car. and cylin.:

Infinitesimal volume elements:




Basis transformations:

According to the basis transformations, all vectorial operators can be transformed

in sph. and cylin. coord. sys. :




Dirac Delta function

Consider the vector function

A point source: Its existence is exactlydefined only at a point and its integration

is finite.

a special function is needed.

- 1d Dirac delta function (x):

xx=0

(x) is not an usual (ordinary) function because it diverges at x=0 and

it is not Riemann-integrable.

(x) picks out the value of f(x) at x=0.

S




-Evaluation with Dirac delta function

- Extending to 3d Dirac delta function (r ):

- Now, back to

definition of (r )

El t ti i E S i 2015 14


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- Supplementary Vector Operations

Chap1 Vector Analysis Stu

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