EEE 224 ELECTROMAGNETIC THEORY: Differential Length & Line Integral, Lecture notes of Electromagnetism and Electromagnetic Fields Theory

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EEE 224 ELECTROMAGNETIC THEORY^ Differential Length & Line Integral

METU -

NCC

Assist. Prof. Dr.

Özlem Özgün Office^

:^ S- Phone^

:^ 661 2972 E-mail^

:^ ozozgun@metu.edu.tr Web^

:^ http://www.metu.edu.tr/~ozozgun/

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Differential Length Vector For example, if a point moves such that its coordinate changes from^ to^

, then the position vector that describes that

point changes from

to

A^ +^ ΔA^

A

r^ r^ +^

ΔA

,^ ,x y z^ ( ) is simply a distance (displacement) vectorΔA

(^

)

• Δ^ + Δ^

• Δ, , x^ x y^

y z^ z

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How to expressdifferential length

vector in Cartesian

coordinates? ˆ ˆ ˆ^

ˆ xy

z x

y

z d^ a d d^

d^ d a dx a dy^

a dz = =^ +

=^

+^

A^ A^ A A^ A

A

y

z ˆa dzz

ˆa dxx

ˆa dyy

dz

ˆa dAA^ ˆaA

Differential Length Vector

(arc-length)^ φ^ d φ 5

dr d φ r dr rd^ φ (arc-length)

dz rd^ φ

change in angled φ change in lengthrd due to change in angle

φ φ

φ

Differential Length Vector ˆ → →

ˆ^

ˆ ˆ^ ˆ rz r r^ ˆ

z^ z r

z

d^ d^

d^ d a d a d^

a d

a dr^ a rd

φ φ^ φ^ a dzφφ

=^ +^

=^

+^

=^ +

A^ A^

A^ A A A^

A How to expressdifferential length

vector in cylindrical

coordinates?

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Line Integral

We will use the differential length

vector

when evaluating a

line^ integral. (^ )A r^ d⋅A∫ C

Line^ integral involves integrating the

“projection”

of

a vector field onto a specified

contour C.

**NOTES: ***^ The^ result of the integral is a

**scalar.

• The integration is over one-dimension**

(1D).

*** The contour C is a line or curve through three-dimensional**

(3D)^ space.

*** The position vector**

denotes only those points that lie

on contour C.

Therefore, the value of this integral

only^ depends on the value of vector

field^

at the points

along this contour. A^ r(^ )

r

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Differential Length & Line Integral The differential vector

is the differential length vector formed when a point moves a small distance

along contour C.

d^ A

A^ r^ d⋅A^ (^ )∫ C

is^ always tangential

to every point of the contour.

d^ A

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Line Integral

(^ )^

(^ )^

(^ ) (^10) N^ lim ii i N

C

C A^ r^ d^

A^ r d

A^ r =→∞Δ → ⋅^ =^

=^

Δ ∑

∫^

A^ ∫

A A A^

A^

(Contour is dividedA into N segments.)

Line^ integral integrates (i.e., “adds up”)

the values of the

tangential component

of a vector field at

each and

every point along contour C.

C

A AA i^ ˆaA ΔA

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Line Integral^ QUESTION ??

If^ the^ vector field

is orthogonal to the contour at every

point, what is the result of the line integral?

(^ )^

A^ r^ C

d⋅ =

∫^

A A

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Some Facts about the Line Integral

(^ )^

Circulation integral A^ r^ d⋅^ C

→ ∫^

A If the contour C is a closed

path, the resulting line Noticethe circle! integral of the vector A is defined as the “circulation”^ of A around C.^ If the circulation integral is zero, the vector field Ais called “irrotational” or “conservative”.

C

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Some Facts about the Line Integral

(^ )^

(^ ) 2 1

C C

A^ r A^ r^ d

d ≠^

⋅ ⋅A^ ∫ ∫

A

BUT, for some special cases, they are equal. This property is known as “path independence property”. We will discuss this property after introducing theconcept of gradient.

In general; C 1 C^2

Integral depends on path.

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Follow board

for the examples!