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EEE 224 ELECTROMAGNETIC THEORY: Gradient, Divergence, Curl, and Stokes' Theorem, Lecture notes of Electromagnetism and Electromagnetic Fields Theory

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EEE 224 ELECTROMAGNETIC THEORY
Gradient –
Divergence –
Curl
Divergence & Stokes’
Theorems
Null Identities –
Helmholtz Theorem
METU -
NCC
Assist. Prof. Dr. Özlem Özgün
Office :S-144
Phone :661 2972
E-mail :ozozgun@metu.edu.tr
Web :http://www.metu.edu.tr/~ozozgun/
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Download EEE 224 ELECTROMAGNETIC THEORY: Gradient, Divergence, Curl, and Stokes' Theorem and more Lecture notes Electromagnetism and Electromagnetic Fields Theory in PDF only on Docsity!

1

EEE 224 ELECTROMAGNETIC THEORY

Gradient –

Divergence –

Curl

Divergence & Stokes’

Theorems

Null Identities –

Helmholtz Theorem

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/

2

“Del”^

or “Nabla”

Operator (

∇^ is a vector operator

that acts upon

T or A.

∇^ is not

a vector

that “multiplies”

T or A.

(.)^

(.)^

(.) ˆ^

ˆ^

ˆ

x^

y^

z

a^

a^

a

x^

y^

z

∂^

∂^

∇ =^

+^

∂^

∂^

T^ : a scalar field A^ : a vector field

4

GRADIENT

5

Question: Suppose we have a function of one variable:

f(x).

What does the derivative,

df^ / dx

, do for us?

Answer: It tells us how rapidly the function

f(x) varies

when we change the argument

x by a tiny amount,

dx.

In words: If we change

x by an amount

dx,

thenf changes by an amount

df.

The derivative is theproportionality factor. The derivative

df^ / dx

is the

slope of the graph of

f versus

f x.

x

First think “ordinary derivative”.

Gradient

7

Gradient

I want to fly in such a direction that I will get warm as soon as possible. In what direction must I

fly?

Δx^ T = 15^ °C T = 25^ °C

Δx

T = 20^ °C

T = 22^ °C^ Δx

T = 12^ °C

BEE Δx T = 18^ °C

Δx

Directionofgradient

8

Gradient

The gradient

∇^ T points in the

direction of maximum

increase of the function

T.

The magnitude |

∇^ T |^ gives the

slope (rate of increase) alongthis maximal

direction.

Directionofgradient

(Function, T is

the^ hill height)

Gradient of

T

(vector quantity

Æ^ has magnitude & direction)

ˆ^

ˆ^

x^

y^

z

T^

T^

T

T^

a^

a^

a

x^

y^

z

∂^

∂^

∇^ =^

+^

∂^

∂^

10

Gradient

From^ these maps,

we can infer the

slope of the Earth’s

surface,

and^ the direction of these slopes

as topographic

contours lie closer together where the surface is

very steep.

Steep cliff

11

Gradient

h (x,y)^ Æ

A scalar field representing the

height

(elevation) of the Earth at some point

denoted by

coordinates x and y.

13 Gradient 1 ˆ^

ˆ^

ˆ

r^

z

V^

V^

V

V^

a^

a^

a

r^

r^

φ z φ ∂^

∂^

∇^ =

+^

∂^

∂^

∂ ˆ^

ˆ^

ˆ x^

y^

z

V^

V^

V

V^

a^

a^

a

x^

y^

z

∂^

∂^

∇^ =

+^

∂^

∂^

∂ 1

1 ˆ^

ˆ^

ˆ sin

V^ R

V^

V

V^

a^

a^

a

R^

R^

θ R

φ

θ^ φ

∂^

∂^

∇^ =^

+^

∂^

∂^

Cylindrical Coordinates:^ ( ∂ Cartesian Coordinates: ,^ , r^ z^ φ^ )^ Spherical Coordinates: ,^ , R^ θ^ φ^ (^ )

14

Path Independence Property

(^ )^

(^ ) 2

1

C C

A^ r

A^ r^

d

d

≠^

∫^

A^ ∫

A

BUT; If

C^1 C^2 REMEMBER

: In general;

Integral depends on path.

(^ )^

(^ )

A^ r^

g^ r= ∇

“gradient of a scalar field g(r)”

(^ )^

(^ ) 2

1

C C

A^ r

A^ r^

d

d

=^

∫^

A^ ∫

A

Then,^ (^ )^

(^ )^

(^ )^

(^ )^ A

B

C

C A^ r^

d

g^ r^

d^ g

r^

g^ r

⋅^

∇^

⋅^ =

=^

∫^

A^ ∫

A^

Depends on onlythe end points

PA
PB

16

DIVERGENCE

DIVERGENCE THEOREM

17

Divergence

Variations in a vector field can be represented by FLUX LINES,which are directed curves indicating the direction of vector fields.

Electric field of a charge

Flux: the rate of flow across a surface

A ds ⋅∫ S

19

Divergence

Consider a volume V enclosed by a surface S When the volume contains a“source”, there will be a net“outward flux” through S.^ (+ divergence)

When the volume contains a“sink”, there will be a net“inward flux

” through S. (-^ divergence)

20

Divergence

Divergence (

)^ measures how much the vector A spreads out (diverges) from the point in question. Divergence measures the outflow per unit volume of avector field at a point. Divergence

is a local measure of outgoingness. Origin: source (+ divergence)

Origin: sink (-^ divergence)

Is it diverging or converging?

A ∇ ⋅