Dr. Champak B. Das ( BITS, Pilani) PHYSICS-II (PHY C132) Introduction to Electrodynamics: by David...

Dr. Champak B. Das ( BITS, Pilani)

PHYSICS-II (PHY C132)

Introduction to Electrodynamics:

by David J. Griffiths (3rd Ed.)

ELECTRICITY & MAGNETISM


VECTOR ANALYSIS

Differential Calculus

Integral Calculus

Curvilinear Coordinates

The Dirac Delta Function

Theory of Vector Fields

Revisited


Derivative of any function f(x,y,z):

Differential Calculus

dzz

fdy

y

fdx

x

fdf

dzkdyjdxiz

fk

y

fj

x

fidf ˆˆˆˆˆˆ

ldfdf


Change in a scalar function f corresponding to a change in position :

ldfdf

f is a VECTOR

z

fk

y

fj

x

fifwhere

ˆˆˆ

Gradient of function f


Geometrical interpretation of GradientZ

X

Y

P Qdl

f

Czyxf ),,(

change in f : ldfdf

=0

=> f dl


Z

X

Y

P

Q

dl

1Cf

12 CCf

ldfCCdf

12


θcosfdl

df

fld

||

f

• The rate of change of f is max. for

• The max. value of rate of change of f is

• f increases in the direction of f

• Grad f is in the direction of the normal to the surface of constant f


slope of the function along the direction of maximum rate of

change of the function

Gradient of a function


If f = 0 at some point (x0,y0,z0)

(x0,y0,z0) is a stationary point of f(x,y,z)

=> df = 0 for small displacements about the point (x0,y0,z0)


Prob. 1.12

The height of a certain hill (in feet) is:h(x,y) = 10(2xy – 3x2 -4y2 -18x + 28y +12)

where x is distance (in mile) east and y north of Pilani.

(a) Where is the top located ?

Ans: 3 miles North & 2 miles West


Prob. 1.12 (contd.)

h(x,y) = 10(2xy – 3x2 -4y2 -18x – 28y +12)

(b) How high is the hill ?

(c) How steep is the slope at 1 mile north and 1 mile east of Pilani? In what direction the slope is steepest, at that point ?

Ans: 311 ft/mile, direction is Northwest

Ans: 720 ft


Prob. 1.13

Let rs is the separation vector from (x,y,z) to (x,y,z) .

2

ˆ

s

s

r

r

sr

2

sn

s rnr ˆ1

2sra

srb

1

nsrc


The Operator

zk

yj

xi

ˆˆˆ

is NOT a VECTOR,

but a VECTOR OPERATORVECTOR OPERATOR

Satisfies: •Vector rules

•Partial differentiation rules


On a scalar function f : f

can act:

GRADIENT

On a vector function F as: . F

DIVERGENCE

On a vector function F as: × F

CURL


Divergence of a vector

z

F

y

F

x

FF zyx

zyx FkFjFiz

ky

jx

iF ˆˆˆˆˆˆ

Divergence of a vector is a scalar.


.F is a measure of how much the vector F spreads out/in (diverges) from/to the point in

question.

Geometrical interpretation of Divergence


Physical interpretation of DivergenceFlow of a compressible fluid:

Z

X

Y

dy

dxdz

A

CD

B

E F

GH

(x,y,z) density of the fluid at a point (x,y,z)

v(x,y,z) velocity of the fluid at (x,y,z)


Net rate of flow out through all pairs of surfaces (per unit time):

dxdydzvz

vy

vx zyx

ρρρ

dxdydzvρ


Net rate of flow of the fluid per unit volume per unit time:

vρ

DIVERGENCE


Example:

nrr

12 nrn

rfr

dr

rdfrrf 3

Calculate,


Prob. 1.16

Sketch the vector function and compute its divergence. Explain the answer !

2

ˆ

r

rv

0 v

!


Curl

zyx FFF

zyx ///

kji

F

y

F

x

Fk

x

F

z

Fj

z

F

y

Fi xyzxyz ˆˆˆ

Curl of a vector is a vector


×F is a measure of how much the vector F “curls around” the point in question.

Geometrical interpretation of Curl


Physical significance of Curl

Circulation of a fluid around a loop about a point :

X

Y

1

4 2

3

ldv

Circulation

yyxx dlvdlv

),( 00 yxy

x


yxy

v

x

vxy ΔΔ

∂

∂-

∂

∂

Circulation per unit area

z-component of CURL

z

v


Sum Rules

2121 ffff ∇∇∇

2121 FFFF

For Gradient:

For Divergence:

For Curl:

2121 FFFF


Rules for multiplying by a constant

fkkf

FkFk

For Gradient:

For Divergence:

For Curl: FkFk


Product Rules

122121 ffffff ∇∇∇

Gradients:

1221

122121

FFFF

FFFFFF

For a Scalar from two functions:


Product Rules

fFFfFf

Divergences:

211221 FFFFFF

For a Vector from two functions:


Product Rules

fFFfFf

Curls:

1221

211221

FFFF

FFFFFF


Prob. 1.21 (b)

Compute: rr ˆˆ

Ans: 0

Prob. 1.21 (a)

What does the expression mean ?

21 FF


Quotient Rules

22

2112

2

1

f

ffff

f

f

2f

fFFf

f

F

2f

fFFf

f

F


Second Derivatives

f

f2

f

Divergence :

Curl :

Laplacian

( Prob. 1.27: Prove it ! )

0

Of a gradient:


Second Derivatives

FGradient :

F

2

Of a divergence:


Curl :

FFF

2

F

Divergence :

Prob. 1.26: Prove it !

0

Second Derivatives

Of a Curl:


Integral Calculus

Line Integral: b

aldv ldv

Surface Integral: S

adv adv

Volume Integral: τ τdf


Fundamental theorem for gradient

afbfldfb

a

Line integral of gradient of a function is given by the value of the

function at the boundaries of the line.


Corollary 1:

tindependenpathisldfb

a

Corollary 2: 0 ldf


Fundamental theorem for Divergence

adFdF

τ

The integral of divergence of a vector over a volume is equal to the value of the function over

the closed surface that bounds the volume.

Gauss’ theorem, Green’s theorem


Fundamental theorem for Curl

Stokes’ theorem

ldFadF

Integral of a curl of a vector over a surface is equal to the value of the function over the closed boundary that encloses the surface.


Corollary 1:

surfaceparticulartheonnot

lineboundarytheondependsadF

Corollary 2: 0 adF


Curvilinear coordinates:

used to describe systems with symmetry.

Spherical Polar coordinates (r, , )

Cylindrical coordinates (s, , z)


Spherical Polar Coordinates

r : distance from origin

A point is characterized by:

: polar angle

: azimuthal angler

Z

X

Y

P


Cartesian coordinates in terms of spherical coordinates:

φθcossinrx

φθ sinsinry

θcosrz r

Z

X

Y

P


Spherical coordinates in terms of Cartesian coordinates:

222 zyxr

2221

221

cos

/tan

zyxzor

zyx

θ

xy /tan 1φ

r

Z

X

Y

P


Prob. 1.37 : Unit vectors in spherical coordinates

θφθφθ cosˆsinsinˆcossinˆˆ kjir

r

Z

X

Y

r

θ

φθ

φθθ

sinˆ

sincosˆ

coscosˆˆ

k

j

i

φ

φφφ cosˆsinˆˆ ji


Line element in spherical coordinates:

φθφθθ drrddrrld sinˆˆˆ

Volume element in spherical coordinates:

φθθτ dddrrd sin2


Area element in spherical coordinates:

rddrad ˆsin21 φθθ

θφ ˆ2 ddrrad

on a surface of a sphere (r const.)

on a surface lying in xy-plane ( const.)


Ranges of r, and

r : 0

: 0

: 0 2


The Operator in Spherical Polar Coordinates

φθφ

θθ

sin

1ˆ

1ˆˆrrr

r


Gradient:

φθφ

θθ

f

r

f

rr

frf

sin

1ˆ

1ˆˆ

φθ

θθθ

φθ

F

rF

r

Frrr

F r

sin

1sin

sin

1

1 22

Divergence:


Curl:

r

r

FrFrr

rFr

Fr

FFr

rF

θφ

φθθ

φθ

θθ

φ

φ

θφ

1ˆ

sin

11ˆ

sinsin

1ˆ


Laplacian:

2

2

22

2

22

2

sin

1

sinsin

1

1

φθ

θθ

θθ

f

r

f

r

r

fr

rrf


Cylindrical Coordinates

Z

X

Y

z

s P

s : distance from z-axis

A point is characterized by:

z : cartesian coordinate

: azimuthal angle


Prob. 1.41 : Unit vectors in cylindrical coordinates

Z

X

Y

z

s

s

φz

jis ˆsinˆcosˆ φφ

ji ˆcosˆsinˆ φφφ

kz ˆˆ


Line element in cylindrical coordinates:

dzzsddssld ˆˆˆ φφ

Volume element in cylindrical coordinates:

dzddssd φτ


Ranges of s, and z

s : 0

: 0 2

z : -


The Operator in Cylindrical Coordinates

zz

sss

ˆ1ˆˆ

φφ


Gradient:

z

fz

f

ss

fsf

ˆ1ˆˆ

φφ

z

FF

ssF

ssF z

s

φφ11

Divergence:


Curl:

φ

φφ

φ

φs

sz

FsF

ssz

s

Fz

z

F

z

FF

ssF

1ˆ

ˆ1ˆ

Laplacian:

2

2

2

2

22 11

z

ff

ss

fs

ssf

φ


General expressions for the Derivatives in different coordinate systems

Coordinate System: u, v, w

Line element :

dwhwdvgvdufuld ˆˆˆ


System u v w f g h

Cartesian x y z 1 1 1

Spherical r 1 r r sin

Cylindrical s z 1 s 1


GRADIENT

w

t

hw

v

t

gv

u

t

fut

1ˆ1ˆ1ˆ


DIVERGENCE

wv

u

fgAw

fhAv

ghAufgh

A1


CURL :

wvu hAgAfA

wvu

hwgvfu

fghA

ˆˆˆ1


LAPLACIAN

w

t

h

fg

wv

t

g

fh

v

u

t

f

gh

ufght

12


Recall Prob. 1.16Sketch the vector function and

compute its Divergence

2

ˆ

r

rv


Calculation of Divergence =>

0 ττ

dv

Divergence theorem =>

πττ

4 dv


Note: as r 0; v ∞

0,0

;,0

rat

buteverywherev

πττ

4 dv

And its integral over ANY

volume containing the point r = 0


THE DIRAC DELTA FUNCTION

0

00

xif

xifxδ

1

dxxwith δ



Dirac Delta Function is NOT a Function

An infinitely high, infinitesimally narrow

“spike” with area 1


The Defining Characteristic Integral :

0fdxxxf

δ

A Generalized Function OR distribution


Delta function is something that is always intended for use under an integral sign.

dxxDxfdxxDxfIf 21

xDxDThen 21,

Let D1(x) & D2(x) are two expressions involving Delta functions and f(x) is any ordinary function


One can show:

xk

kx δδ||

1

xx δδ

………..for a proof, see Example 1.15



Shifting the singularity from 0 to a;

axif

axifax

0δ

1

dxaxwith δ



afdxaxxf

δ

& the Defining Characteristic Integral :


Prob. 1.43:

dxxxxa 3123:6

2

2 δ

20

dxxxc 1:3

0

3 δ 0


Prob. 1.45 :

xxdx

dxovea δδ :Pr

00

01

xif

xifxb θ

xdx

dshowTo δ

θ:


THE DIRAC DELTA FUNCTION

0

00

xif

xifxδ

1

dxxwith δ



Shifting the singularity from 0 to a;

axif

axifax

0δ

1

dxaxwith δ


The Dirac Delta Function(in three dimension)

0,0,0

;03

at

buteverywherer

δ

13 τδ drspaceall


Why such a function ?

• Describe a point charge in terms of a charge density

• Describe a point particle in terms of a mass density

• Describe very short range forces as nuclear force


Prob. 1.46:

Charge density of a point charge q at r :

rrqr 3δρ

Charge density of a dipole with -q at 0 and +q at a:

rqarqr 33 δδρ


Charge density of a thin spherical shell of radius R and total charge Q:

RrR

Qr δ

πρ 24

Prob. 1.46: (contd.)


From calculation of Divergence:

0ˆ2

τ

τd

r

r

By using the Divergence theorem:

The Paradox of Divergence of

πττ

4ˆ2

d

r

r

2

ˆ

r

r


So now we can write:

rr

r 3

2 4ˆ

πδ

πττ

4ˆ2

d

r

rSuch that:


Theory of Vector Fields

By specifying appropriate boundary conditions,

Helmholtz theorem implies that the field can be uniquely

determined from its divergence and curl.


Potentials

b

a

tindependenpathldF

everywhereF 0

pathclosedldF 0

( For Curl-less fields )THEOREM 1:

potentialscalaraV

VF

:


Conclusions from theorem 1

VFF

0If curl of a vector field vanishes,

(everywhere), then the field can always be written as the gradient of a scalar potential

( not unique )


Potentials

tindependensurfaceadFs

everywhereF 0

surfaceclosedadF 0

For Divergence-less fields

THEOREM 2:

potentialvectoraA

AF

:


Conclusions from theorem 2

AFF

0If divergence of a vector field vanishes,

(everywhere), then the field can always be written as the curl of a vector potential

( not unique )


Helmholtz theorem:

Any vector field F with both source and circulation densities vanishing at infinity may be written as the sum of two parts: one of which is curl-less

and the other is divergence-less.

AVF

(Always)

Dr. Champak B. Das ( BITS, Pilani) PHYSICS-II (PHY C132) Introduction to Electrodynamics: by David...

Documents

Transcript of Dr. Champak B. Das ( BITS, Pilani) PHYSICS-II (PHY C132) Introduction to Electrodynamics: by David...