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SPECIAL TECHNIQUES. To find the electric field of a stationary charge distribution :. Find the potential of the distribution. To Solve : Poisson’s / Laplace’s Equation. SPECIAL TECHNIQUES. ► to determine the Potential uniquely …. Poisson's / Laplace’s equation. +.

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Presentation Transcript
slide1

SPECIAL TECHNIQUES

To find the electric field of a stationary charge distribution :

Find the potential of the distribution

To Solve :

Poisson’s / Laplace’s Equation

Dr. Champak B. Das (BITS, Pilani)

slide2

SPECIAL TECHNIQUES

► to determine the Potential uniquely…

Poisson's / Laplace’s equation

+

A set of boundary conditions

Proof ?

………. Uniqueness Theorem

Dr. Champak B. Das (BITS, Pilani)

slide3

First Uniqueness Theorem :

The solution to Laplace’s equation in some volume is uniquely determined if the potential is specified on the boundary surface.

Dr. Champak B. Das (BITS, Pilani)

slide4

Example :

Answer:

Dr. Champak B. Das (BITS, Pilani)

slide5

Corollary of 1st Uniqueness Theorem :

The potential in a volume is uniquely determined if (a) the charge density throughout the region, and (b) the value of the potential on all boundaries, are specified.

Dr. Champak B. Das (BITS, Pilani)

slide6

Second Uniqueness Theorem :

In a volume surrounded by conductors and containing a specified charge density, the electric field is uniquely determined if the total charge on each conductor is given.

(The entire region can be unbound/bounded by another conductor).

Dr. Champak B. Das (BITS, Pilani)

slide7

E is uniquely determined

q2

q1

 known

q3

Dr. Champak B. Das (BITS, Pilani)

slide8

SPECIAL TECHNIQUES

  • The Method of Images
  • Multipole expansion

Dr. Champak B. Das (BITS, Pilani)

slide9

METHOD OF IMAGES

P

( x, y, z)

z

q

d

y

Grounded conducting plane

x

To find out the potential in the region above the plane

Dr. Champak B. Das (BITS, Pilani)

slide10

Solution of Poisson’s equation:

(in the region z > 0 ),

WITH

  • A point charge q at (0,0,d)
  • Boundary conditions:
  • V = 0 when z = 0
  • V  0 when x2+y2+z2 >> d2

Dr. Champak B. Das (BITS, Pilani)

slide11

Given :

  • charge density in the region on interest
  • value of the potential on the boundary

Corollary of First Uniqueness Theorem

ONLY one function that meets the requirement

Dr. Champak B. Das (BITS, Pilani)

slide12

Z

P

( x, y, z)

+q

d

Y

d

-q

X

A new problem:

Dr. Champak B. Das (BITS, Pilani)

slide13

satisfies

  • the poisson’s equation in the region of interest
  • the boundary conditions

… of the original problem

Final answer

(By virtue of Uniqueness Theorem)

Dr. Champak B. Das (BITS, Pilani)

slide14

Induced Surface Charge

Total induced charge :

Dr. Champak B. Das (BITS, Pilani)

slide15

Induced Surface Charge on the grounded conducting plane as a function of

(for x2+y2 >> d2)

Dr. Champak B. Das (BITS, Pilani)

slide16

Force

Force of attraction on q towards the plane

Force of attraction on +q towards -q

Dr. Champak B. Das (BITS, Pilani)

slide17

ENERGY

Two point charges and no conductor :

Single point charge and conducting plane :

Dr. Champak B. Das (BITS, Pilani)

slide18

Another example :

R

a

q

V=0

A point charge and a grounded conducting sphere :

Dr. Champak B. Das (BITS, Pilani)

slide19

Image charge :

Location of image charge :

(to the right of the centre of the sphere)

Dr. Champak B. Das (BITS, Pilani)

slide20

b

q

q'

a

Two point charges q and q and no conductor

Dr. Champak B. Das (BITS, Pilani)

slide21

rs

r

rs´

Prob. 3.7(a):

θ

z

q

b

q'

a

 V=0

r = R

Dr. Champak B. Das (BITS, Pilani)

slide22

Prob. 3.7(b) :

Induced surface charge on the sphere :

Dr. Champak B. Das (BITS, Pilani)

slide23

Induced Surface Charge on the grounded conducting sphere as a function of  :

Dr. Champak B. Das (BITS, Pilani)

slide24

Prob. 3.7(b) :

Total Induced surface charge :

Dr. Champak B. Das (BITS, Pilani)

slide25

Prob. 3.7(c) :

Force on q :

Energy of the configuration :

Dr. Champak B. Das (BITS, Pilani)

slide26

rs

d'

r

θ'

r'

MULTIPOLE EXPANSION

To characterize the potential of an arbitrary charge distribution, localized in a rather small region of space

Dr. Champak B. Das (BITS, Pilani)

slide27

Law of cosines 

Dr. Champak B. Das (BITS, Pilani)

slide28

Legendre polynomials

More on this next sem. in Maths - III

Dr. Champak B. Das (BITS, Pilani)

slide29

Systematic expansion for the potential of an arbitrary localized charge distribution, in powers of 1/r

Multipole expansion of V in powers of 1/r

Dr. Champak B. Das (BITS, Pilani)

slide30

Monopole term

Dipole term

Quadrupole term

Dr. Champak B. Das (BITS, Pilani)

slide31

The Monopole Term:

…… is the most dominant term for r >>

Dr. Champak B. Das (BITS, Pilani)

slide32

What does Vmon describe ?

  • APPROXIMATELY ;

… the potential of any distribution (if looked from a very far point)

  • EXACTLY ;

… the potential of a point charge at origin (everywhere)

Dr. Champak B. Das (BITS, Pilani)

slide33

The Dipole Term:

…… is the most dominant term if total charge is zero

Dr. Champak B. Das (BITS, Pilani)

slide34

dipole moment of the distribution

Dr. Champak B. Das (BITS, Pilani)

slide35

z

-q

d

r'_

+q

r'+

y

x

For a collection of point charges,

For aphysicaldipole:

Dr. Champak B. Das (BITS, Pilani)

slide36

P

rs+

+q

r

d

rs-

-q

Potential of a physical dipole

Dr. Champak B. Das (BITS, Pilani)

slide37

Potential due to a point charge ~ 1/r

Potential due to a dipole ~ 1/r2

Dr. Champak B. Das (BITS, Pilani)

slide38

physical

dipole

Dr. Champak B. Das (BITS, Pilani)

slide39

Potential for a pure dipole (d  0)

Physical dipole

Pure dipole

for d  0, q   , with p=qd kept fixed

Dr. Champak B. Das (BITS, Pilani)

slide40

What does Vdip describe ?

  • APPROXIMATELY ;

… the potential of a physical dipole (if looked from a very far point)

  • EXACTLY ;

… the potential of a pure dipole (everywhere)

Dr. Champak B. Das (BITS, Pilani)

slide41

z

r

rs

y

d

O

q

x

Role played by ORIGIN of coordinate system in multipole expansion

A point charge away from origin :

 Posses a non zero dipole contribution

Dr. Champak B. Das (BITS, Pilani)

slide42

Dipole moment changes when origin is shifted :

d'

y

r'

a

x

Dr. Champak B. Das (BITS, Pilani)

slide43

If Q = 0, then

If net charge of the configuration is zero, then the dipole moment is independent of the choice of origin.

Dr. Champak B. Das (BITS, Pilani)

slide44

3q

a

a

a

-2q

-2q

a

q

Prob. 3.27:

In the charge configuration shown, find a simple approximate formula for potential, valid at points far from the origin. Express your answer in spherical coordinates.

Answer:

Dr. Champak B. Das (BITS, Pilani)

slide45

Field due of a dipole

Potential at a point due to a pure dipole:

z

r

p

y

x

Dr. Champak B. Das (BITS, Pilani)

slide46

Field due of a dipole (contd.)

Recall:

Dr. Champak B. Das (BITS, Pilani)

slide47

z

r

y

x

Prob 3.33:

Electric field in a coordinate free-form :

p

Dr. Champak B. Das (BITS, Pilani)

slide48

z

y

Field lines of a pure dipole

Dr. Champak B. Das (BITS, Pilani)

slide49

Field lines of a physical dipole

Dr. Champak B. Das (BITS, Pilani)

slide50

Prob. 3.31:

A “pure” dipole p is situated at the origin, pointing in the z-direction.

(a) What is the force on a point charge q at (a,0,0) (Cartesian coordinates) ?

(b) What is the force on a point charge q at (0,0,a) ?

(c) How much work does it take to move q from (a,0,0) to (0,0,a) ?

Dr. Champak B. Das (BITS, Pilani)