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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. To determine V. Poisson's / Laplace’s equation. +. A set of boundary conditions. Proof ?. Uniqueness Theorem.

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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

To determine V

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.

Corollary

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)

slide4

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)

slide5

SPECIAL TECHNIQUES

  • The Method of Images
  • Multipole expansion

Dr. Champak B. Das (BITS, Pilani)

slide6

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)

slide7

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)

slide8

 one function that meets the requirement

Guaranteed by :

Corollary of First 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)

slide9

Z

P

( x, y, z)

+q

d

Y

d

-q

X

A new problem:

Dr. Champak B. Das (BITS, Pilani)

slide10

Z = 0

x2+y2+z2 >> d2

Final answer

(By virtue of Uniqueness Theorem)

Dr. Champak B. Das (BITS, Pilani)

slide11

Induced Surface Charge

Total induced charge :

Dr. Champak B. Das (BITS, Pilani)

slide12

Force

Force of attraction on q towards the plane

Force of attraction on +q towards -q

Dr. Champak B. Das (BITS, Pilani)

slide13

ENERGY

Two point charges and no conductor :

Single point charge and conducting plane :

Dr. Champak B. Das (BITS, Pilani)

slide14

R

a

q

V=0

Another example :

A point charge and a grounded conducting sphere :

Dr. Champak B. Das (BITS, Pilani)

slide15

Image charge :

Location of image charge :

(to the right of the centre of the sphere)

Dr. Champak B. Das (BITS, Pilani)

slide16

rs

r

rs´

b

q

q'

a

Two point charges q and q and no conductor

Dr. Champak B. Das (BITS, Pilani)

slide17

rs

r

rs´

Prob. 3.7(a):

θ

z

q

b

q'

a

 V=0

r = R

Dr. Champak B. Das (BITS, Pilani)

slide18

Prob. 3.7(b) :

Induced surface charge on the sphere :

Total Induced surface charge :

Dr. Champak B. Das (BITS, Pilani)

slide19

Prob. 3.7(c) :

Force on q :

Energy of the configuration :

Dr. Champak B. Das (BITS, Pilani)

slide20

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)

slide21

Law of cosines 

Dr. Champak B. Das (BITS, Pilani)

slide22

Legendre polynomials

More on this next sem. in Maths - III

Dr. Champak B. Das (BITS, Pilani)

slide23

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)

slide24

Monopole term

Dipole term

Quadrupole term

Dr. Champak B. Das (BITS, Pilani)

slide25

The Monopole Term:

…… is the most dominant term for r >>

Potential of any distribution  Vmon ,

(if looked from very far point)

For a point charge at origin,

V = Vmon, everywhere

Dr. Champak B. Das (BITS, Pilani)

slide26

The Dipole Term:

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

Dr. Champak B. Das (BITS, Pilani)

slide27

dipole moment of the distribution

Dr. Champak B. Das (BITS, Pilani)

slide28

z

-q

d

r'_

+q

r'+

y

x

For a collection of point charges,

For aphysicaldipole:

Dr. Champak B. Das (BITS, Pilani)

slide29

P

rs+

+q

r

d

rs-

-q

Potential of a physical dipole

Dr. Champak B. Das (BITS, Pilani)

slide30

Potential due to a point charge ~ 1/r

Potential due to a dipole ~ 1/r2

Dr. Champak B. Das (BITS, Pilani)

slide31

physical

dipole

Dr. Champak B. Das (BITS, Pilani)

slide32

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)

slide33

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)

slide34

Dipole moment changes when origin is shifted :

d'

y

r'

a

x

Dr. Champak B. Das (BITS, Pilani)

slide35

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)

slide36

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)

slide37

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)

slide38

Field due of a dipole (contd.)

Recall:

Dr. Champak B. Das (BITS, Pilani)

slide39

z

r

y

x

Prob 3.33:

Electric field in a coordinate free-form :

p

Dr. Champak B. Das (BITS, Pilani)

slide40

z

y

Field lines of a pure dipole

Dr. Champak B. Das (BITS, Pilani)

slide41

Field lines of a physical dipole

Dr. Champak B. Das (BITS, Pilani)