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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)

► to determine the Potential uniquely…

Poisson's / Laplace’s equation

+

A set of boundary conditions

Proof ?

………. Uniqueness Theorem

Dr. Champak B. Das (BITS, Pilani)

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)

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)

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)

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)

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)

- 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)

- 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)

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

(for x2+y2 >> d2)

Dr. Champak B. Das (BITS, Pilani)

Force of attraction on q towards the plane

Force of attraction on +q towards -q

Dr. Champak B. Das (BITS, Pilani)

Two point charges and no conductor :

Single point charge and conducting plane :

Dr. Champak B. Das (BITS, Pilani)

R

a

q

V=0

A point charge and a grounded conducting sphere :

Dr. Champak B. Das (BITS, Pilani)

Location of image charge :

(to the right of the centre of the sphere)

Dr. Champak B. Das (BITS, Pilani)

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

Dr. Champak B. Das (BITS, Pilani)

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)

Dr. Champak B. Das (BITS, Pilani)

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)

- 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)

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

Dr. Champak B. Das (BITS, Pilani)

dipole moment of the distribution

Dr. Champak B. Das (BITS, Pilani)

-q

d

r'_

+q

r'+

y

x

For a collection of point charges,

For aphysicaldipole:

Dr. Champak B. Das (BITS, Pilani)

Potential due to a point charge ~ 1/r

Potential due to a dipole ~ 1/r2

Dr. Champak B. Das (BITS, Pilani)

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)

- 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)

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)

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)

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)

Potential at a point due to a pure dipole:

z

r

p

y

x

Dr. Champak B. Das (BITS, Pilani)

Field lines of a physical dipole

Dr. Champak B. Das (BITS, Pilani)

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)

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