SPECIAL TECHNIQUES

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SPECIAL TECHNIQUES
To find the electric field of a stationary charge
distribution
Find the potential of the distribution
To Solve Poissons / Laplaces Equation
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SPECIAL TECHNIQUES

• The Method of Images

• Multipole expansion

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METHOD OF IMAGES
z
q
d
y
Grounded conducting plane
x
To find out the potential in the region above the
plane
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Solution of Poissons equation
(in the region z gt 0 ),
WITH

• A point charge q at (0,0,d)

• Boundary conditions

• V 0 when z 0

• V ?? 0 when x2y2z2 gtgt d2

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A new problem
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Z 0
x2y2z2 gtgt d2
Final answer
(By virtue of Uniqueness Theorem)
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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.
( Corollary of First Uniqueness Theorem )
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Induced Surface Charge
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Total induced charge
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Force
Force of attraction on q towards the plane
Force of attraction on q towards -q
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ENERGY
Two point charges and no conductor
Single point charge and conducting plane
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Another example
A point charge and a grounded conducting sphere
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Image charge
Location of image charge
(to the right of the centre of the sphere)
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Two point charges q and q? and no conductor
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? V0
r R
Prob. 3.7(a)
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Prob. 3.7(b)
Induced surface charge on the sphere
Total Induced surface charge
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Prob. 3.7(c)
Force on q
Energy of the configuration
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MULTIPOLE EXPANSION
To characterize the potential of an arbitrary
charge distribution, localized in a rather small
region of space
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Law of cosines ?
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Legendre polynomials
More on this next sem. in Maths - III
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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
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Monopole term
Dipole term
Quadrupole term
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The Monopole Term
is the most dominant term for r gtgt
Potential ?of any distribution ? Vmon ,
(if looked from very far point)
For a point charge at origin,
V Vmon, everywhere
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The Dipole Term
is the most dominant term if total charge is
zero
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dipole moment of the distribution
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For a collection of point charges,
For a physical dipole
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Potential of a physical dipole
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Potential due to a point charge 1/r Potential
due to a dipole 1/r2
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Potential for a physical dipole
Also
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Potential for a pure dipole (d ? 0)
Physical dipole
Pure dipole
for d ? 0, q ? ?? , with pqd kept fixed
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Role played by ORIGIN of coordinate system in
multipole expansion
A point charge away from origin
? Posses a non zero dipole contribution
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Dipole moment changes when origin is shifted
d?'
y
r'
a
x
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If Q 0, then
If net charge of the configuration is zero, then
the dipole moment is independent of the choice of
origin.
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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
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Field due of a dipole
Potential at a point due to a pure dipole
z
r
?
p
y
?
x
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Field due of a dipole (contd.)
Recall
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Prob 3.33
Electric field in a coordinate free-form
p
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Field lines of a pure dipole
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Field lines of a physical dipole