Unit-2Poission’s equations

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Mr. Himanshu DiwakarAssistant Professor

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Poisson’s and Laplace Equations

A useful approach to the calculation of electric potentialsRelates potential to the charge density. The electric field is related to the charge density by the divergence relationship

The electric field is related to the electric potential by a gradient relationship

Therefore the potential is related to the charge density by Poisson's equation

In a charge-free region of space, this becomes Laplace's equation

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Potential of a Uniform Sphere of Charge

outside

inside

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Poisson’s and Laplace Equations

Poisson’s Equation

From the point form of Gaus's Law

Del_dot_D v

Definition D

D E

and the gradient relationship

E DelV

Del_D Del_ E Del_dot_ DelV v

Del_DelV v

L a p l a c e ’ s E q u a t i o n

if v 0

Del_dot_ D v

Del_Del Laplacian

T h e d i v e r g e n c e o f t h e g r a d i e n t o f a s c a l a r f u n c t i o n i s c a l l e d t h e L a p l a c i a n .

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LapRx x

V x y z( )dd

dd y y

V x y z( )dd

dd

z z

V x y z( )dd

dd

LapC1

V z d

d

dd

1

2

V z dd

dd

z z

V z dd

dd

LapS1

r2 rr2

rV r d

d

dd

1

r2 sin sin

V r d

d

dd

1

r2 sin 2 V r d

ddd

Poisson’s and Laplace Equations

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Given

V x y z( )4 y z

x2 1

x

y

z

1

2

3

o 8.85410 12

V x y z( ) 12Find: V @ and v at P

LapRx x

V x y z( )dd

dd y y

V x y z( )dd

dd

z z

V x y z( )dd

dd

LapR 12

v LapR o v 1.062 10 10

Examples of the Solution of Laplace’s Equation

D7.1

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Examples of the Solution of Laplace’s Equation

Example 7.1

Assume V is a function only of x – solve Laplace’s equation

VV o x

d

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Examples of the Solution of Laplace’s Equation

Finding the capacitance of a parallel-plate capacitor

Steps

1 – Given V, use E = - DelV to find E2 – Use D = E to find D3 - Evaluate D at either capacitor plate, D = Ds = Dn an4 – Recognize that s = Dn5 – Find Q by a surface integration over the capacitor plate

CQ

Vo

Sd

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CAPACITANCE

• The ratio of electric charge to electric potential of a conductor or a device is called capacitance

• Capacitance C = Q/V• Unit is farad (F)• 1 farad = 1 coulomb / 1 volt

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PRINCIPLE OF A CAPACITOR

• Capacitor is based on the principle that the capacitance of an isolated charged conductor increases when an uncharged earthed conductor is kept near it and the capacitance is further increased by keeping a dielectric medium between the conductors.

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CAPACITANCE OF A PARALLEL PLATE CAPACITOR

Electric field between the plates,E = /0

But =Q/AE=Q/A0

Potential difference between the two plates , V = Ed = Qd/A 0

Capacitance, C = Q/VC=A 0/d

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CAPACITANCE OF A PARALLEL PLATE CAPACITOR WITH A DIELECTRIC SLAB

When a dielectric slab is kept between the plates COMPLETELY filling the gap

E’ = E0/K where K is the dielectric constant of the medium. Potential difference V’ = E’d = E0d/K=Qd/K 0A Capacitance C’ = Q/V’ = K 0A/d = KC

when a dielectric medium is filled between the plates of a capacitor, its capacitance is increased K times.

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DIELECTRIC STRENGTH

• Dielectric strength of a dielectric is the maximum electric field that can be applied

to it beyond which it breaks down.

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PRACTICE PROBLEMS

• Calculate the number of electrons in excess in a body with 1 coulomb of negative charge.

• Q = ne• Q = 1C• e = 1.6 X 10-19C• n = Q/e= 1/(1.6 X 10-19C) = 6.25 X 1018

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Resistance

• The electrical resistance of an electrical conductor is a measure of the difficulty to pass an electric current through that conductor.

• The inverse quantity is electrical conductance, and is the ease with which an electric current passes.

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RESISTORS IN SERIES AND PARALLEL

nRRRR ...21eq

RESISTORS IN SERIES The magnitude of the charge is constant. Therefore, the

flow of charge, current I is also constant. The potential of the individual resistors are in general

different.

The equivalent resistance of resistors in series equals the sum of their individual resistances.

RESISTORS IN PARALLEL The upper plates of the capacitors are connected together to

form an equipotential surface – they have the same potential. The lower plate also have equal potential.

The charges on the plates may not necessarily be equal.

RESISTORS IN SERIES AND PARALLEL

1

21eq

1...11

nRRRR

RESISTORS INSERIES AND PARALLEL

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