D. Lukáš 2010

Physical principles of nanofiber production

3. Theoretical background of electrospinning (1) Electrostatics

D. Lukáš2010

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Literature:

Feynman R P, Leighton R B, Sands M, Feynmans lectures from physics, Part 2, Fragment, Havlíčkův Brod, 2001.

Chapter 4, Elektrostatics, str. 63 – 81 (=18 pages)

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Electrospinning may be thought to be a member of larger group of physical phenomena, classified as electrohydrodynamics. This important group of electrical appearances concerns the nature of ion distribution in a solution, caused by the influence of electric field, generated by organized groups of charges, to give a wide range of solution behaviour, such as, electrophoresis, electroosmosis, electrocapillarity and electrodiffusion, as recorded by Bak and Kauman [25].

In this lecture will be briefly described how the theory of electrohydrodynamics has been evolving since the initial pioneering experimental observations. To start with, it is convenient to introduce an overview of the basic principles of electrostatics and capillarity to enable deeper understanding of physical principles of electrospinning.

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Electrostatics

Historically, the basic law of electrostatics is the Coulomb law, describing a force by which a charge acts on a charge on a distance in a space with electric permittivity,

r

r

r

qqF

2

21

04

1

(3.1)

r

12

F r

re

x

y

Feynmans lectures from physics, Part2, chapter 4.2 4

Coulomb force per unitary charge is called field strength or, field intensity and is commonly denoted as

1q

FE

(3.2)

r

r

r

qrE

204

1)(

r

x

y

E

5

For electrostatic field, holds the superposition principle. For charges and that generate electrostatic fields with intensities and respectively, the resultant / joint field is determined by the following sum,

21 EEE

(3.3)

r

x

y

121R

2R

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The space dependence of intensity generated by a point charge together with the superposition principle, leads to an alternative formulation of Coulomb law that is called Gauss Theorem of electrostatics.

According to this theorem, the scalar product of intensity, E, with a surface area element ds, integrated along a closed surface S , is equal to a charge, q , trapped inside the close surface by permittivity, .

The surface area element is considered here as a vector normal to the surface element.

q

sdES

(3.4)

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24 rS

r

24

1)(

r

qrE

22

44

1r

r

qSEsdE

S

q

sdES

S q

0S

sdE

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Gauss’s principle in electrostatics describes electrostatic field property from macroscopic point of view. It has also a microscopic variant, given by:

E

(3.5)

zyx /,/,/

Hamilton Operator

This equation is also known as the First Maxwell Law for electrostatics.

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x

y

z

0xE xEx

xx

EExE x

xx

0

dVdVE

yxzz

Ezxy

y

Ezyx

x

EsdE zyx

S

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Another consequence of Coulomb law is the fact, that electrostatic field is the conservative one and, hence, there exists a potential that determines unequivocally the field intensity by means of the following relation

E (3.6)

x

y

z

0 E

0S

rdE

rd

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Maxwell equations for electrostatics

E

0 E

12

The substitution from the relation (3.6) into the equation (3.5) provide us with so-called Poisson Equation

E

E

Laplace Operator

222222 /// zyx

0Laplace Equation

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Summary:

• Coulomb law• Field Intensity• Superposition Principle• Gauss Theorem• First Maxwell Law for electrostatics• Hamilton Operator• Electrostatic field is the conservative one – potential• Poisson Equation and Laplace Equation• Laplace Operator

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Homework:

?,,,, zyxzyx bbbaaaba

?3,4,18,5,2

1. Scalar product

2.

3. Show that pays:

?ba4. Vector product

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