Transcript Electrostatics

Physical principles of nanofiber
production
3. Theoretical background of
electrospinning
(1) Electrostatics
D. Lukáš
2010
1
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)
2
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.
3
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,

1 q1q2
F
40 r 2
2

r

r

r
(3.1)
y
1

F
x
Feynmans lectures from physics, Part2, chapter 4.2

 r
e 
r
4
Coulomb force
per unitary charge is called field strength or, field

intensity E and is commonly denoted as

 F
E
q1
y
(3.2)

r
x

 
1 q r
E (r ) 
2 
40 r r
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,
 

E  E1  E2

R2
y
1
(3.3)

R1

r
2
x
6
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
 E ds 
S

(3.4)
7
  q
 E ds 
S

1 q
E (r ) 
4 r 2

 
 E ds  0
S  4r 2
S
S
q

r
 
1 q
2
S E ds  E  S  4 r 2  4r
8
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)
This equation is also known as the First Maxwell Law for
electrostatics.

   / x,  / y,  / z 
Hamilton Operator
9
E x
E x x   E x 0  
x
x
z
Ex 0
Ex x 
x
y
  Ex
E y
Ez
E
d
s


x


y

z


y


x

z

z  xy 
S
x
y
z

 dV
 E  dV 

10
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
z


E  

dr
y
(3.6)
x
 
 E dr  0
S
 
 E  0
11
Maxwell equations for electrostatics
 
E 

 
 E  0
12
The substitution from the relation (3.6) into the equation (3.5)
provide us with so-called Poisson Equation
 
E 


  



E  
  0
Laplace Equation

  
Laplace Operator
   / x   / y   / z
2
2
2
2
2
2
13
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
14
Homework:
1. Scalar product
 
a  b  ax , a y , az  bx , by , bz   ?
2.
2,5,8 1,4,3  ?

  
4. Vector product 

a b  ?
3. Show that pays:
15