chapter 6
DESCRIPTION
EmaaagTRANSCRIPT
by: Organis, Zenaida D.
Chapter 6: capacitance
Capacitance Defined
Capacitance of two conductor systems as the ratio of the magnitude of the total charge on either conductor to the ratio of the potential difference between conductors
the capacitance is independent of thepotential and the total charge, for their ratio constant
C = measured in farads (F) - one coulomb per volt
CQ
V0
C
SEdot
d
neg
pos
LEdot
d
6.2 Parallel Plate Capacitor
V0upper
lower
LEdot
d
d
0
zS
dS
d
Q S S
CQ
V0
Sd
6.3 Several Capacitance Examples
Coaxial Cable Vab
L
2 0ln
b
a
potential difference between points= a and = b (equation 11 sec. 4.3)
C2 L
lnb
a
Two concentric spheresVab
Q
4
1
a
1
b
CQ
Vab
4 1
a
1
b
Parallel-plate capacitor – two dielectrics
C1
1
C 1
1
C 2
6.4 Capacitance of A Two-Wire Line
VL
2 ln
R0
R
VL
2 ln
R10
R1
lnR20
R2
L
2 ln
R10 R2
R20 R2
CL L
V
2 L
ln hh
2b
2b
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
Potential of a Uniform Sphere of Charge
outside
inside
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 .
LapRx x
V x y z( )d
d
d
d y yV x y z( )d
d
d
d
z zV x y z( )d
d
d
d
LapC1
V z d
d
d
d
1
2
V z d
d
d
d
z z
V z d
d
d
d
LapS1
r2 r
r2
rV r d
d
d
d
1
r2
sin sin
V r d
d
d
d
1
r2
sin 2 V r d
d
d
d
Poisson’s and Laplace Equations
Given
V x y z( )4 y z
x2
1
x
y
z
1
2
3
o 8.8541012
V x y z( ) 12Find: V @ and v at P
LapRx x
V x y z( )d
d
d
d y yV x y z( )d
d
d
d
z zV x y z( )d
d
d
d
LapR 12
v LapR o v 1.062 1010
Examples of the Solution of Laplace’s Equation
D7.1
Examples of the Solution of Laplace’s Equation
Example 6.1
Assume V is a function only of x – solve Laplace’s equation
VVo x
d
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
Examples of the Solution of Laplace’s Equation
Example 6.2 - Cylindrical
V Vo
lnb
lnb
a
C2 L
lnb
a
Examples of the Solution of Laplace’s Equation
Example 6.3
Examples of the Solution of Laplace’s Equation
Example 6.4 (spherical coordinates)
V Vo
1
r
1
b
1
a
1
b
C4 1
a
1
b
Examples of the Solution of Laplace’s Equation
Example 6.5
V Vo
ln tan2
ln tan2
C2 r1
ln cot2
Examples of the Solution of Poisson’s Equation
10 5 0 5 101
0
11
1
v x( )
1010 x
0
1
E x( )
1010 x
10 5 0 5 100.5
0
0.50.5
0.5
V x( )
1010 x
charge density
E field intensity
potential
References
Engineering Electromagnetics, Hayt, 8th ed.