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Electrostatic Boundary-Value Problems

Chapter 6
1

BoundaryChapter 6 Boundary-Value Problems

6
...
2 Poisson’s and Laplace Equations
6
...
4 Resistance and Capacitance

2

Poisson’s and Laplace
Equations

Poisson’s and Laplace Equations

Poisson’s
Equation

Laplace
Equation

4

Poisson’s and Laplace Equations

Laplace Equation

5

Uniqueness Theorem

If the solution to Laplace equation can be
found that satisfies the boundary conditions,
then the solution is unique

6

Capacitance and Resistance

Resistance
Uniform
Cross Section

ρc ℓ
ℓ
R=
=
σS
S

NonUniform
Cross Section
V
R= =
I

∫ E id ℓ
σ E id S
∫

Capacitance

Capacitance

Capacitor
Ratio of the magnitude of the charge
on one of the plates to potential
difference between them

Q ε ∫ E id S
C= =
V
∫ E id ℓ

Q εS
C= =
V
d

Parallel Plate
Capacitance

d
R=
σS

2

1
1
Q
2
WE = CV = QV =
2
2
2C

Coaxial
Capacitor

Q 2πε L
C= =
b
V
ln
a

b
ln
a
R=
2πσ L

4πε
Q
C= =
V 1−1
a b

1 1
−
R= a b
4πσ

Spherical
Capacitor

Example

Example1a (Cross section of spherical Capacitor)

Determine Their
Capacitance

Example1b (Cross section of spherical Capacitor)

Determine Their
Capacitance

Example 2a

Determine Their
Capacitance

Example 2b

Determine Their
Capacitance

Example3a (Cross section Cylindrical Capacitor)

Determine Their
Capacitance

Example 3b (Cross section Cylindrical Capacitor)

Determine Their
Capacitance

Example 4

Determine Their Resistance

Example 5



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