Exercise – 3 –
Example.1
Consider the parallel conductors of Fig.1, where
V at z = d.
Figure 1
Solution:
Laplace’s equation, assuming the region between the plates is charge-free, in
rectangular coordinates:
Equation 1
Since the potential is varied in
coordinates:
Equation 2
Integrating equation.2:
Equation 3
By Appling the boundary condition on equation. 3
Equation 4
Equation 5
The electric field intensity is given by ⃗
Surface charge is given by:
⃗
⃗
̂
The total charge is given by:
| ⃗ || |
∬ ⃗ ⃗⃗⃗⃗
̂
̂
Example.2
The parallel conducting disks in Fig.2 are separated by 5 mm and contain a dielectric
for which
. Determine the charge densities on the disks.
Figure 2
Solution:
Laplace’s equation, assuming the region between the plates is charge-free, in
cylindrical coordinates:
Equation 6
Since the potential is varied in
coordinates:
Equation 7
Integrating equation.2:
Equation 8
By Appling the boundary condition on equation. 3
Equation 9
Equation 10
Equation 11
The electric field intensity is given by ⃗
̂
Surface charge is given by:
⃗
⃗
̂
∬⃗
⃗⃗⃗⃗
(
) ̂
The total charge is given by:
| ⃗ || |
The electric field intensity ⃗ and electric flux density ⃗ :
⃗
⃗
̂
⃗
The surface charge on the upper plate and lower plate is given by:
Example.3
Find the potential function and the electric field intensity for the region
between two concentric right circular cylinders, where
and
Neglect fringing. See Fig.3
Figure 3
Solution
Laplace’s equation in cylindrical coordinates:
Equation 12
Since the potential is varied in
coordinates:
Equation 13
Integrating equation.13:
Equation 14
By Appling the boundary condition on equation. 14
Equation.14
Equation
15
Equation.14
Equation
16
By substituting by equation 16 from equation 15:
By substituting
in equation 15
The potential is given by:
V
⃗
(
̂
Equation 17
̂
Equation 18
)
The total charge on the inner surface is:
∬⃗
⃗⃗⃗⃗
⃗
|⃗ |
(
(
)
)