Lecture Notes: EEEC6430310 Electromagnetic Fields And Waves - Cylindrical Capacitor And Solenoid
1. EEEC6430310 ELECTROMAGNETIC FIELDS AND WAVES
Cylindrical Capacitor And Solenoid
FACULTY OF ENGINEERING AND COMPUTER TECHNOLOGY
BENG (HONS) IN ELECTRICALAND ELECTRONIC ENGINEERING
Ravandran Muttiah BEng (Hons) MSc MIET
2. Cylindrical Capacitor
Example 1:
Design a cylindrical capacitor consisting of two coaxial cylinders of radius
𝑟1 and 𝑟2 meters. With the value of electric flux density on the surface of the
imaginary cylinder, 𝐷 =
𝛹
𝐴
, the electric intensity, 𝐸 =
𝐷
𝜀o𝜀r
, the capacitance
of l meter length must be, 𝐶 =
2π𝜀o𝜀r𝑙
2.3 log10
𝑟2
𝑟1
Farad.
The design of a cylindrical capacitor consisting of two coaxial cylinders is
shown in figure 1. Now, let us find the value of electric intensity at any point
distant 𝑥 meters from the axis of the imaginary coaxial cylinder. Consider
an imaginary coaxial cylinder of radius 𝑥 meters and length one meter
between the two given cylinders. The electric field between the two
cylinders is radial as shown in figure 1. Total flux coming out radially from
the curved surface of this imaginary cylinder is 𝑄 coulomb. Area of this
curved surface = 2π𝑥 × 1 = 2π𝑥 m2
. Hence, the value of electric flux
density on the surface of the imaginary cylinder is,
𝐷 =
flux in coulomb
area in meter2
=
𝛹
𝐴
=
𝑄
𝐴
Cm−2
1
3. 2
∴ 𝐷 =
𝑄
2π𝑥
Cm−2
The value of electric intensity is,
𝐸 =
𝐷
𝜀o𝜀r
or
𝐸 =
𝑄
2π𝜀o𝜀r𝑥
Vm−1
Now, d𝑉 = −𝐸 d𝑥
or
𝑉 =
𝑟2
𝑟1
−𝐸 d𝑥
=
𝑟2
𝑟1
−
𝑄
2π𝜀o𝜀r𝑥
d𝑥
=
−𝑄
2π𝜀o𝜀r 𝑟2
𝑟1 1
𝑥
d𝑥
6. 5
Example 2:
Design a solenoid having a length 𝑙 and radius of turns 𝑟 be uniformly
wound with 𝑁 turns each carrying a current of 𝐼. The winding density
number of turns per unit length of the solenoid must be
𝑁
𝑙
. Hence, in a
small element of length d𝑥, there must be
𝑁d𝑥
𝑙
turns.
A very short element of length of the solenoid can be regarded as a
concentrated coil of very short axial length and having
𝑁d𝑥
𝑙
turns. Let d𝐻
be the magnetizing force contributed by the element d𝑥 at any axial point
𝑃 as shown in figure 2. Then, substituting d𝐻 for 𝐻 and
𝑁d𝑥
𝑙
for 𝑁 in 𝐻 =
𝑁𝐼
2𝑟
sin2
𝜃, we get,
d𝐻 =
𝑁d𝑥
𝑙
∙
𝐼
2𝑟
sin2
𝜃
Solenoid
7. 6
Now, d𝑥 ∙ sin 𝜃 = 𝑟
d𝜃
sin 𝜃∗ ∴ d𝑥 = 𝑟
d𝜃
sin2 𝜃
Substituting this value of d𝑥 in the above equation, we get
d𝐻 =
𝑁𝐼
2𝑙
sin 𝜃 d𝜃
Total value of the magnetizing force at 𝑃 due to the whole length of the
solenoid may be found by integrating the above expression between proper
limits.
∴ 𝐻 =
𝑁𝐼
2𝑙 𝜃1
𝜃2
sin 𝜃 d𝜃
=
𝑁𝐼
2𝑙
− cos 𝜃 𝜃1
𝜃2
=
𝑁𝐼
2𝑙
cos 𝜃1 − cos 𝜃2
8. 7
The above expression may be used to find the value of 𝐻 at any point of
the axis, either inside or outside the solenoid. At the mid-point,
𝜃2 = π − 𝜃1
Hence, cos 𝜃2 = − cos 𝜃1
∴ 𝐻 =
2𝑁𝐼
2𝑙
cos 𝜃1
=
𝑁𝐼
𝑙
cos 𝜃1
Obviously, when the solenoid is very long, cos 𝜃1 becomes nearly unity.
In that case,
𝐻 =
𝑁𝐼
𝑙
ATm−1