Capacitance
Capacitance is a fundamental concept in electronics and electrical engineering. At its core, capacitance is the ability of a system to store electrical energy in an electric field. It’s most commonly associated with capacitors, components used in countless electronic devices—from smartphones and computers to electric vehicles and power grids.Measured in farads (F), capacitance plays a key role in controlling voltage, filtering signals, and maintaining power supply stability.
1.0Definition of Capacitance
- Capacitance of conductor is defined as charge required to increase the potential of conductor by one unit.
- When a conductor is charged ,its electric potential increases.
- For an isolated conductor with finite size the potential at infinity is considered zero.
- The potential V of the conductor is directly proportional to the charge q on it.
q∝V⇒q=CV
- The constant C is called the capacitance of the conductor, it measures the conductor ability to store charge per unit potential.
Graph between Q and V
Understanding Capacitance: Units & Dependencies
- It is a scalar quantity.
- Unit of capacitance is farad in SI units and its dimensional formula is[M−1L−2I2T4].
- 1 Farad: 1 Farad is the capacitance of a conductor for which 1 coulomb charge increases potential by 1 volt.
1 Farad =1 Volt 1 Coulomb
1μF=10−6F,1nF=10−9F or 1pF=10−12F
1F=9×1011 state −F
Capacitance of an isolated conductor depends on following factors:
(a) Shape and size of the conductor: On increasing the size, capacitance increases.
(b) On the surrounding medium: With an increase in dielectric constantK, capacitance increases.
(c) Presence of other conductors: When a neutral conductor is placed near a charged conductor, capacitance of conductors increases.
Capacitance of a conductor does not depend on
(a) Charge on the conductor
(b) Potential of the conductor
2.0Capacitance of Parallel Plate Capacitor
- It consists of two metallic plates M and N each of area A at separation d. Plate M is positively charged and plate N is earthed. If εr is the dielectric constant of the material medium and E is the field at a point P that exists between the two plates then
Step-1: Finding Electric Field
E=E++E−=2Eσ+2Eσ=Eσ=ϵ0ϵRσ[ϵ=ϵ0ϵR]
Step-2: Finding Potential Difference
V=Ed=ϵ0ϵRσd=Aϵ0ϵRqd(∵E=dV and σ=Aq)
Step-3: Finding Capacitance
C=Vq=dAϵ0ϵR
Capacitance of a capacitor depends on
(a) Area of plates.
(b) Distance between the plates.
(c) Dielectric medium between the plates.
3.0Capacitance of Spherical Capacitor
Capacitance of an Isolated Spherical Conductor
Let there is charge Q on sphere of radius R
Potential V=RKQ
Hence by formula,
Q=CV→Q=RCKQ→C=4πϵ0R
(a) If the medium around the conductor is a vacuum or air
Cvacuum =4πϵ0R
R=Radius of spherical conductor may be solid or hollow
If the medium around the conductor is a dielectric of constant K from surface of sphere to infinity then,
CMedium =4πϵ0KR
(b) CAir/vacuum CMedium =K=DielectricConstant
Capacitance of a spherical capacitor
- Outer sphere is earthed
When a charge Q is given to inner sphere it is uniformly distributed on its surface,A charge -Q is induced induced on inner surface of outer sphere. The charge +Q induced on outer surface of outer sphere. The charge +Q induced on the outer surface of the outer sphere flows to earth as it is grounded.
E=0 for r<a
E=0 for r>b
This arrangement is known as a spherical capacitor.
Potential of inner sphere,
V1=4πϵ0aQ+4πϵ0b−Q⇒4πϵ0Q(abb−a)
As outer surface is earthed so potential V2=0
V1−V2=[aKQ−bKQ]−[bKQ−bKQ]=aKQ−bKQC=V1−V2Q=aKQ−bKQQ=K(b−a)ab=b−a4πϵ0abC=b−a4πϵ0ab
If b>>a then C=4πϵ0a (Like an isolated spherical capacitor)
If dielectric mediums are filled as shown then C=b−a4πϵ0ϵr2ab
(b) Inner Sphere is Earthed
Here the system is equivalent to a spherical capacitor of inner and outer radii a and b respectively and a spherical conductor of radius b in parallel. This is because charge Q given to outer sphere distributes in such a way that for the outer sphere charge on the inner sides is baQ and charge on the outer side is
Q−baQ=b(b−a)Q
So total capacitance of the system
C=4πϵ0b−aab+4πϵ0b
C=b−a4πϵ0b2
4.0Capacitance of Cylindrical Capacitor
There are two co-axial conducting cylindrical surfaces where l>>a and l>>b, where a and b are the radius of cylinders.
When a charge Q is given to inner cylinder it is uniformly distributed on its surface.A charge -Q is induced on inner surface of outer cylinder.The charge +Q induced on the outer surface of the outer cylinder flows to Earth as it is grounded.
Electrical Field between cylinder, E=2πϵ0rλ=2πϵ0rQ/l
Potential Difference between plates,
V=∫ab2πϵ0rlQdr=2πϵ0lQln(ab)
Capacitance per unit length
C=Vλ=2Kλlnabλ=2lnab4πϵ0=lnab2πϵ0
Capacitance per unit length =lnab2πϵ0F/m