Each capacitor has a special property. Due to this property, a capacitor can store electrical energy. We call the property as the capacitance. Again sometimes, we call the capacitance as the capacity of the capacitor.
What is capacitance?
A capacitor requires a certain charge to establish a potential difference across it. The quantity electrical charge establishes 1-volt potential difference across the capacitor is its capacitance.
Expression of Capacitance
Suppose, one plate of a parallel plate capacitor accumulates Q coulomb charge. Due to this charge, V volt potential difference appears across the capacitor. Then we can write the capacitance as follows.
The above equation tells us the capacitance in terms of charge and voltage.
What is the Unit of Capacitance?
The unit of electrical charge is coulomb. Again, the unit of potential difference is volt. Hence, the unit of capacitance as coulomb per volt. Farad is the name of the unit of capacitance. The unit farad came after the name of Michael Faraday. Definitely, one farad equals to 1 coulomb per volt.
Definition of Farad
So, 1 farad is such amount capacitance which establishes 1 volt potential difference for 1 coulomb charge in the capacitor.
Actually, the practical capacitance is much smaller than 1 Farad. Because of that, we generally express the unit of capacitance in micro Farad or nano Farad instead of a Farad.
Capacitance of Isolated Sphere
Let us place a charged spare of radius R meters in the air. Also, we assume that this sphere has a charge of Q coulomb.
We know that the expression of the surface potential of such a sphere with respect to the earth is
As per the definition capacitance, C is
Now, you can ask how an isolated sphere can be a capacitor. Because, at first sight, it appears to have one surface only. The question arises where is the second surface. But if you remember that the expression of the surface potential of a sphere has the reference of earth. Hence, obviously, the second surface is the earth. Hence, the capacitance 4πεoR exists between the surface of the sphere and the earth.
Capacitance of Spherical Capacitor
When the outer surface is grounded
Let us consider two concentric spheres of radii A and B as shown. This arrangement forms a spherical capacitor.
Let us apply the charge of + Q coulomb in the inner sphere. It induces – Q coulomb on the inner surface of the outer sphere. Again, the charge of the inner surface of the outer surface induces + Q charge on the outer surface of the outer sphere. Then the earth connection on the outer surface of the outer sphere discharges its charge to the earth.
Expression of Capacitance
The dielectric medium between two spheres has the relative permeability of εr. Therefore, the expression of the free surface potential on the outer surface of the inner sphere due to its own charge is,
The potential of the outer sphere due to – Q coulomb charge on its inner surface is,
The total potential between these two surfaces is
When the inner surface is grounded
The figure below is showing a spherical capacitor. The charge of + Q coulombs is the charge of the outer sphere. It distributes itself over both the inner and outer surface of the sphere. Suppose, the charge of + q1 coulombs remains on the outer surface of the outer sphere. Because there is the earth surface surrounding the sphere. Also, some charge of + q2 coulombs shifts to the inner surface. Because there is an earthed sphere inside.
Expression of Capacitance
The electrical charge of + q2 coulombs on the inner surface of outer sphere induces – q2 coulombs on the surface of the inner sphere. But this induced charge goes to the earth through the earth connection. Consequently, the arrangement forms two parallel capacitors. Among which, one capacitor consists of the outer surface of the inner sphere and the inner surface of the outer sphere. Hence, its capacitance is
Then another capacitor consists of the outer surface of the outer sphere and the earth. Hence, its capacitance is
The total capacitance is
Capacitance of Parallel Plate Capacitor
There are two parallel plates of effective area A m2. There is a dielectric medium of relative permittivity εr in between the parallel plates. The distance between the parallel plates or the thickness of the dielectric medium is of d meters.
Formula of Capacitance
One of the parallel plates has the charge of + Q coulombs. Another plate has ground potential. The charge density on the charged plate is
The flux density in the medium is
This is because the total flux of a body has the same value as its total charge.
Hence, the electric field intensity is
The potential difference between the two plates is
Hence, the formula of capacitance is
Capacitance of Parallel Plate Capacitor with Two Dielectric Mediums
The figure below is showing a parallel plate capacitor with two dielectric mediums.
The distance between two parallel plates is of d meters. The thickness of two dielectrics are d1 and d2 (d = d1 + d2). Here, one plate has the charge of + Q coulombs. Then, we connect the other plate with the earth.
The flux density in both of the mediums is the same. It is
In the first medium potential gradient is
In the second medium potential gradient is
The potential between the plates is