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Capacitance of capacitors in electricity and electronics

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What is capacitance in a circuit?

“Capacitance is the amount of charge that a capacitor can store per unit of voltage across its plates.”It is denoted by C.That is capacitance is a measure of a capacitor’s ability to store charge.The more charge per unit of voltage that a capacitor can store,the greater its capacitance.

Formula of capacitance of a capacitor

,its formula is given as:

C=Q/V

Where C is capacitance,Q is voltage,and V is voltage.We can also find charge Q and voltage V by rearranging the above formula as:

Q=CV

V=Q/C

 What is the unit of capacitance?

Farad is the unit of capacitance.One Farad is the amount of capacitance when one coulomb of charge is stored with one volt across its plates.

Most capacitors that are used in electronics work have capacitance values that are specified in micro farad (µF) and pico farads(pF) .A micro farad is one millionth of a farad, and a pico farad is one trillionth of a farad.

What are the factors that effect the capacitance of capacitor?

The capacitance of a capacitor depends on the following factors:

Plate Area:

plate area

Capacitance is directly proportional to the physical size of the plates as determined by the plate area,A.A larger plate area produces a larger capacitance ,and a smaller capacitance .Fig(a) shows that the plate area of a parallel plate capacitor is the area of one of the plates.If the plates are moved in relation to each other,as shown in fig(b),the overlapping area determines the effective plate area.This variation in effective plate area is the basic for a certain type of variable capacitor.

Plate separation:

plate seperation

`Capacitance is inversely proportional to the distance between the plates.The plate separation is designated d,as shown in fig(a). A greater separation of the plates produces a smaller capacitance ,as illustrated in fig(b).As previously discussed,the breakdown voltage is directly proportional to the plate separation.The further the plates are separated ,the greater the breakdown voltage.

Dielectric Constant:

As you know,the insulating material between the plates of a capacitor is called the dielectric.Dielectric materials tend to reduce the voltage between plates for a given charge and thus increase the capacitance.If the voltage is fixed ,more charge can be stored due to the presence of a dielectric than can be stored without a dielectric.The measure of a material’s ability to established an electric field is called dielectric constant or relative permittivity,symbolized by ∈r.

Capacitance is directly proportional to the dielectric constant.The dielectric constant of a vacuum is defined as 1 and that of air is very close to 1.These values are used  as a reference,and all other materials have values of ∈r specified with respect to that of a vacuum or air.For example,a material with ∈r=8 can result in a capacitance eight times greater than that of air with all other factors being equal.

The dielectric constant ∈r is dimensionless because it is a relative measure.It is a ratio of the absolute permittivity of a material ,∈r,to the absolute permittivity of a vacuum ,∈0,as expressed by the following formula:

r=∈/∈0

Formula of capacitance in terms of physical parameters

You have seen how capacitance is directly related to plate area,A,and the dielectric constant,∈r,and inversely related to plate separation ,d.An exact formula for calculating the capacitance in terms of these three quantities is:

C=A ∈r∈/d

where ∈= ∈r0=∈r(8.85×10-12F/m)

Capacitance of parallel plate capacitor derivation

Consider a parallel plate capacitor.The size of the plate is large and the distance between the plates is very small,so the electric field between the plates is uniform.

capacitance of parallel plate capacitor

The electric field ‘E’ between the parallel plate capacitor is:

relation of parallel plate capacitor

Capacitance of cylindrical capacitors physics

Consider a cylindrical capacitor of length L,formed by two coaxial cylinders of radii ‘a’ and ‘b’.Suppose L >>b ,such that there is no fringing field at the ends of cylinders.

capacitance of cylindrical capacitor

Let ‘q’ is the charge in the capacitor and ‘V’ is the potential difference between plates.The inner cylinder is positively charged while the outer cylinder is negatively charged.We want to find out the expression of capacitance for the cylindrical capacitor.For this we consider a cylindrical Gaussian surface of radius ‘r’ such that a<<b.

If ‘E’ is the electric field intensity on any point of the cylindrical Gaussian surface,then by Gauss’s law:

If ‘V’ is the potential difference between plates,then

relation of capacitance of cylindrical capacitor

This is the relation for the capacitance of a cylindrical capacitor.

Capacitance of a spherical capacitor

relation of capacitance of spherical capacitor

Capacitance of an isolated spherical capacitor

capacitance of isolated sphere

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