Formula for capacitance of a capacitor | definition types

Capacitor is one of the most important circuit components. A capacitor is nothing but a conductor or the combination of two conductors having equal and opposite charges. Now, what is the difference between a capacitor and a conductor? What makes the capacitor different from a conductor? A capacitor is made of two conductors that can store electric charge or electric energy in it. In this article, we are going to discuss the definition, formula, unit, dimension for capacitance of a capacitor and all related facts of different type of capacitors.

Contents in this article:

  1. Difference between a Conductor and a Capacitor
  2. General formula of capacitance of a capacitor
  3. Definition for capacitance of a capacitor
  4. Unit of capacitance
  5. Dimension of capacitance
  6. Graphs
  7. On what parameters the capacitance depends?
  8. Types of capacitors
  9. Formula for capacitance of parallel plate capacitor
  10. Capacitance of spherical capacitor
  11. Capacitance of cylindrical capacitor
  12. Use of a Capacitor

What is the difference between a conductor and a capacitor?

We know that a conductor allows to flow electric charge through it to carry electric current. It cannot store the electric charge or electrical energy in it. But if two conductors are placed at a small separation, the combination can store electric charges or electrical potential energy inside it. This combination of conductors is the Capacitor. The effect of a capacitor is known as the capacitance of the capacitor.

Capacitor
Capacitor

So, a capacitor is the combination of two equal and oppositely charged conductors placed at a small distance of separation. A capacitor can store electric charge or electrostatic energy. Sometime, a single isolated conductor behaves like a capacitor. In this case, we consider that another similar conductor is present at infinity.

Formula for capacitance of a capacitor

If a conductor is charged, its electric potential increases. Thus, the electrostatic potential of a conductor is directly proportional to its charge. If the potential of a conductor becomes V due to its charge Q, then, \small V\propto Q

or, \small V=\frac{1}{C}Q

or, Q = CV………………. (1)

This is the general formula of capacitance of all type capacitors. This is true for all type of capacitors. But there are some formulae for some specific types of capacitors like spherical capacitors, cylindrical capacitors, parallel plate capacitors, etc.

Definition of capacitance of a capacitor

The capacitance of a capacitor indicates its charge storing capacity. More charge will rise the potential more and hence more potential energy. One can define the capacitance of a capacitor in terms of its charge and potential by using equation-(1).

The capacitance of a capacitor is defined as the amount of electric charge required to raise its electric potential by unity.

Unit of Capacitance

The SI unit of capacitance is Farad (F) and the CGS unit of capacitance is Stat-Farad.

1 Farad = 9×1011 Stat-Farad.

Practically used units are micro-Farad, nano-farad and pico-farad. Because, the value of capacitance greater than these range are not possible to achieve on the Earth’s surface. why the capacitance of a capacitor cannot be in the range of 1 farad?

  • 1 milli-Farad (mF) = 10-3 Farad
  • 1 micro-Farad (\small \muF) = 10-6 Farad
  • 1 nano-Farad (nF) = 10-9 Farad
  • 1 pico-Farad (pF) = 10-12 Farad

Definition of 1 Farad Capacitance

One can establish the definition of one Farad from the equation-(1). If 1 coulomb electric charge is required to raise the electric potential of a capacitor by 1 volt, then the capacitance of the capacitor is 1 Farad.

Dimension of Capacitance:

One can determine the dimensional formula of capacitance from equation-(1). Electric charge has the dimension of [TI] and the electric potential has the dimension of [ML2T-3I-1].

So, dimensional formula for the capacitance is [M-1L-2T4I2].

C vs V, C vs Q, V vs Q Graphs for a Capacitor

One can get three graphs for a capacitor – Capacitance vs Charge graph (C-Q graph), Capacitance vs Voltage graph (C-V graph) and Voltage vs charge graph (V-C graph). Here, I am going to draw each graph one by one and will discuss the nature of all graphs. These variations of capacitance is valid for all capacitors.

Variation of Capacitance with the charge of capacitor

Graph for variation of capacitor with charge
Graph for variation of capacitor with charge (C vs Q graph)

The capacitance of the capacitor is independent on the charge on its plate. Therefore, the Capacitance vs charge graph is a straight line with constant value of capacitance.

Variation of capacitance with the voltage across it

Graph for variation of Capacitance with voltage
Graph for variation of Capacitance with voltage (C vs V graph)

The capacitance of a capacitor is independent on the voltage across its plates also. Therefore, capacitance versus voltage graph is similar to that of capacitance vs charge graph. The nature of the C vs V graph is a straight line with constant value of capacitance.

Variation of Voltage with the charge of a capacitor

Voltage vs Charge graph for a capacitor
Voltage vs Charge graph for a capacitor (V vs C graph)

The variation of voltage of a capacitor with its charge can be observed from equation-(1). Voltage across the plates of a capacitor is directly proportional to the charge on it. Therefore, the Voltage vs charge graph gives a straight line passing through the origin. The slope of this graph is 1/C.

On which parameters the Capacitance of a Capacitor depends?

The capacitance is the property of the capacitor. It do not depend on the amount of charge and the value of applied voltage. The factors affecting the capacitance of a capacitor are –

  1. Shape or the surface area of the conductors
  2. Nature of the surrounding medium
  3. Presence of other conductors
  4. Distance of two conductors in the capacitor

Greater surface area allows to store more charges and hence the capacitance of the capacitor increases. The capacitance is inversely proportional to the distance between the conductors in a capacitor. The presence of an uncharged conductor near a capacitor increases its capacitance. If the surrounding medium is a dielectric medium, then the capacitance becomes greater.

Different type of Capacitors

There are different type of capacitors. These are –

  1. Single Isolated conductor
  2. Parallel plate capacitor
  3. Spherical capacitor
  4. Cylindrical capacitor

For a single isolated conductor it is imagined that there is another similar conductor at infinity. Hence, it is also a capacitor. Spherical conductors are examples of single isolated conductors. In the following sections, I have discussed the formula for capacitance of different type capacitors.

Formula for capacitance of an isolated spherical conductor

Let a spherical conductor of radius R and charge on its surface Q. Then electric potential on its surface is, \small {\color{Blue} V=\frac{Q}{4\pi \epsilon _{0}R}}

Then, capacitance, \small {\color{Blue} C=\frac{Q}{V}}

or, \small {\color{Blue} C=4\pi \epsilon _{0}R} ……… (2)

Thus, in SI system the capacitance of an isolated spherical conductor is, \small {\color{Blue} C=4\pi \epsilon _{0}R} and in CGS system the capacitance of the charged isolated conductor is equal to its radius (C=R). Thus the conductor with greater volume will have greater capacitance.

Check this article for the calculation of capacitance of Earth.

Formula for capacitance of a Parallel plate capacitor

A parallel plate capacitor consists of two parallel plates at some distance of separation. Plates can be rectangular or circular in shape. The plates should have equal and opposite charges on its surface.

Parallel Plate Capacitor with rectangular plates
Parallel Plate Capacitor with rectangular plates
Parallel plate Capacitor with circular plate
Parallel plate Capacitor with circular plates

If A be the cross-section area of each plate and d be the distance between the plates, then the formula for capacitance of the parallel plate capacitor is,

\small {\color{Blue} C=\frac{k\epsilon _{0}A}{d}} …….(3) ( see the derivation of this formula)

where, k is the dielectric constant of the medium between the plates and is the \small \epsilon _{0} permittivity of free space. If the medium between the plates is air medium, then the value of k is 1.

Formula for capacitance of a Spherical capacitor

Spherical Capacitor
Spherical Capacitor

Two co-centric spherical conductors of different radii can act like a capacitor. Spheres should have equal and opposite charges. If r1 and r2 be the radii of the inner and outer sphere respectively, then the formula for the capacitance of the spherical capacitor is,

\small {\color{Blue} C=\frac{4\pi\epsilon _{0} r_{1}r_{2}}{r_{2}-r_{1}}} ………….(4)

Formula for capacitance of a Cylindrical capacitor

Cylindrical Capacitor
Cylindrical Capacitor

The system of two co-axial cylindrical conductors of same length but different radius behave is also a capacitor. Let two co-centric cylinders have radii a and b respectively (b>a). Then the formula of the capacitance of the cylindrical capacitor is,

{\color{Blue} C=\frac{2\pi \epsilon _{0}}{ln\frac{b}{a}}} ………..(5)

Use of a Capacitor

A capacitor is used as a circuit component in different purposes.

  1. A capacitor can store electric charges and releases those charges the supplied voltage becomes less than the voltage across the capacitor. Therefore, a capacitor is useful in rectifier circuit.
  2. Capacitor can bypass alternating current (AC) through it. Because, it offers low resistance to AC.

This is all about the “formula for capacitance of different type capacitors”. If you have any doubt on this topic, you can ask me in the comment section.

Thank you!

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  2. Energy stored in a Capacitor
  3. Capacitance of Earth and other planets
  4. Electric potential and potential energy
  5. Gauss’s law of electrostatics
  6. Electric field and electric field intensity