Combination of Capacitors

Do you know why sometimes you get an electric shock? Has it happened with a wooden covering? No! But, why? It is all the magic of science! The concepts of conductors and insulators control this phenomenon. In this chapter, we will dig deeper into it to know all about capacitors and their various combinations arrangements. However, before we proceed, let us have a quick review of what conductors and insulators are.

Suggested Videos

 

What are Conductors and Insulators?

Conductors are those substances through which electric charge can travel easily. On the other hand, electric charges can’t pass easily in insulators. This is the basic difference between the two.

Capacitors

The potential V of a conductor depends upon the charge Q given to it. According to observations, the potential of a conductor is proportional to the charge on it.

Q ∝V or Q = CV

The proportionality constant ‘C’ is known as the capacitance of the conductor. Thus,

C = Q/V

The capacity of a conductor is the ratio between the charge of the conductor to its potential. If V = 1, then C = Q. The capacity of a conductor is the charge required to raise it through a unit potential.

Units

• S.I Unit: Farad (coulomb/volt). The capacity of a conductor is said to be 1 farad if a charge of 1 coulomb is required, to raise its potential through 1 volt.
• C.G.S – stat farad (stat-coulomb/stat-volt). The capacity of a conductor is said to be 1 stat farad is a charge of 1 statcoulomb is required, to raise its potential through 1 statvolt.

Dimension of C:- [M^-1L^-2T⁴A²]

Browse more Topics under Electrostatic Potential And Capacitance

• Electric Potential Energy and Electric Potential
• Capacitors and Capacitance
• Electrostatics of Conductors
• The Parallel Plate Capacitor
• Energy Stored in a Capacitor
• Dielectrics and Polarisation
• Effect of Dielectric on Capacitance
• Van De Graaff Generator

Capacitance of an Isolated Spherical Conductor

Consider a spherical conductor of radius ‘r’ completely isolated from other charged bodies and situated in the air. Let a charge ‘q’ be given to it. For calculation purposes, the charge ‘q’ can be supposed to be concentrated at the centre of the sphere. The capacity of a conductor can be obtained as follows:

Charge on the sphere = q
Potential of the surface of the sphere = (1/4πε₀r) (q/r)
Capacitance, C = charge/potential = [q/(q/4πε₀r)] = 4πε₀r
But 1/4πε₀r = 910⁹
So, 4πε₀r = 1/910⁹
Thus, C = r/910⁹
Here, ‘C’ is in farad and ‘r’ is taken in the meter.

A capacitor or a condenser is an arrangement which provides a larger capacity in a smaller space.

The Principle of a Capacitors

The principle of a conductor is that an earthed conductor when placed in the neighborhood of a charged conductor, the capacity of the system increases considerably.

Capacitors in Series

Capacitors are said to be connected in series if the second plate of one is connected with the first plate of the next and so on. This leaves the first plate of the first capacitor and the second plate of the last capacitors free plates.

Assuming, as seems reasonable, that these plates carry zero charges when zero potential difference is applied across the two capacitors, it follows that in the presence of a non-zero potential difference the charge +Q on the positive plate of capacitor 2 must be balanced by an equal and opposite charge -Q on the negative plate of capacitor 1.

Since the negative plate of capacitor 1 carries a charge -Q, the positive plate must carry a charge +Q. Likewise, since the positive plate of capacitor 2 carries a charge +Q, the negative plate must carry a charge -Q.

The net result is that both capacitors possess the same stored charge Q. The potential drops, V₁ and V₂, across the two capacitors are, in general, different. However, the sum of these drops equals the total potential drop V applied across the input and output wires: i.e., V = V₁+V₂. The equivalent capacitance of the pair of capacitors is again C_eq = Q/V. Thus,

1/C_eq = V/Q = (V₁+V₂)/Q = V₁/Q + V₂/Q = 1/C₁ + 1/C₂

Thus, the reciprocal of the resultant capacity of a number of capacitors, connected in series, is equal to the sum of the reciprocals of their individual capacities.

Capacitors in Parallel

Consider two capacitors connected in parallel: i.e., with the positively charged plates connected to a common “input” wire, and the negatively charged plates attached to a common “output” wire–see in the figure. What is the equivalent capacitance between the input and output wires?

In this case, the potential difference V across the two capacitors is the same and is equal to the potential difference between the input and output wires. The total charge Q, however, stored in the two capacitors is divided between the capacitors, since it must distribute itself such that the voltage across the two is the same.

Since the capacitors may have different capacitances, C₁ and C₂, the charges Q₁ and Q₂ may also be different. The equivalent capacitance C_eq of the pair of capacitors is simply the ratio Q/V, where Q = Q₁+Q₂ is the total stored charge. It follows that

C_eq = Q/V =[Q₁+Q₂ /V] = Q₁/V + Q₂/V = C₁ + C₂

Thus, the resultant capacity of a number of capacitors, connected in parallel, is equal to the sum of their individual capacitors.

Solved Example for You

Question: How many 6μF, 200V condensers are needed to make a condenser of 18μF, 600V?

1. 9
2. 18
3. 3
4. 27

Solution: Option (D) 27. Place three 200V, 6μF capacitors in series to get 1 equivalent 600V, 2μF capacitor. Now place 9 of these equivalent 600V, 2μFcapacitors in parallel to obtain an equivalence of 18μF at 600 Volts. All this requires a total of 27 6μF capacitors. Nine rows connected in parallel with 3 capacitors connected in series in each row.