Capacity

 Capacity:

Capacity is a word very commonly used in our daily life. Before we go into the concept of capacity in reference to electrostatics let us consider following examples. 

1) We have a beaker with 200 ml inscribed over it. It us capable of containing of 200 ml of a liquid. A drop more than that will overflow. So, its capacity is 200 ml. 

2) We have a cycle tube and a tyre fitted over it. As we push air into it, with the help of a pump, more and more air can be filled in it. The pressure of air in the tune shall keep on rising. So, what is the capacity of that tube? Of course, there are two limitations. 

(a) The tyre may now withstand the high pressure inside and may burst open. 

(b) The pump may not be that strong so as to push air against the already high pressure inside. 

Suppose, somehow or the other, we are able to overcome these two limitations. How much air can be put into the tube? What is the capacity of the tube? The answer, obviously, is infinite. So, the concept of capacity, which we use in our daily life, does not hold good here. 

The meaning of capacity in reference to electrostatic is analogous to the second example quoted above. If we supply charge (air in above example) to a body, its potential (pressure in above example) rises. More the charge, More will be potential (i) A situation may arise when the body's potential is so high that it is not able to hold charge. It leaks to the surroundings. (ii) we may not be able to add fresh charge due to a very strong repulsion between the incoming charge and the charge already project on the body. 

If we are able to overcome both these limitations, how much charge can be added to the body? What is the capacity of body ? Obviously, the situation resembles the second example quoted above. 

If 'V' is the potential of the conductor due to a charge Q given to it, then

       Q ∝ V          or         Q = CV

The proportionality constant 'C' is known as the capacity of the conductor. Thus, 

                     C = Q/V

The capacity of a capacitor is defined as the ratio between the charge on the conductor to its potential. 

       If V = 1, then       C = Q. 

The capacity of a capacitor is also defined as the charger required to raise it through a unit potential. 

Units of Capacity:

In S.I. units, the capacity of a capacitor is measured in farad. 

The capacity of a capacitor is said to be 1 farad if a charge of 1 coulomb is sufficient to raise its potential through 1 volt. 

       1 farad (F) = 1 coulomb/1 volt

Following units are use in practice. Their relationship with farad is also given. 

       1 micro-farad (μF) = 10-6 F 

       1 micro-micro farad (μμF)    or    

       1 pico farad = 10-12 F

In C.G.S. units. There are two types of units in C.G.S. system. 

(i) e.s.u. of capacity ('statfarad'):

Capacity of a capacitor is said to be 1 statfarad if a charge of 1 statcoulomb is required to raise its potential through 1 statvolt. 

       1 statfarad = 1 statcoulomb/1 statvolt

(ii) e.m.u. of capacity ('abfarad'):

Capacity of a capacitor is said to be 1 abfarad if a charge of 1 abcoulomb is required to raise its potential through 1 abvolt. 

       1 abfarad = 1 abcoulomb/1 abvolt

Relation between farad and statfarad

       1 farad = 1 coulomb/1 volt

               = 3 × 10⁹ statcoulomb/1/300 statvolt

       or 1 farad = 9 × 10¹¹ statfarad

Relationship between farad and abfarad

       1 farad = 1 coulomb/1 volt

       = 1/10 abcoulomb/10⁸ abvolt

       or 1 farad = 1/10⁹ abfarad

Dimensions of capacitance:

Capacity (in farad) = coulomb/volt

                                  = coulomb/joule/coulomb

                                  = (coulomb)²/joule

                                  = C²/Nm

                                  = [A2 T2]/[M L T-2] [L]

= [M-1 L-2 T4 A2]

Therefore, dimensions of capacity are -1, -2, 4 and 2 in mass, length, time and electric current respectively. 

Example-

Capacity of Earth

Radius of earth, R = 6.38 × 106   

Therefore, capacity of earth C is (considering it to be an isolated sphere). 

         C = 4πε0 × R = R/9 × 10⁹

         C = 6.38 × 106 / 9 × 10⁹

         C = 7.088 × 10-4  F 

or     C ≃ 709 μF

Evidently, the capacity of earth is very large. Therefore, any amount of charge added to it or taken away from it does not bring about a change in its potential. Therefore, earth is always considered to be at zero potential. Zero potential of earth is analogous to zero level of sea. The sea being a vast reservoir of water does not show a change in its level due to addition or subtraction of any amount of water from it. 

When a positively charged body is connected to earth, electrons flow from earth to body to reduce its potential to zero. When a negativity charged body is connected to earth, electrons flow from body to earth to increase its potential to zero. 

Important Notes:

1. Capacity of a capacitor is a scalar quantity since both Q and V are scalar. 
2. Capacity of a capacitor is independent of both the charge and the potential of the conductor. 
3. Theoretically, infinite charge can be stored in a capacitor. Practically, however, it is not possible due to some limitations. 
4. Large capacity of a capacitor means that the capacitor can store a large amount of charge for a small difference of potential. 
5. Capacity of a capacitor is given by the slope of a graph between charge (along y-axis) and potential (along x-axis) of the conductor.