Resistance

 Resistance:

If  'V' be the potential difference between the two terminals of a conductor and 'i' be the current through it,then,

          V/i = Constant =R

'R' is called the resistance of the material

         i = V/R

An increase in the value of R results in a decrease in the value of  'i'.

Qualitative definition:

Resistance is the opposition offered by conductor to the flow of  electricity through it.Its value is given by equation R = ρ l/A.

Quantitative definition:

Resistance of a conductor is defined as the ratio between potential difference between the two ends of the conductor to the current flowing through it.

If          i = 1, R = V

resistance of a conductor can also be defined as the difference of potential across the two ends of the  conductor required to pass a unit electric current through it.

Concept of resistance:

Every  conductor contains a large number of free electrons. When a difference of potential applied between the two ends of the conductor, an electric field is set up inside the material of the conductor. A free electron (being a negatively charged particle) experiences a force, due to this field, which accelerates it from higher to lower potential side. After acquiring some velocity it suffers collision with other free electrons of the material and  loses the acquired energy. It, again, it accelerated and goes through the above process repeatedly. Thus, motion of the electron cannot be termed as free. It experiences resistance forward motion. This resistance is termed as electrical resistance.

Important notes:

Electric Energy (W)

Electric Energy:

 Electric energy is defined as the amount of work done in sending a current through a conductor. 

We know, V= W/Q 

W= VQ 

 W= VIT

Here, I is the current flowing through a conductor for a time T. When V is the potential difference across the two ends of a conductor. This is the expression for electric energy. 

Units of electric energy:

1 joule= 1 volt×1 ampere×1 sec

There fore, 1 joule= 1 watt × 1 sec (1 volt × 1 ampere = 1 watt) 

Bigger unit of electric energy= 1 watt hour    

 1 watt hour = 1 watt × 1 hour

                       = 1 watt × 3600 second

                       = 3600 watt second = 3600 joule

                       = 3600 × 10⁷ erɡ.

In practice, a still bigger unit, kilowatt hour, is used. 

  1 kilowatt hour = 1000 watt hour. 

This is the unit which is used for calculations of electricity for domestic consumption and is called B.O.T. (Board of Trade) unit. 

   1 B.O.T. = 36 × 10⁵ J = 36 × 10¹² erg. 

 Sources of Electric Energy:                 

      Electricity is a form of energy which is not available in material form like kinetic energy and potential energy. We have to obtain it by converting some other form of energy into electricity. For this, we shall require a source of energy. There are two types of sources of energy.                       

1. Non-renewable sources of electricity 

2. Renewable sources of electricity

1. Non-renewable sources of electricity. These are the sources of energy which once consumed are finished. Examples for this category are fossil fuels and nuclear energy. 

(a) Electricity from chemical energy. In some chemical reaction, ions are created. Movement of these ions across a cross-section results in flow of electric current. In this process, chemical energy is said to have been converted into electric energy. We can also arrange to store electrical energy, thus, constituting a battery. This technique can be used to develop sources of electricity to be used on a small scale. These sources can be used to perform experiment in science lab, to power a T.V. remote or to work an electric toy, etc. 

(b) Electricity from fossil fuels. Heat produced by burning coal is given to water. Steam, this, produced is fed to an engine which operates a generator to produce electricity. Similarly, petrol and diesel is used as a fuel for engines to produce electricity. We do not have an unlimited stock of fossil fuel. The rate at which we are using these reserves tells us that the stock may get exhausted in about 50 years. This is the right time to plan for future sources of energy once fossil fuel is not available. 

(c) Electricity from nuclear energy. Fission reaction of uranium is highly exothermic. This energy can also be utilised for production of electricity. Fission reaction is allowed to proceed in a controlled manner in a nuclear reactor. The heat produced is used to produce steam which works a turbine to get electricity. Uranium is extracted from the ground and we do not have unlimited stock of uranium also. Estimate tell us that stock of uranium may last about 60 - 70 years. Harnessing of nuclear energy is very expensive and is also not hazard free. It accompanied with lot of radioactive radiations which, in the event of a leakage, may prove to be disastrous. 

2. Renewable sources of electricity. Sources which are available at all times and are capable of sustaining themselves are called Renewable sources e.g. water, air, sun, etc. We shall have to develop some techniques to harness these sources for long time production of electricity. Some of them have already been put to use while techniques are being developed for others. 

(a) Hydropower plant. Water flowing through the rivers is being used for production of electricity. Water us allowed to fall from a height. It's potential energy is converted into kinetic energy. This falling water is made to spin a turbine which in turn operates the armature of a dynamo, this, producing electricity. For the construction of a hydropower plant we need to build a big reservior of water (in the form of a lake) at a certain height. For this, we shall have get a number of villages vacated resulting in large scale displacement of population. For this reason some governments do not favour building up big hydropower plants in spite of the fact that this a self sustaining source of electricity. 

Hydropower generation plays a major role in countries which have abundant supply of water resources. Norway gets 89%, Canada 58%, Switzerland 55% of its electricity for this sources. 16% of the total global consumption is obtained from hydropower plants. 

(b) Wind mill. Blowing wind has kinetic energy associated with itself. This energy can be converted into electric energy using a wind mill. 

A wind mill [in Fig.] consists of a rotor having three blades capable of rotation about a horizontal axis. The rotor is mounted on a high steel tower whose base consists of a strong reinforced concrete. As wind strikes against the blades, which gets slowed down by imparting some of its kinetic energy to the blades which starts rotating. By means of a shaft rotational kinetic energy is conveyed to the armature of an electric generator. As the armature rotates in between the two poles pieces of a strong electromagnet, an e.m.f. is induced in it. 

Wind mill. 

The power harnessed by the wind mill depends upon following factors:

(i) density 'ρ' of wind 

(ii) area 'A' swept by the turbine blades

(iii) wind velocity v. 

Power P developed by the wind mill is given by,                                                                                                P = 1/2 ρ × A × v³

If r = radius of revolution of turbine blades            Power = 1/2 ρ × πr² × v³

It can be seen that :

(i) Increasing the length of blades 2 times results in increase in power by 4 times. 

(ii) Increasing the velocity of wind 2 times results in increase in power by 8 times. 

The operation of such turbine requires wind speed to be in the range (4 - 25) ms-1. The maximum output is obtained at speed range (12 - 25) ms-1 . 

It is never possible to convert 100% energy of wind into electricity due to following reasons. 

(i) There is a limit known as 'Betz limit' according to which turbine blades cannot extract more than 59.26% of energy from the wind. Beyond this limit more and more air tends to go around the turbine rather than going through it with air pooling up in front. 

(ii) Electric generator is also not 100% efficient to convert whole input (rotational kinetic energy) into output (electric energy). 

(iii) Some energy is listed in overcoming friction between parts of the assembly. 

Power conversion by a wind mill is measured in terms of coefficient of power denotes by 'Cp'. It has been shown that coefficient of power greater than 35% has not been attained as yet. 

Large scale electricity generation through wind mill requires installation of huge wind turbines (about 50 m in large) in a region where we have large wind velocity. Naturally, the region cannot be in a density populated town. So, electricity generated has to be transmitted to the distant places. Now a days there has been a greater awareness of harnessing wind power. Some people have installed smaller wind turbines at their house tops for using electricity for domestic purpose. 

(c) Solar power plant. God is showering immense amount of energy on our planet for the benefit of human being irrespective of cast and creed. It is for us to harness these energy for our practical use. 

(i) Small scale use of solar energy. Sun rays, when incident on a photosensitive material, liberate electrons from it. The phenomenon is known as 'photoelectric effect'. Based upon this principle, photo cells have been developed which can be used as sources of electricity for small scale purpose i.e. for operating small electrical devices like calculator, camera, etc. 

(ii) Large scale use of solar energy. To harness solar energy for industrial purpose we have to build power plants using energy from sun. To achieve this, we shall have to concentrate solar energy by using large parabolic reflectors. Efforts are being made in this direction in a number of ways. 

(i) Concentrated heat from reflectors is made to heat an observer which is placed on the focus of the reflector. The observer is, generally, a synthetic oil which gets heated to about 700°C. This heated oil transfer energy to a secondary circuit which produces steam to drive a conventional turbine or generator to produce electricity. Solar one project in Nevada costing about $ 250 million projected to produce 124 million kWh of energy per year is spread over 160 hectares using 760 mirrored throughs to concentrate heat. 

(ii) In another type of solar power plant an array of reflecting mirrors concentrate energy on tunes containing hydrogen. This energy pressurizes hydrogen to power the four cylinder reciprocating solar sterling engine which in turn drives a generator, thus, producing electricity. 

(iii) In another type of plant, a huge chimney is built. Near the base of the chimney reflectors are used to heat air which rises through the chimney. Hot air, as it rises up, operates a turbine in conjunction with a generator to produce electricity. 

Solar input may be, both diffuse and intermittent (due to night and clouds). Therefore, generation of electricity from solar power has a low capacity factor. It is, generally, less than 15%. Moreover the power cost is also two to three times that of conventional sources. 

(d) Electricity from geothermal energy. Inside of the earth is very hot. This heat is, generally, due to radioactive decay taking place at a depth of a few kilometres below the surface of earth. 

(i) There are places where hot underground steam can be trapped. This steam is used to operate a turbine to produce electricity. 

(ii) There is another technique known as 'hot dry rock technique' to harness geothermal energy. There are regions on earth where fractures in rocks are observed. These may be natural or artificial one. Water pumped into these fractures, gets converted into the steam and is subsequently used for production of electricity. 

(e) Electricity from biofuel. Wood or any other crop which can be burnt to produce heat is a biofuel. Production of ethanol and biodiesel from fermentation of sugarcane is also an example of biofuel. Ethanol and biodiesel can be directly used as fuel for transport vehicles. Using biofuel is another way of utilizing the solar energy stored them. This source of energy has its own hazard. Crops are, generally, situated large distances away from the situation of power plant. So, we shall have to use our water resources to grow these crops. Again burning of biofuel is accompanied with emission of gases which is harmful to the environment. Considering the cost of transport, use of water resources etc. We find that our inputs will be much more costly than the outputs. 

(f) Some other renewable sources of energy. A part from those discussed above there are some sources of energy of which can also be utilised for production of electricity. 

1. Wave energy. 
2. Ocean thermal energy. 
3. Tidal energy.      

         

                            

Electric Current (i)

 Electric current (i):

We know that electric potential of a body is defined as the degree of electrification and it determines the direction of flow of charge. Consider two bodies 'A' and 'B' charged to potentials 'V1' and 'V2', respectively, such that V1 > V2. If they are not connected to each other [Fig. 1(i)], no charge flows and hence their potentials remain constant. If they are connected by means of good conductor R [Fig. 1(ii)], positive charge flows from higher potential to the lower potential or negative charge, i.e., electrons move from lower potential to higher potential till the potentials become equal. This is an instantaneous process. Within a fraction of a second the charge adjusts itself. There is no movement of charge after the potentials have been equalised. Now connect 'A' and 'B' to the two terminals of a source 'S' of e.m.f. having it's two terminals at potential difference 'V1 - V2' [Fig. 1(iii)]. Source 'S' helps in maintaining the difference of potentials between 'A' and 'B'. As a result of this, the charge keeps moving through the conductor R. In other words, a current 'i' keeps on flowing through the conductor.

Fig. 1. Flow of charge due to potential difference. 

 Current strength, in a conductor, is defined as the rate of flow of charge across any cross-section of the conductor. 

If a charge 'q' flows across any cross-section in 't' second, current i is given by           

                       i = q/t                 ..... (1) 

Electric current flowing through a conductor is associated with magnitude as well as direction. In spite of this, the electric current is considered to be a scalar quantity. Whenever we have to find the resultant current through a wire we have to take the algebraic sum of all the individual currents. Since the rule is valid only for scalar quantities, we take current to be a scalar quantity. 

Relation (1) holds good if the flow of charge is uniform with time. In case of non-uniform flow, let '∆q' be the small amount of charge flowing across any cross-section of the conductor in a small interval of time '∆t', then current 'i' is given by

                i = ∆q/∆t

If the time interval is choosen to be very small, i.e., ∆t →0

               i = Lt   ∆q/∆t = dq/dt                                                        ∆t→0

Electric current is time derivative of charge. 

Unit of current in S.I. is coulomb/sec or ampere. 

Type of electric currents:

1) steady current. A current is said to be steady if it's magnitude is constant and direction is always the same. Fig. 2(i) shows the current-time graph for d.c. or steady current. 

Fig. 2. Various types of electric currents. 

2) Variable current. A variable current, in general, is defined as the current which changes in magnitude with time while its direction may or may not change. Fig. 2(ii) shows a current whose magnitude changes within 1 A and 3 A, while its direction is always same (positive). Such a current is called variable direct current. Fig. 2(iii) shows a current whose magnitude changes between +1 A and -1 A, while its directions gets reversed after equal intervals of time. This current is called alternating current (a.c.). 

Conventional Current:

By convention the direction of flow of current is taken to be the direction of flow of positive charge. 

The current in that sense is called conventional current. The direction of conventional current is from A to B if positive charge flows from A to B. A negative charge moving from A to B is equivalent to a current flowing from B to A [in Fig. 3].

Fig. 3. Convention regarding direction of current. 

Units of Electric Current:

1) C.G.S. electro-static unit (e.s.u.). The current flowing through a conductor is said to be one e.s.u. if a charge of 1 e.s.u. flows across any of its cross-section in one second. 

 ∴  1 e.s.u. of current = 1 e.s.u of charge/1sec    

The e.s.u. of current is also called statampere. 

2) C.G.S. electro-magnetic unit (e.m.u). The current flowing through a conductor is said to be one e.m.u. if a charge of 1 e.m.u. flows across any of its cross-section in one second. 

∴ 1 e.s.u. of current = 1 e.m.u. of charge/1sec

The e.m.u. of current is also called abampere. 

3) S.I. unit (ampere). The current flowing through a conductor is said to be one ampere if a charge of 1 coulomb flows across any of its cross-section in one second

  ∴     1 ampere = 1 coulomb/1 second

Relation between ampere and statampere (e.s.u):

We know that, 

     1 coulomb = 3 × 10⁹ e.s.u. of charge

∴   1 ampere = 1 coulomb/1 second

                    = 3 × 10⁹ e.s.u. of charge/1 second

1A = 3 × 10⁹ e.s.u. of current or statampere

Relation between ampere and abampere:

We know that, 

      1 coulomb = 1/10 e.m.u. of charge, 

      1 ampere = 1 coulomb/1 second

1A = 1/10 e.m.u. of charge/1 second

      = 1/10 e.m.u. of current or ampere. 

Important notes:

1. A current taken to be along the direction of motion of electrons is called electronic current. 
2. A current taken to be opposite to the direction of motion of electrons is called conventional current. In physics, we often deal with conventional currents.

Capacitor and its principle

 Capacitor : 

A capacity of the order of 1 μF is extensively used in the assembly of radio sets. The capacity can be achieved by means of a simple spherical capacitor. The radius of that capacitor can be calculated as follows 

  Since       C = r/9 × 10⁹

              10−6  =  r/9 × 10⁹

               r = 9 × 10³ m       or       r = 9 km.                   

 This value of 'r' is so large that it is impossible to assemble a radio set using capacities from a single conductor. Therefore, some method has to be devised to obtain a large capacity in a smaller space. 

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

Principle of a capacitor : 

It is based on the principle that when, an earthed conductor is placed in the neighbourhood of a charged conductor, the capacity of the system increases considerably, this can be shown by the following simple experiment. 

Consider a plate P charged positively and connected to the knob of a gold leaf electroscope. Note the divergence of the leaves [Fig.(i)]. The divergence is a measure of potential 'V' of the plate. Now place another plate Q near it [Fig.(ii)]. Negative charge is induced on the inner side of Q. Positive charge is induced on the outer side Q. Induced negative charge on inner side of Q tries to decrease the potential of P while induced positive charge on the outer side of Q tries to increase it. On the whole, there is a net decrease in the potential of P because negative charge is nearer to P than the positive charge. This is indicated by a decrease in the divergence of leaves. 

  Principle of a capacitor     

But            C = Q/V

Therefore, the capacity of the system increases. 

Now connect the plate Q to earth [Fig. (iii)]. The free positive charge on its outer surface disappears, thereby, causing a further reduction in the potential of P. Hence the capacity of the system increases further. 

Types of Capacitors:

1. Paper capacitor. It consists of a pair of longer tin foils having a piece of wax paper in between them [Fig. (i)]. The wax paper acts as a di-electric medium. For the sake of convenience, the foils along with wax paper are rolled into the form of a cylinder [Fig. (ii)].                

Paper capacitor. 

Two leads connected to the foils are taken out from the two sides of the roll. 

2. Mica capacitor. It consists of a number of mica sheets arranged one above the other in such a way that there is a mica sheet in between every two copper plates [in fig.]. Alternate copper plates are connected to terminals A and B. This way different condensers get connected in parallel with each other. Thus net capacity of the capacitor is equal to the sum of the individual capacities.                                                       

Mica capacitor. 

3. Variable condenser. It is a condenser whose capacity can be varied at will. It consists of two sets X and Y of parallel plates. One set of plates, say Y, is fixed while the other set X can be turned with the help of knob k [in fig.]. As the knob turns, the set X either moves into the spacings of set Y Or comes out of them, thereby changing the common area of the plates. Since capacity also undergoes a change. Generally a pointer attached to the knob moves over a graduated scale enabling us to read the capacity directly.                          

Variable condenser. 

4. Electrolytic capacitor. It consists of a pair of aluminium plates A and C dipping in a solution of aluminium borate. A and C are connected to the positive and negative terminals of a source of steady current [in fig.]. Due to electrolysis a very fine lyer (≈ 10^-6 cm) of aluminium oxide is formed on plate A. This layer acts as a di-electric medium while rhe solution along with plate C acts as cathode. Since the thickness of the di-electric is very small, the capacitor has a large capacity.             

Electrolytic capacitor. 

                                                                                      In a wet type electrolytic condenser, the anode is taken in the form of a cylinder immersed in the electrolyte contained in a metal can which works as a cathode. In case of a dry type electrolytic capacitor, two long strips of aluminium (one of which has a thin layer of a aluminium oxide) have cotton or paper gauze soaked in electrolyte in between them.                                                                        

 The oxide layer of aluminium has a low resistance in one direction and a high resistance in the other. So the electrolytic capacitors are used only in cases of sources of unidirectional current in such a way that the oxide plate is always positive with respect to the other plate. 

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.