Ranking And Time Sequence

Ranking And Time Sequence - Short-cut Tricks And Examples

Dear Reader, below are five simple types of ‘ranking and time sequence’ problems. You will find detailed solutions with each of the problems

As far as ranking problems are concerned, you have to remember one simple formula.

Total number of people = Rank of a person from START + Rank of person from the END – 1

Let we start with our tutorial. At the end of the tutorial, you will find a short practice test. Please do take the test so that you can be double sure that you have understood well.

Type I: Finding Rank From the Start (or the End)

In type 1, you will know the rank of a person from either the start (or the end). Using that data, you have to find the rank of that person from the end (or the start). Below example will help you.

Example Question 1: Reena is 10 ranks ahead of Priya in a class of 40. If Priya`s rank is 20th from the last, what is Reena`s rank from the start?
Answer: 31th

Solution:
Priya’s rank from the last is 20. Reena is 10 ranks ahead. Therefore, Reena’s rank from the last = Rank of Reena from the last = 20 – 10 = 10th
Now you have to apply the formula that you saw in the introduction.
Total number of students = Rank of Reena from the start + Rank of Reena from the end – 1
40 = Rank of Reena from the start + 10 – 1
Rank of Reena from the start = (40 – 10) +1
= 31th

Type II: Finding Total Number of People in a Sequence

In type 2, you will find ranks of a person from the start and the end. You have to find the total number of people. Let us see an example.

Example Question 2: Venkatesh ranks 8th from the top and 24th from the bottom in the class. How many students are there in total?
Answer: 31

Solution:
Total number of Students = Rank of Venkatesh from the top + Rank of Venkatesh from the bottom – 1
= [8 + 24] – 1= 31

Type III: Interchanging Positions

In this type, positions of the people in a sequence will be interchanged. You have to solve these problems after processing the data given.

Example Question 3: In a row of girls, Uma is 10th from the left and Meena is 20th from the right. If they interchange their positions, Uma becomes 15th from the left. How many girls are there in the row?
Answer: 34

Solution:
After interchange, rank of Uma from the left = 15
But, before interchange, Meena was 20th from the right. After interchange, Uma would have occupied the same position of Meena’s earlier spot. Therefore,
Present rank of Uma from the right = 20
Now you know the current ranks of Uma from left as well as right. Therefore,
Total number of girls = Uma’s rank from the left + Uma’s rank from the right – 1
= (15+20)-1
=34.

Type IV: Intervention in a Frequent Event

Though the heading of this type looks complicated, this is one of the easiest. There will be intervention at certain time of a frequent event. Based on that time data you have to solve the problem logically.

Let us see an example.

Example Question 4: A bus leaves to Chennai from Bangalore every 30 minutes. A passenger inquired about the next bus to Bangalore, and he was informed that the bus left 15 minutes before, and the next bus will be at 5.00 pm. Find at what time the passenger had enquired?
Answer: 4.45 pm

Solution:
For every 30 minutes, there is a bus. The next bus will be at 5.00 pm, so the last bus must have left at 4.30 pm.
The informer said that the bus had left 15 minutes before his inquiry. So the time of inquiry is 4.30 + 0.15 = 4.45 pm.

Type V: Day of Week Based on Frequency

This type is very easy just like the previous one. You can answer this with little effort. (You may not expect questions simple in your exam. However, this will be a first step in understanding more difficult questions.)

Example Question 5: Gita went to the temple five days ago. If she goes to the temple every Friday, then what day of the week is today?
Answer: Wednesday

Solution:
She went to temple five days ago i.e., Friday.
Five days from Friday is Wednesday.

परमाणु की सरंचना एक संक्षिप्त जानकारी

परमाणु की सरंचना एक संक्षिप्त जानकारी 

परमाणु तत्त्वों के रचनात्मक भाग होते हैं। ये तत्त्व के एेसे छोटे भाग हैं, जो रासायनिक क्रिया में भाग लेते हैं। प्रथम परमाणु सिद्धांत, जिसे जॉन डॉल्टन ने सन् 1808 में प्रतिपादित किया, के अनुसार परमाणु पदार्थ के एेसे सबसे छोटे कण होते हैं, जिन्हें और विभाजित नहीं किया जा सकता है। उन्नीसवीं शताब्दी के अंत में प्रयोगों द्वारा यह प्रमाणित हो गया कि परमाणु विभाज्य है तथा वह तीन मूल कणों (इलेक्ट्रॉन, प्रोटॉन तथा न्यूट्रॉन) द्वारा बना होता है। इन अव-परमाणविक कणों की खोज के बाद परमाणु की संरचना को स्पष्ट करने के लिए बहुत से परमाणु मॉडल प्रस्तुत किए गए।

सन् 1898 में थॉमसन ने कहा कि परमाणु एक समान धनात्मक विद्युत् आवेश वाला एक गोला होता है, जिस पर इलेक्ट्रॉन उपस्थित होते हैं। वह मॉडल, जिसमें परमाणु का द्रव्यमान पूरे परमाणु पर एक समान वितरित माना गया था, सन् 1909 में रदरफोर्ड के महत्त्वपूर्ण α-कण के प्रकीर्णन प्रयोग द्वारा गलत सिद्ध हुआ। रदरफोर्ड ने यह निष्कर्ष निकाला कि परमाणु के केंद्र में बहुत छोटे आकार का धनावेशित नाभिक होता है और इलेक्ट्रॉन इसके चारों ओर वृत्ताकार कक्षों में गति करते हैं। रदरफोर्ड मॉडल, जो सौरमंडल से मिलता-जुलता था, निश्चित रूप से डाल्टन मॉडल से बेहतर था, परंतु यह परमाणु की स्थिरता की, अर्थात् यह इस बात की व्याख्या नहीं कर पाया कि इलेक्ट्रॉन नाभिक में क्यों नहीं गिर जाते हैं? इसके अलावा यह परमाणु की इलेक्ट्रॉनिक संरचना, अर्थात् नाभिक के चारों ओर इलेक्ट्रॉनों के वितरण और उनकी ऊर्जा के बारे में कुछ नहीं बता सका। रदरफोर्ड मॉडल की इन कठिनाइयों को सन् 1913 में नील बोर ने हाइड्रोजन परमाणु के अपने मॉडल में दूर किया तथा यह प्रस्तावित किया कि नाभिक के चारों ओर वृत्ताकार कक्षों में इलेक्ट्रॉन गति करता है। केवल कुछ कक्षों का ही अस्तित्व हो सकता है तथा प्रत्येक कक्षा की निश्चित ऊर्जा होती है। बोर ने विभिन्न कक्षों में इलेक्ट्रॉन की ऊर्जा की गणना की और प्रत्येक कक्षा के लिए नाभिक और इलेक्ट्रॉन की दूरी का आकलन किया। हालाँकि बोर मॉडल हाइड्रोजन के स्पेक्ट्रम को संतोषपूर्वक स्पष्ट करता था, लेकिन यह बहु-इलेक्ट्रॉन परमाणुओं के स्पेक्ट्रमों की व्याख्या नहीं कर पाया। इसका कारण बहुत जल्द ही ज्ञात हो गया। बोर मॉडल में इलेक्ट्रॉन को नाभिक के चारों ओर एक निश्चित वृत्ताकार कक्षा में गति करते हुए आवेशित कण के रूप में माना गया था। इसमें उसके तरंग जैसे लक्षणों के बारे में नहीं सोचा गया था। एक कक्षा एक निश्चित पथ होता है और इस पथ को पूरी तरह तभी परिभाषित माना जा सकता है, जब एक ही समय पर इलेक्ट्रॉन की सही स्थिति और सही वेग ज्ञात हो। हाइज़ेनबर्ग  के ‘अनिश्चितता सिद्धांत’ के अनुसार एेसा संभव नहीं है। इस प्रकार हाइड्रोजन परमाणु का बोर मॉडल न केवल इलेक्ट्रॉन के दोहरे व्यवहार की उपेक्षा करता है, बल्कि हाइज़ेनबर्ग अनिश्चितता सिद्धांत का भी विरोध करता है।

सन् 1926 में इरविन श्रोडिंजर ने एक समीकरण दिया, जिसे ‘श्रोडिंजर समीकरण’ कहा जाता है। इसके द्वारा त्रिविम में इलेक्ट्रॉन के वितरण और परमाणुओं में अनुमत ऊर्जा स्तरों का वर्णन किया जा सकता है। यह समीकरण न केवल दे ब्रॉग्ली के तरंग-कण वाले दोहरे लक्षण की संकल्पना को ध्यान में रखता है, बल्कि हाइज़ेनबर्ग के ‘अनिश्चितता सिद्धांत’ के भी संगत है। जब इस समीकरण को हाइड्रोजन परमाणु में इलेक्ट्रॉन के लिए हल किया गया, तो इलेक्ट्रॉन के संभव ऊर्जा-स्तरों और संगत तरंग फलनों (जो गणितीय फलन होते हैं) के बारे में जानकारी प्राप्त हुई। ये क्वांटित ऊर्जा-स्तर और उनके संगत तरंग-फलन जो तीन क्वांटम संख्याओं– मुख्य क्वांटम संख्या n, दिगंशीय क्वांटम संख्या l, और चुंबकीय क्वांटम संख्या ml के द्वारा पहचाने जाते हैं, श्रोडिंजर समीकरण के हल के परिणामस्वरूप प्राप्त होते हैं। इन तीन क्वांटम संख्याओं के मानों पर प्रतिबंध भी श्रोडिंजर-समीकरण के हल से स्वतः प्राप्त होते हैं। हाइड्रोजन परमाणु का क्वांटम यांत्रिकीय मॉडल उसके स्पेक्ट्रम के सभी पहलुओं की व्याख्या करता है और उसके अतिरिक्त कुछ एेसी परिघटनाओं को भी समझाता है, जो बोर मॉडल द्वारा स्पष्ट नहीं हो सकीं।

परमाणु के क्वांटम यांत्रिकीय मॉडल के अनुसार बहु-इलेक्ट्रॉन परमाणुओं के इलेक्ट्रॉन-वितरण को कई कोशों में बाँटा गया है। ये कोश एक या अधिक उप-कोशों के बने हुए हो सकते हैं तथा इन उप-कोशों में एक या अधिक कक्षक हो सकते हैं, जिनमें इलेक्ट्रॉन उपस्थित होता है। हाइड्रोजन और हाइड्रोजन जैसे निकायों (उदाहरणार्थ– He+, Li2+ आदि) में किसी दिए गए कोश के सभी कक्षकों की समान ऊर्जा होती है, परंतु बहु-इलेक्ट्रॉन परमाणुओं में कक्षकों की ऊर्जा n और l के मानों पर निर्भर है। किसी कक्षक के लिए (n + l) का मान जितना कम होगा उसकी ऊर्जा भी उतनी ही कम होगी। यदि कोई दो कक्षकों का (n + l) मान समान है, तो उस कक्षक की ऊर्जा कम होगी, जिसके लिए n का मान कम है। किसी परमाणु में एेसे कई कक्षक संभव होते हैं, तथा उनमें ऊर्जा के बढ़ते क्रम में इलेक्ट्रॉन पाउली के अपवर्जन सिद्धांत (किसी परमाणु में किन्हीं दो इलेक्ट्रॉनों की चारों क्वांटम-संख्या का मान समान नहीं हो सकता है) और हुंड के अधिकतम बहुकता नियम (एक उपकोश के कक्षकों में इलेक्ट्रॉनों का युग्मन तब तक प्रारंभ नहीं होता, जब तक प्रत्येक कक्षक में एक-एक इलेक्ट्रॉन न आ आए) के आधार पर भरे जाते हैं। परमाणुओं की इलेक्ट्रॉनिक संरचना इन्हीं विचारों पर आधारित है।

A Concise Details About Structure of Atom

A Concise Details About Structure of Atom

Atoms are the building blocks of elements. They are the smallest parts of an element that chemically react. The first atomic theory, proposed by John Dalton in 1808, regarded atom as the ultimate indivisible particle of matter. Towards the end of the nineteenth century, it was proved experimentally that atoms are divisible and consist of three fundamental particles: electrons, protons and neutrons. The discovery of sub-atomic particles led to the proposal of various atomic models to explain the structure of atom.

Thomson in 1898 proposed that an atom consists of uniform sphere of positive electricity with electrons embedded into it. This model in which mass of the atom is considered to be evenly spread over the atom was proved wrong by Rutherford’s famous alpha-particle scattering experiment in 1909. Rutherford concluded that atom is made of a tiny positively charged nucleus, at its centre with electrons revolving around it in circular orbits. Rutherford model, which resembles the solar system, was no doubt an improvement over Thomson model but it could not account for the stability of the atom i.e., why the electron does not fall into the nucleus. Further, it was also silent about the electronic structure of atoms i.e., about the distribution and relative energies of electrons around the nucleus. The difficulties of the Rutherford model were overcome by Niels Bohr in 1913 in his model of the hydrogen atom. Bohr postulated that electron moves around the nucleus in circular orbits. Only certain orbits can exist and each orbit corresponds to a specific energy. Bohr calculated the energy of electron in various orbits and for each orbit predicted the distance between the electron and nucleus. Bohr model, though offering a satisfactory model for explaining the spectra of the hydrogen atom, could not explain the spectra of multi-electron atoms. The reason for this was soon discovered. In Bohr model, an electron is regarded as a charged particle moving in a well defined circular orbit about the nucleus. The wave character of the electron is ignored in Bohr’s theory. An orbit is a clearly defined path and this path can completely be defined only if both the exact position and the exact velocity of the electron at the same time are known. This is not possible according to the Heisenberg uncertainty principle. Bohr model of the hydrogen atom, therefore, not only ignores the dual behaviour of electron but also contradicts Heisenberg uncertainty principle. 

Erwin Schrödinger, in 1926, proposed an equation called Schrödinger equation to describe the electron distributions in space and the allowed energy levels in atoms. This equation incorporates de Broglie’s concept of wave-particle duality and is consistent with Heisenberg uncertainty principle. When Schrödinger equation is solved for the electron in a hydrogen atom, the solution gives the possible energy states the electron can occupy [and the corresponding wave function(s) (ψ) (which in fact are the mathematical functions) of the electron associated with each energy state]. These quantized energy states and corresponding wave functions which are characterized by a set of three quantum numbers (principal quantum number n, azimuthal quantum number l and magnetic quantum number ml) arise as a natural consequence in the solution of the Schrödinger equation. The restrictions on the values of these three quantum numbers also come naturally from this solution. The quantum mechanical model of the hydrogen atom successfully predicts all aspects of the hydrogen atom spectrum including some phenomena that could not be explained by the Bohr model.

According to the quantum mechanical model of the atom, the electron distribution of an atom containing a number of electrons is divided into shells. The shells, in turn, are thought to consist of one or more subshells and subshells are assumed to be composed of one or more orbitals, which the electrons occupy. While for hydrogen and hydrogen like systems (such as He+, Li2+ etc.) all the orbitals within a given shell have same energy, the energy of the orbitals in a multi-electron atom depends upon the values of n and l: The lower the value of (n + l ) for an orbital, the lower is its energy. If two orbitals have the same (n + l ) value, the orbital with lower value of n has the lower energy. In an atom many such orbitals are possible and electrons are filled in those orbitals in order of increasing energy in accordance with Pauli exclusion principle (no two electrons in an atom can have the same set of four quantum numbers) and Hund’s rule of maximum multiplicity (pairing of electrons in the orbitals belonging to the same subshell does not take place until each orbital belonging to that subshell has got one electron each, i.e., is singly occupied). This forms the basis of the electronic structure of atoms.

Measurement of Length

Measurement of Length

You are already familiar with some direct methods for the measurement of length. For example, a metre scale is used for lengths from 10–3 m to 102 m. A vernier callipers is used for lengths to an accuracy of 10–4 m. A screw gauge and a spherometer can be used to measure lengths as less as to 10–5 m. To measure lengths beyond these ranges, we make use of some special indirect methods.

 Measurement of Large Distances

Large distances such as the distance of a planet or a star from the earth cannot be measured directly with a metre scale. An important method in such cases is the parallax method.

When you hold a pencil in front of you against some specific point on the background (a wall) and look at the pencil first through your left eye A (closing the right eye) and then look at the pencil through your right eye B (closing the left eye), you would notice that the position of the pencil seems to change with respect to the point on the wall. This is called parallax. The distance between the two points of observation is called the basis. In this example, the basis is the distance between the eyes. 

To measure the distance D of a far away planet S by the parallax method, we observe it from two different positions (observatories) A and B on the Earth, separated by distance AB = b at the same time as shown in Fig. 2.2. We measure the angle between the two directions along which the planet is viewed at these two points. The ∠ASB in Fig. 2.2 represented by symbol θ is called the parallax angle or parallactic angle.

As the planet is very far away,

b/D <<1

and therefore, 

D = b/θ

is very small. Then we approximately take AB as an arc of length b of a circle with centre at S and the distance D as the radius AS = BS so that AB = b = D θ  where θ  is in radians.

(2.1)

Having determined D, we can employ a similar method to determine the size or angular diameter of the planet. If d is the diameter of the planet and α  the angular size of the planet (the angle subtended by d at the earth), we have 

α = d/D (2.2)

Fig. 2.2  Parallax method.

The angle α  can be measured from the same location on the earth. It is the angle between the two directions when two diametrically opposite points of the planet are viewed through the telescope. Since D is known, the diameter d of the planet can be determined using Eq. (2.2).

Estimation of Very Small Distances: Size of a Molecule

To measure a very small size, like that of a molecule (10–8 m to 10–10 m), we have to adopt special methods. We cannot use a screw gauge or similar instruments. Even a microscope has certain limitations. An optical microscope uses visible light to ‘look’ at the system under investigation. As light has wave like features, the resolution to which an optical microscope can be used is the wavelength of light (A detailed explanation can be found in the Class XII Physics textbook). For visible light the range of wavelengths is from about 4000 Å to 7000 Å 

(1 angstrom = 1 Å = 10-10 m). Hence an optical microscope cannot resolve particles with sizes smaller than this. Instead of visible light, we can use an electron beam. Electron beams can be focussed by properly designed electric and magnetic fields. The resolution of such an electron microscope is limited finally by the fact that electrons can also behave as waves . The wavelength of an electron can be as small as a fraction of an angstrom. Such electron microscopes with a resolution of 0.6 Å have been built. They can almost resolve atoms and molecules in a material. In recent times, tunnelling microscopy has been developed in which again the limit of resolution is better than an angstrom. It is possible to estimate the sizes of molecules. 

A simple method for estimating the molecular size of oleic acid is given below. Oleic acid is a soapy liquid with large molecular size of the order of 10–9 m.

The idea is to first form mono-molecular layer of oleic acid on water surface. 

We dissolve 1 cm3 of oleic acid in alcohol to make a solution of 20 cm3. Then we take 1 cm3 of this solution and dilute it to 20 cm3, using alcohol. So, the concentration of the solution is equal to  

(1/20×20) cm³

  of oleic acid/cm3 of solution. Next we lightly sprinkle some lycopodium powder on the surface of water in a large trough and we put one drop of this solution in the water. The oleic acid drop spreads into a thin, large and roughly circular film of molecular thickness on water surface. Then, we quickly measure the diameter of the thin film to get its area A. Suppose we have dropped n drops in the water. Initially, we determine the approximate volume of each drop (V cm3).

Volume of n drops of solution 

= nV cm3 

Amount of oleic acid in this solution

= 

nV(1/20×20) cm³

This solution of oleic acid spreads very fast on the surface of water and forms a very thin layer of thickness t. If this spreads to form a film of area A cm2, then the thickness of the film

t = Volume of the film /Area of the film

or, 

t = nV(1/20×20A) cm

(2.3)

If we assume that the film has mono-molecular thickness, then this becomes the size or diameter of a molecule of oleic acid. The value of this thickness comes out to be of the order of 10–9 m.

The International System of Units

The International System of Units

In earlier time scientists of different countries were using different systems of units for measurement. Three such systems, the CGS, the FPS (or British) system and the MKS system were in use extensively till recently.

The base units for length, mass and time in these systems were as follows :

• In CGS system they were centimetre, gram and second respectively.

• In FPS system they were foot, pound and second respectively.

• In MKS system they were metre, kilogram and second respectively. 

The system of units which is at present internationally accepted for measurement is the Système Internationale 

d’ Unites (French for International System of Units), abbreviated as SI. The SI, with standard scheme of symbols, units and abbreviations, developed by the Bureau International des Poids et measures (The International Bureau of weights and measures, BIPM) in 1971 were recently revised by the General Conference on weights and measures in November 2018. The scheme is now for international usage in scientific, technical, industrial and commercial work. Because SI units used decimal system, conversions within the system are quite simple and convenient. We shall follow the SI units in this book.

In SI, there are seven base units as given in  Table 2.1. Besides the seven base units, there are two more units that are defined for (a) plane angle dθ as the ratio of length of arc ds to the radius r and (b) solid angle dΩ as the ratio of the intercepted area dA of the spherical surface, described about the apex O as the centre, to the square of its radius r, as shown in Fig. 2.1(a) and (b) respectively. The unit for plane angle is radian with the symbol rad and the unit for the solid angle is steradian with the symbol sr. Both these are dimensionless quantities.

Fig.  2.1 Description  of  (a)  plane  angle  dθ  and  (b)  solid  angle

Table 2.1  SI Base Quantities and Units*

* The values mentioned here need not be remembered or asked in a test. They are given here only to indicate the extent of accuracy to which they are measured. With progress in technology, the measuring techniques get improved leading to measurements with greater precision. The definitions of base units are revised to keep up with this progress.

  

Table 2.2 Some units retained for general use (Though outside SI)

Note that when mole is used, the elementary entities must be specified. These entities may be atoms, molecules, ions, electrons, other particles or specified groups of such particles. 

We employ units for some physical quantities that can be derived from the seven base units (Appendix A 6). Some derived units in terms of the SI base units are given in (Appendix A 6.1). Some SI derived units are given special names (Appendix A 6.2 ) and some derived SI units make use of these units with special names and the seven base units (Appendix A 6.3). These are given in Appendix A 6.2 and A 6.3 for your ready reference. Other units retained for general use are given in Table 2.2.

Common SI prefixes and symbols for multiples and sub-multiples are given in Appendix A2. General guidelines for using symbols for physical quantities, chemical elements and nuclides are given in Appendix A7 and those for SI units and some other units are given in Appendix A8 for your guidance and ready reference.

Coulomb’s Law

Coulomb’s Law

Coulomb’s law is a quantitative statement about the force between two point charges. When the linear size of charged bodies are much smaller than the distance separating them, the size may be ignored and the charged bodies are treated as point charges. Coulomb measured the force between two point charges and found that it varied inversely as the square of the distance between the charges and was directly proportional to the product of the magnitude of the two charges and acted along the line joining the two charges. Thus, if two point charges q1, q2 are separated by a distance r in vacuum, the magnitude of the force (F) between them is given by 

F =|q1×q2|/r²

How did Coulomb arrive at this law from his experiments? Coulomb used a torsion balance* for measuring the force between two charged metallic spheres. When the separation between two spheres is much larger than the radius of each sphere, the charged spheres may be regarded as point charges.

Basic Properties of Electric Charge

1.5 Basic Properties of Electric Charge

We have seen that there are two types of charges, namely positive and negative and their effects tend to cancel each other. Here, we shall now describe some other properties of the electric charge. 

If the sizes of charged bodies are very small as compared to the distances between them, we treat them as point charges. All the  charge content of the body is assumed to be concentrated at one point  in space.

1.5.1 Additivity of charges

We have not as yet given a quantitative definition of a charge; we shall follow it up in the next section. We shall tentatively assume that this can be done and proceed. If a system contains two point charges q1 and q2, the total charge of the system is obtained simply by adding algebraically q1 and q2 , i.e., charges add up like real numbers or they are scalars like the mass of a body. If a system contains n charges q1, q2, q3, …, qn, then the total charge of the system is q1 + q2 + q3 + … + qn . Charge has magnitude but no direction, similar to mass. However, there is one difference between mass and charge. Mass of a body is always positive whereas a charge can be either positive or negative. Proper signs have to be used while adding the charges in a system. For example, the  total charge of a system containing five charges +1, +2, –3, +4 and –5,  in some arbitrary unit, is (+1) + (+2) + (–3) + (+4) + (–5) = –1 in the  same unit.

1.5.2 Charge is conserved

We have already hinted to the fact that when bodies are charged by rubbing, there is transfer of electrons from one body to the other; no new charges are either created or destroyed. A picture of particles of electric charge enables us to understand the idea of conservation of charge. When we rub two bodies, what one body gains in charge the other body loses. Within an isolated system consisting of many charged bodies, due to interactions among the bodies, charges may get redistributed but it is found that the total charge of the isolated system is always conserved. Conservation of charge has been established experimentally.

It is not possible to create or destroy net charge carried by any isolated system although the charge carrying particles may be created or destroyed in a process. Sometimes nature creates charged particles: a neutron turns into a proton and an electron. The proton and electron thus created have equal and opposite charges and the total charge is zero before and after the creation. 

1.5.3 Quantisation of charge

Experimentally it is established that all free charges are integral multiples of a basic unit of charge denoted by e. Thus charge q on a body is always given by

q = ne

where n is any integer, positive or negative. This basic unit of charge is the charge that an electron or proton carries. By convention, the charge on an electron is taken to be negative; therefore charge on an electron is written as –e and that on a proton as +e. 

The fact that electric charge is always an integral multiple of e is termed as quantisation of charge. There are a large number of situations in physics where certain physical quantities are quantised. The quantisation of charge was first suggested by the experimental laws of electrolysis discovered by English experimentalist Faraday. It was experimentally demonstrated by Millikan in 1912. 

In the International System (SI) of Units, a unit of charge is called a coulomb and is denoted by the symbol C. A coulomb is defined in terms the unit of the electric current which you are going to learn in a subsequent chapter. In terms of this definition, one coulomb is the charge flowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 2 of Class XI, Physics Textbook , Part I). In this system, the value of the basic unit of charge is 

e = 1.602192 × 10–19 C

Thus, there are about 6 × 1018 electrons in a charge of –1C. In electrostatics, charges of this large magnitude are seldom encountered and hence we use smaller units 1 µC (micro coulomb) = 10–6 C or 1 mC (milli coulomb) = 10–3 C. 

If the protons and electrons are the only basic charges in the universe, all the observable charges have to be integral multiples of e. Thus, if a body contains n1 electrons and n2 protons, the total amount of charge on the body is n2 × e + n1 × (–e) = (n2 – n1) e. Since n1 and n2 are integers, their difference is also an integer. Thus the charge on any body is always an integral multiple of e and can be increased or decreased also in steps of e. 

The step size e is, however, very small because at the macroscopic level, we deal with charges of a few µC. At this scale the fact that charge of a body can increase or decrease in units of e is not visible. In this respect, the grainy nature of the charge is lost and it appears to be continuous.

This situation can be compared with the geometrical concepts of points and lines. A dotted line viewed from a distance appears continuous to us but is not continuous in reality. As many points very close to each other normally give an impression of a continuous line, many  small charges taken together appear as a continuous charge  distribution.

At the macroscopic level, one deals with charges that are enormous compared to the magnitude of charge e. Since e = 1.6 × 10–19 C, a charge of magnitude, say 1 µC, contains something like 1013 times the electronic charge. At this scale, the fact that charge can increase or decrease only in units of e is not very different from saying that charge can take continuous values. Thus, at the macroscopic level, the quantisation of charge has no practical consequence and can be ignored. However, at the microscopic level, where the charges involved are of the order of a few tens or hundreds of e, i.e., they can be counted, they appear in discrete lumps and quantisation of charge cannot be ignored. It is the magnitude of scale involved that is very important.