Physical Laws

The mode of inquiry, and the form of reasoning, that physics is based upon can only be grasped by following this process in the actual construction of the leading ideas of physics. The choice of subjects in these lecture notes is intended to both demonstrate the methodology of physics by going through the chain of reasoning that leads to the central ideas in physics, while at the same time explaining in some detail the structure of these leading ideas. Methodology cannot be separated from the content of knowledge, and hence one has to actually reason the reasoning of physics by engaging with the ideas from which this reasoning emerges.

Rationale and Outline of Lecture Notes

The two most important and far reaching ideas in physics are the concepts of the atom, and that of the forces between atoms. Having an understanding of these two key ideas is sufficient for explaining all of physical phenomenon. An atom is composed of negatively charged electrons and a positively charged nuclei. Large conglomerations of atoms compose all of the physical entities in the universe, from simple materials such as the atmosphere, to more complex biological molecules, all the way to planets and stars. Atoms can form large conglomerations of matter due to the forces holding them together. The concept of force leads naturally generalizes to the more powerful physical concept of a field. For example, atoms can combine to form a biological molecule due to electromagnetic field between the electrons and nuclei of the atoms. Large collection of atoms form a planet or a star due to the attraction caused by the gravitational field. Atomic nuclei are composed of more fundamental particles called quarks, and which themselves belong to a larger family of fundamental particles. In the twentieth century, a fairly comprehensive understanding was reached as to what are the underlying laws of the fundamental particles , together with the fields responsible for their interactions. It is not possible to study the vast terrain of physics in a short introductory course meant for the non-specialists. The organizing principle of the subjects in these lecture notes is hence to focus a few of the most important ideas in physics, and present the simplest possible formulation of the fundamental ideas. The choice of subjects in these notes is based on the universal applicability of these concepts. The lecture notes start with the concept of energy, as this is probably the most important and universal idea in our understanding of nature. In Chapter 3 the energy of a particle is discussed. An essential characteristic of a particle is its energy due to motion as well as its energy due to its position. The idea of a particle is a precursor to the idea of an atom. Just like an ordinary particle, say a piece of stone, an atom also has energy, position and so on. A brief introduction is made in Chapter 4 of probability theory, since understanding the idea of randomness and chance is not only fundamental to our understanding of nature, but is also a necessary exercise for preparing one's mind for the counter-intuitive behaviour that one encounters in quantum mechanics. Chapter 5 discusses the phenomenon of waves for two reasons, firstly because waves are encountered in a vast array of natural phenomena, and secondly to foreground the ideas needed to the understand the concept of a field. Chapter 6 discusses electromagnetic radiation and light, and is the first time in the lecture notes that the fundamental concept of a field is introduced. One of the most important examples of a field is light. Chapter 7 studies the behaviour of the electric and magnetic fields, and follows logically from the study, in Chapter 6, of the electromagnetic field . Chapters 8 and 9 are concerned with the concept of entropy, which is as important as energy in understanding the behaviour of large collections of atoms. Chapter 8 introduces the idea of entropy, and in Chapter 9 the Second Law of Thermodyamics is illustrated using the example of heat engine and refrigerators. In Chapter 10 we discuss the discipline of statistical mechanics, which studies large collections of atoms using statistical methods. All the microscopic properties of entropy and thermodynamics can be derived from the postulates of statistical mechanics. In Chapter 11 we embark on the trail of quantum physics, and discuss the counter-intuitive and bizarre reasoning that forms the logical basis and template of this branch of physics. Quantum physics, together with the theory of relativity, forms the cornerstone of present day physics. Quantum physics reigns supreme in the sense that it has stood the test of all experiments till date, and continues to predict and explain all sorts of novel and unexpected phenomenon. In Chapter 12 we explain the concept and phenomenon of the atom using the principles of quantum mechanics. We then go on to show how all the atoms found in nature follow in a relatively straightforward manner from the ideas of quantum mechanics. In Chapters 2-9 all the phenomenon that were discussed are compatible with classical physics. In Chapter 13, to give a flavor of phenomena that are explicable only in terms of quantum theory, various interesting and striking examples of quantum phenomena are discussed, and which are also of great technological importance. The concepts discussed in the lecture notes are meant to open up the entire world of science to the readers. Unlike popular books on science which avoid any attempt at a quantitative understanding of science, the emphasis in these notes in to illustrate, by demonstration, that a precise understanding of the physical laws is possible with a minimal level of mathematics. The lecture notes focus on a quantitative approach to the leading ideas of physics, and is presented at a level of complexity in between the qualitative mode of presentation adopted in the book The Physical Sciences by Hazen and Trefil and the technical level appropriate for physics undergraduates found in the book Fundamentals of Physics by Halliday, Resnick and Walker. A mathematical formulation of the physical laws, no matter how rudimentary, allows the reader to grapple with relatively advanced ideas of science.

Domains of Physical Phenomena

To have a meaningful understanding of the vast range of physical phenomena, we need to know at what length scales and time scales these phenomena occur. All physical bodies are characterized by dimensional quantities of which the dimensions of length and time are of fundamental importance. Length is measured in meters and time in seconds. For example, we know that objects that we are familiar with have a size (called scale) of around 1-100 meters. Similarly events that we are witness in daily life have a duration from one second (a heartbeat) to days and years.

Table 2.1: Some typical Length and Distance Scales

Length (or distance)	Meters (approximate)
Radius of nucleus	$10^{-15}$
Atom	$10^{-10}$
A virus	$10^{-7}$
Grain of salt	$10^{-4}$
Height of a child	1
Distance to the Sun	$10^{11}$
Distance to the nearest star	$10^{16}$
Distance to the center of our Galaxy	$10^{20}$
Distance to the nearest Galaxy	$10^{22}$
Distance to the furthest Galaxy	$10^{26}$

However, we have no direct experience to decide for example, what is the size of a typical atom. Or how many atoms and molecules constitute a living cell. Or how far are the distant galaxies. To obtain a more complete classification of nature, we need recourse to scientific experiments. As one can imagine, to even arrive at such a classification of physical phenomena, we need to know what discipline studies the phenomena in question.

Table 2.2: Some typical Time Scales

Time interval	Seconds (approximate)
Light crosses a nucleus	$10^{-24}$
Lifetime of an unstable subnuclear particle	$10^{-23}$
Period of an atomic vibration	$10^{-15}$
Light travels 1 m	$3.3\times 10^{-9}$
Time of one heartbeat	1
One year	3 $\times 10^{7}$
Typical human life span	$2 \times 10^{9}$
Age of pyramids	$10^{11}$
Life on earth	$10^{17}$
Age of Universe	$10^{18}$

Table 2.1 gives a sample of phenomena on different length scales and Table 2.2 does the same for different time scales. Various theories of physics are applicable for different domains of phenomena. All theories of physics are an approximate description of nature, although it is worth noting the remarkable fact that so far no experiment has been able to find any inaccuracy with quantum mechanics. For objects that are not too big or too small, Newtonian, or classical physics is a valid approximation. In the Figure 2.1 we chart out the various length and time scales, or equivalently, length and velocity scales for which the various theories are good approximations.

Figure 2.1: Domains of Applicability

Figure 1.1 lists all the theories of physics, and as one moves towards the right, the theories encompass a greater range of phenomena and are more accurate as well. The crowning glory of theoretical physics is embodied in string theory. String theory has as yet no experimental evidence in support of its correctness. Nevertheless, it has made a number of theoretical advances, the most important being the successful synthesis of quantum theory with the geometry of spacetime. Einstein's theory of gravitation is incorrect at short distances, and emerges from string theory as an approximation valid at large distances of the more fundamental equations of string theory. A remarkable prediction of string theory is that the physical universe is at least ten dimensional, with only four dimensions being visible to our senses, and the remaining six dimensions existing in the unseen.

Dimensions and Units

All physical quantities are described by parameters such as length, mass and so on, all of which have dimensions. In general we will denote the dimension of length by $L$, of time by $T$ and of mass by $M$. The dimension of a quantity $x$ will be denoted by square brackets as $[x]$. For example, the dimension of the volume of an object $V$ is denoted by $[V]=L \times L \times L = L^3$. An equation such at $X+Y=Z$ is consistent only if the dimension of both the sides match, in this case only if all the three quantities $X, Y$ and $Z$ have the same dimensions. Checking that the dimensions agree for an equations is one of the first steps that one should take when analyzing a new set of equations. Dimensional analysis can lead to fairly sophisticated results; however, dimensional analysis can only lead to the general form of the equation, and cannot fully specify the solution. For example, if one has two dimensional quantities $X$ and $Y$, and we know that the solution $A$ is a dimensionless combination of $X$ and $Y$, then we can straight away conclude that the solution must be a function only of the ratio $X/Y$; however, based only on dimensional considerations the solution can be either $A=1+X/Y$ or equivalently $A=100-Y/X$, and hence the complete answer for $A$ cannot simply be obtained from dimensional analysis. The units that one uses for the various dimensional quantities are specific choices of what is one unit of length, time and so on. Units are arbitrary, and any set of consistent units can be used. For example, we can use meters or equivalently inches to measure length, grams or pounds to measure mass, and so on. Of course one can always convert from one set of units to another, and is analogous to converting from one currency to another. We will use the SI (Systeme International)units, which previously was known as the MKS system of units. In these units distance is measured in meters (m), mass is measured in terms of kilogram (kg), and time is measured in seconds (s). Hence the dimension of velocity $v$ is given by $[v]=LT^{-1}$, and in SI units we have $[v]=ms^{-1}$.

Experimental Uncertainties

As discussed above, experiments are fundamental to science, and experimental verification of scientific theories is a unique feature of scientific truths. In addition to verifying scientific theories, experiments freely explore nature very much in the way a cook experiments with new recipes. New conditions are created in the laboratory, and the behaviour of nature is then observed. Experiments and observations are carried out using instruments and experimental apparatus. The precision and accuracy of a measurement is determined by the instrument being used. For example, consider cutting a piece of cloth after measuring out a certain length. Suppose the smallest division in the measuring tape is say 0.01m. Since this is the most accurate measurement we can make, our inability to resolve the length of the cloth to any greater degree results is one form of experimental error, called a systematic error. We may measure the cloth to be

$$\mathrm{Length}=7.32 \pm 0.01 m$$ (2.1)

Systematic errors in general result from inherent limitations of the instrument being used. Systematic errors also arise from an incorrect calibration of an instrument, and every measurement made will contain this error. There is another kind of error, called random error.This error arises because the object being measured may not be in exactly the same condition every time we measure it. In the example of the cutting a piece of cloth, the length of the cloth might not be measured in exactly the same manner, resulting in random errors. Random errors can be reduced by repeatedly making the measurement, in the example of the cloth by measuring the length of the piece to be cut. If we measure the length say a 100 times, then the error can be further reduced by measuring the length say 10,000 times and on so.

Order of Magnitude Estimates

Before embarking on any calculation or experiment, it is a good habit to first do a crude estimate of the magnitude of the result that one expects. Suppose one needs to estimate the mass of water in a water tank which is a cylinder of radius 9m and height 2m. Since we are only interested in an order of magnitude analysis, we can make many approximations. We hence do the following estimate.

$$\mathrm{Volume} = \mathrm{Height}\times \pi \mathrm{radius}^2$$ (2.2)

$$\sim 1m \times 3 \times 80 m^2$$ (2.3)

$$\sim 250m^3$$ (2.4)

Given the mass of $1m^3$ of water is $1,000 kg$, we have that the mass of water in the tank is approximately $250,000 kg=250$ tons. Although this is not an exact answer, we have an estimate of what to expect. Hence it would be disastrous if we designed the tank to hold say $1$ ton of water.

Mode of Presentation

No prior knowledge of physics is required for reading the material presented. A working knowledge of the following mathematics will be helpful.

1. Elementary algebra, in particular the ability to use and manipulate symbols. For example, symbols $v_1, v_2$ or $p,p'$ and so on normally mean that these two quantities are in some way related, and that is why they have been named in a manner to reflect their connection.
2. Elementary trigonometry, in particular familiarity with sines and cosines.
3. Elementary geometry, in particular, the properties of triangles.
4. Elementary knowledge of vectors. Throughout the text, vectors will always be denoted by boldface, namely a symbol v means that v is a vector and that ${\bf v}\equiv(v_1,v_2,v_3)$.
5. Complex numbers are necessary in the study of quantum theory, and we will review the essential properties of these.
6. We will almost always use the symbol $x,y,z$ to denote points of space and $t$ to parametrize time.

Results that are more complicated - such as those derived using knowledge of calculus - or that involve more advanced ideas, as well discussions on scientific methodology, are placed in a box to separate it from the main text. These boxes can be skipped without affecting the reasoning being followed in the main text. All equations not in boxes can be derived without the use of calculus, and all readers are encouraged to do so. We will often have to study small changes in a system when some independent parameter specifying the system undergoes a change. We will denote changes which are finite by the Greek symbol delta, denoted by $\Delta$. So, for example, change in time from instant $t_1$ to $t_2$ will be denoted by $\Delta t=t_2-t_1$. The limit of calculus is to take, for example, $\Delta t$ to be vanishingly small, that is take $\Delta t \rightarrow 0$. When we do take such a limit, we will place these equations in a box.