## Wednesday, July 30, 2008

### Blogging or Study - Priority

Between blogging and study, my priority is study. My fluency in the subject has to go up with every effort that I am making. I have a private family commitment and blogging enhances that commitment through a public commitment. My relearning process has to result in some useful material, which is of some benefit in the learning process for IIT aspirants.

## Monday, July 28, 2008

### July - December 2008 Revision for JEE 2009

I plan to go through each chapter in Chemistry, Physics and Mathematics in revision mode during July-December 2008 apart from reading or studying chapter which I have not read so far.

My thinking is that from 1st January 2009 onwards, the aspirants should focus on memorizing things and a revision during July-Dec at leisurely pace, one chapter per day would help in that. From 1 January 2009, the memorization process should take up three chapters per day.

### Chapter 1 Introduction - July-Dec Revision

Physics is the study of nature and its laws.

The nature around us is like a big chess game played by Nature. Various events that happen are like the moves made by a chess players. We are allowed to watch the events that happen, and guess the rules, and then play to derive benefits that we want from nature. We may across new events which do not follow old rules that we have formulated and we need to guess the new rules.

Great scientists or scientists in general guess the rules from the observations available at that time and prove the usefulness of those rules by further experiments or events that happen subsequently. These rules may require modification subsequently if they are not able to explain some events observed that happen subsequently.

The description of nature becomes easy if we have the freedom to use mathematics.

Mathematics is the language of physics.

Units: Fundamental and derived

While there are a large number of physical quantities to be measured, only seven fundamental quantities are found to be sufficient. All other quantities can be measured using these seven fundamental quantities.

The set of fundamental quantities must have the following properties.

a. The fundamental quantities should be independent of each other; and
b. All other quantities may be expressed in terms of the fundamental quantities.

Fundamental quantities are also referred to as base quantities.

Who decides the units?

A body named Conference Generale des Poids or CGPM (General Conference on Weight and Measures in English) has been gibven the authority by international agreement.

Definitions of base units

Length metre (m)
Mass kilogram (kg)
Time second (s)
Electric current ampere (A)
Thermodynamic temperature kelvin (K)
Amount of substance mole (mol)
Luminous intensity candela (cd)

Metre

The distance travelled by light in vacuum in 1/[299,792,458] second is called 1 m.

Kilogram

the mass of cylinder made of planitum-iridium alloy kept at International Bureau of Weights and Measures is defined as 1 kg.

Second

The time duration in 9,192,631,770 time periods of the selected transition of radiation of Cesium-133 atom is defined as 1 s. (To understand this idea see Bohr model chapter)

Ampere

To under stand the definition of this unit, you have to read the chapter on electric field.

Two long straight wires with negligible cross section are to be placed parallel to each other at a separation of 1 m and electric current in the same amout is sent through them in the same direction. If the attractive force between the two wires adjusted, the current in each wire (one of the wires) that gives a force between them of 2*10^-7 newton per metre of the wires, is defined as 1 A of current.

Kelvin

The fraction 1/9273.16) of the thermodynamic temperature of triple point of water is called 1 K.

Mole

The amount of a substance that contain as many elementary entities (molecules or atoms if the substance is monatomic) as there are number of atoms in 0.012 kg of carbon-12 is called a mole.

Candela

It is the luminous intensity of a blackbody of surface area 1/[600,000] m² placed at the temperature of freezing platinum and at a pressure of 101,325 N/m², in the direction perpendicular to its surface.

Dimensions of physical quantities and Dimensional Formula

When a physical quantity is expressed in terms of the base quantities, it is written as a product of different powers of the base quantities.

[Force] = MLT-2

M,L, and T are base quantities – mass, length and time

The exponent of a base quantity that enters into the expression (MLT-2) is called the dimension of the quantity in that base.

The dimensions or force are 1 in mass, 1 in length and -2 in time. The dimensions of all other base quantities are zero.

The base quantities are denoted as follows in writing expressions. The symbols used are M for mass, L of length, T for time, I for current, K for temperature, mol for mole and cd for candela.

The physical quantity that is expressed in terms of the base quantities is enclosed in square brackets to inform that it is expressed in dimensions of base quantities. Such expression of a physical quantity in terms of dimensions of base quantities is called the dimensional formula.

Order of magnitude

The structure of the world

## Sunday, July 13, 2008

### The Nucleus - Laws

N = N0e- λ t

where
N = number of active nuclei at time t

N0 = number of active nuclei at t = 0.

λ = decay constant

-dN/dt = λN

-dN/dt give the number of decays per unit time and is called the activity (A) of the sample

A = λN

A = A0e- λ t

Lawson criterion for fusion reactor

In order to get an energy output greater than the energy input, a fusion reactor should achive

n τ >10^14 s/cm³

where
n = the density of the interacting particles

τ = confinement time

the quantity n τ in s/cm³ is called Lawson number

### X-rays - laws and theories

Moseley’s law

Moseley measured the frequencies of characteristic X-rays from large number of elements and plotted the square root of th frequency against its position number in the periodic table.

The observations can be mathematically expressed as

√(v) = a(Z-b)

v = the frequency of characteristic X-rays from the elements\
Z = atomic number
a, b are constants

Bragg’s law

2d sin θ = n λ

d = interplanar spacing of the crystal on which X-rays are incident
θ = is the incident angle at which X-rays are strongly reflected.
n = 1,2,3 …
λ = wave length of X-rays

Application of Bragg’s law:By using a monochromatic X-ray beam (having a single wave length) and noting the angles of strong reflection, the interplanar spacing d and several information about the structure of the solid can be obtained.

### Bohr’s model - laws and models

Thomson model

Lenard’s suggestion

Rutherford’s model

Balmer’s equation for wavelengths of hydrogen atomic spectra

Rydberg equation for wavelengths of hydrogen atomic spectra

Bohr model

The proposals of Bohr are termed as postulates

1. The electron revolves round the nucleus in circular orbits.

2. the circular orbits take only some special values of radius. In these orbits, the electron does not radiate energy, even though it is rotating around the nuclues. Radiation is expected from Maxwell's laws. But Bohr's conception is that electrons in the special radius orbits do not radiate energy.

3. The energy of the atom has a definite value when electrons of the atom are in specified stationary orbits. The electrons can jump from one stationary orbit another. If an electron jumps from an orbit of higher energy E2 to an orbit of lower energy E1, it emits a photon with energy equal to E2-E1.

The wavelength of the radiation of the photon (in wave nature theory) will be determined according to E2-E1 = hc/λ.

The electron can absorb energy from some source and jump from a lower energy orbit to a higher energy orbit also.

4. In the stationary orbits (orbits are stationary not the electrons) the angular momentum l of the electron around the nucleus is an integral multiple of the Planck constant h divided by 2π,

l = nh/2π

The last postulate is called Bohr's quantization rule.

Schrodinger’s wave equation

Pauli exclusion principle

### Photoelectric effect and dual nature - Laws

Einstein’s photoelectric equation

Kmax = hc/λ – φ = h υ - φ

Where
Kmax = Maximum kinetic energy of an electron that comes out due to collision with a photon
h = Planck constant 6.626*10^-34 J-s or 4.136*10^-15 eV-s
c = speed of light in vacuum 299,792,458 m/s or 3*10^8 m/s
λ = wave length of light represented by photon
φ = work function of the material from which the electron is coming
υ = frequency of the wave of light represented by photon

de Broglie wavelength equation for matter waves

λ = h/p

where
λ = wave length
h = Planck constant 6.626*10^-34 J-s or 4.136*10^-15 eV-s
p = momentum of the particle

### Huygen’s principle

Various points of an arbitrary surface, when reached by a wavefront, become secondary sources of light emitting secondary wavelets. The disturbance beyond the surface results from the superposition of these secondary wavelets.

(chapter: light waves)

### Sound waves – Laws and Theories

Revision notes

Newton’s formula for speed of sound in a gas

v = √(P/ρ)

The density of air at temperature 0°C and pressure 76 cm of mercury column is ρ = 1.293 kg/m³

So P = .76m*(13.6*10^3 kg/ m³)*(9.8 m/s²) = 101292.8

Hence P/ ρ = 78339.37
√(P/ρ) = 279.8917 m/s

The velocity of sound in air comes as 280 m/s.

But the measured value of speed of sound in air is 332 m/s

Laplace suggested a correction. With Laplace’s correction the formula is

v = √( γ P/ρ)

where γ = Cp/Cv (Cp and Cv are molar heat capacities at constant pressure and constant volume respectively)
With this new formula the value comes out to be 331.1723 closer to 332 m/s.

Doppler effect

The apparent change in frequency of the wave due to motion of the source or the observer is called Doppler Effect.

Mach number

Macn Number = µs/v

µs = speed of source creating the sound wave

v = velocity of sound wave

### Wave Motion - Waves on a String - Laws

Principle of Superposition of waves

When two or more waves simultaneously pass through a point, the disturbance at the point is given by the sum of disturbances each wave would produce in absence of the other wave(s).

Laws of Transverse Vibrations of a String: Sonometer

1. Law of length: the fundamental frequency of vibration of a string (fixed at both ends) is inversely proportional to the length of the string provided its tension and its mass per unit length remain the same.

v α 1/L if F and µ are constant.

2. Law of tension; the fundamental frequency of a string is proportional to the square root of its tension provided its length and the mass per unit length remain the same.

v α √(F) if L and µ are constant.

3. Law of mass: The fundamental frequency of a string is inversely proportional to the square root of the linear mass density, i.e., mass per unit length provided the length and the tension remain the same.

v α 1/ √(µ) if L and F are constant.

### Permanent magnets - Laws

Tangent law of perpendicular fields

External magnetic field B may be written in terms horizontal component of earth’s magnetic field BH and deflection of compass needle θ.

B = BH tan θ

Gauss’s law for magnetism

B.dS = µ0*minside0

But minside0 is zero as we do not have an isolated magnetic pole in nature. The smallest unit of the source of magnetic field is a magnetic dipole where the net magnetic pole is zero.

Hence Gauss’s law for magnetism states that

B.dS = 0

### Magnetic field due to a current - Laws

Magnetic field due to a current

Biot Savart Law

dB = (µ0/4 π)*(idl sin θ/r²)

Ampere’s law

The circulation ∫B.dl over closed surface = µ0*i

### Gauss’s law

Gauss’s law

The flux of the net electric field through a lcosed surface equals the net charge enclosed by the surface divided by ε0

E.ds over the closed surface = charge enclosed by the surface/ ε0

### Electromagnetic Waves - Laws

Revision notes

Maxwell’s displacement current

Maxwell generalised Ampere’s law to

B.dl = µ0(i + id)

id = ε0*(d ΦE/dt)

Where

ΦE/ = the flux of the electric field through the area bounded by the closed curve along which the circulation of B is calculated.

Maxwell termed id as displacement current

Maxwell’s Equations

Gauss’s laws for electricity and magnetism, Faraday’s law and Ampere’s are collectively known as Maxwell’s equations

### Electromagnetic induction - Laws

Whenever the flux of magnetic field through the area bounded by a closed conducting loop changes, an emf is produced in the loop. The emf is given by

E = -dΦdt

Where
Φ = ∫B.dS is the flux of the magnetic field through the area.

Lenz’s law

The direction of the induced current is such that it opposes the change that has induced it.

### Paschen’s law

V = f(pd)

V = sparking potential of a gas

P = pressure of the gas

D = separation between the electrodes

### Coulomb’s law

Force exerted by a charged particle on the other charged particle is

F = kq1q2/r²

k = 8.98755*109 N-m²/C²
q1,q2 = charges
r = distance between them

### Equipartition of Energy

Equipartition of energy states that the average of energy of a molecule in a gas associated with each degree of freedom is ½ kT where k is the Boltzmann constant and T is its absolute temperature.

(Chapter; Sp Heat capacities of gases)

## Saturday, July 12, 2008

### Heat Transfer Laws

Prevost Theory of Exchange

According to this theory all bodies radiate at all temperatures. The amount of thermal radiation radiated per unit time depends on the nature of the emitting surface, its area and its temperature.

Every body absorbs part of the thermal radiation emitted by the surrounding bodies when this radiation falls on it.

If a body radiates more than what it absorbs, its temperature falls.

Kirchoff’s law

The ratio of emissive power to absorptive power is the same for all bodies at a given temperature and is equal to the emissive power of a blackbody at that temperature.

Thus
E(body)/a(body) = E(blackbody)

Stefan-Boltzmann Law

The energy of thermal radiation emitted by per unit time by a black body of surface area A is given by

U = σAT4

Where
σ = Stefan Boltzmann constant = 5.67*10-8 W/m²-K4

Newton’s law of cooling

dT/dt = -bA(T-T0)

b = a constant depends on the nature of the surface involved
a = surface area exposed of the body
T- T0) = temperature difference between the body and surrounding

### Mechanical equivalent of heat

Mechanical equivalent of heat

W = JH

### Principle of Calorimetry

Principle of Calorimetry

Neglecting any heat exchange with the surroundings, the principle of calorimetry states that the total heat given by the hot objects equals the total heat received by the cold objects.

### Laws of reflection

Laws of reflection

1. The angle of incidence is equal to the angle of reflection.
2. The incident ray, the reflected ray and the normal to the reflecting surface are coplanar.

(Chapter Geometrical optics)

### Huygen’s principle

Huygen’s principle

Various points of an arbitrary surface, when reached by a wavefront, become secondary sources of light emitting secondary wavelets. The disturbance beyond the surface results from the superposition of these secondary wavelets.

(chapter: light waves)

### Mach number

µs/v

µs = speed of source creating the sound wave

v = velocity of sound wave

### Doppler effect

Doppler effect

The apparent change in frequency of the wave due to motion of the source or the observer is called Doppler Effect.
(chapter: sound waves)

### Newton’s formula for speed of sound in a gas

Newton’s formula for speed of sound in a gas

v = √(P/ρ)

The density of air at temperature 0°C and pressure 76 cm of mercury column is ρ = 1.293 kg/m³

So P = .76m*(13.6*10^3 kg/ m³)*(9.8 m/s²) = 101292.8

Hence P/ ρ = 78339.37
√(P/ρ) = 279.8917 m/s

The velocity of sound in air comes as 280 m/s.

But the measured value of speed of sound in air is 332 m/s

Laplace's Correction

Laplace suggested a correction. With Laplace’s correction the formula is

v = √( γ P/ρ)

where γ = Cp/Cv (Cp and Cv are molar heat capacities at constant pressure and constant volume respectively)
With this new formula the value comes out to be 331.1723 closer to 332 m/s.

### Laws of Transverse Vibrations of a String: Sonometer

1. Law of length: the fundamental frequency of vibration of a string (fixed at both ends) is inversely proportional to the length of the string provided its tension and its mass per unit length remain the same.

v α 1/L if F and µ are constant.

2. Law of tension; the fundamental frequency of a string is proportional to the square root of its tension provided its length and the mass per unit length remain the same.

v α √(F) if L and µ are constant.

3. Law of mass: The fundamental frequency of a string is inversely proportional to the square root of the linear mass density, i.e., mass per unit length provided the length and the tension remain the same.

v α 1/ √(µ) if L and F are constant.

### Principle of Superposition of waves

When two or more waves simultaneously pass through a point, the disturbance at the point is given by the sum of disturbances each wave would produce in absence of the other wave(s).

(Chapter; Wave motion - String)

### Stoke’s law - Reynold's Number

Stoke’s law

Suppose that a spherical body of radius r moves at a speed v through a fluid of viscocity η. The viscous force F acting on the body is:

F = 6 π ηrv

Reynold’s number

The quantity

N = ρvD/ η

Where
ρ = density of the liquid
v = velocity of the liquid
D = diameter of the tube through which the liquid is flowing
η = coefficient of viscocity of the liquid

is called Reynold’s number.

If the Reynold’s number is less than 2000, the flow is steady.
If the Reynold’s number is more than 3000, the flow is turbulent
If the Reynold’s number is more than 2000 but less than 3000, the flow is unstable.

### Poiseulle’s equation

Poiseuille derived a formula for the rate of flow of viscous fluid through a cylindrical tube.

The formula is

V/t = (πPr4)/8ηl

Where
V = volume flowing in time t
t = time
P = pressure difference in the liquid at the two ends
r = radius of cylindrical tube
l = length of cylindrical tube
η = coefficient of viscocity of the fluid

(Chapter Fluid mechanics)

### Hooke’s law

If the deformation is small, the stress in a body is proportional to the corresponding strain.

By Hooke’s law, for small deformations,

Tensile stress/Tensile strain = Y is a constant for a given material
Y is termed as Young’s modulus.

Y = (F/A)/ (ΔL/L) = FL/(A ΔL)

(chapter: Mechanical Properties)

## Friday, July 11, 2008

### Bernoulli’s equation

Revison material

Bernoulli’s equation relates the speed of a fluid at a point, the pressure at that point and the height of that point above a reference level. It is an application of work-energy theorem in the case of fluid flow.

P1 + ρgh1 + ½ ρv1² = P2 + ρgh2 + ½ ρv2²

P + ρgh + ½ ρv² = constant

Where
P = pressure
ρ = density of liquid
g = acceleration due to gravity
h = height of above the reference level
v = velocity of the fluid

### Equation of continuity

Revision material

The product of the area of cross-section and the speed remains the same at all points of a tube of flow.

A1v1 = A2v2

A1, A2 = areas of cross section
v1, v2 = speed of fluid at A1 and A2

Equation of continuity expresses the law of conservation of mass in fluid dynamics.

(Chapter: Fluid mechanics)

### Archimedes’ principle

It states that when a body is partially or fully dipped into a fluid at rest, the fluid exerts an upward force of buoyancy equal to the weight of the displaced fluid.

### Pascal’s Law

If the pressure in a liquid is changed at particular point, the change is transmitted to the entire liquid without being diminished in magnitude.

(chapter: Fluid mechanics)

### Kepler’s Laws of Planetary Motion

1. All planets move in elliptical orbits with the sun at a focus.

2. The radius vector from the sun to the planet sweeps equal area in equal time.

3. The square of the time period of a plant is proportional to the cube of the semimajor axis of the ellipse.

(Chapter: Gravitation)

### Theorems related to Moment of Inertia

Theorem of Parallel Axes

We have to obtain the moment of inertia of a body with mass M and with the centre of mass at C about a given line AB. Visualise a line CZ parallel to AB through C.

Let I0 be the moment of inertia of the body about CZ respectively.
If the perpendicular distance between AB and CZ is d
Then I, the moment of inertia of the body about AB is going to be

I = I0 + Md²

Theorem of Perpendicular axes

This theorem is applicable only to the plane bodies. Let X and Y axes be chosen in the plane of the body and Z=axis perpendicular to this plane, and the three axes are mutually perpendicular.

Ix, Iy, Iz are moment of inertia of the body about x,y, and z-axes respectively.

According to the theorem
Iz = Ix + Iy

(Chapter: Rotational Mechanics)

### Work-energy theorem

The work done on a particle by the resultant force is equal to the change in its kinetic energy.

Principle of conservation of energy

The total mechanical energy of a system remains constant if the internal forces are conservative and the external forces do not work.
(The sum of the kinetic energy and the potential energy is called the total mechanical energy.)

Generalised law of conservation of energy

Energy can never be created or destroyed, it can only be changed from one form to another.

(Chapter: Work, power, energy)

## Thursday, July 10, 2008

### Law of Malus

If a linearly polarized light is incident on a Polaroid with the E-vector (electric field vector) parallel to the transmission axis, the light is completely transmitted by the Polaroid. If the E-vector (electric field vector) perpendicular to the transmission axis, the light is completely stopped by the Polaroid. If the E-vector (electric field vector) is at an angle θ to the transmission axis, the light is partially transmitted. The intensity of partially transmitted light (by the Polaroid) is

I = I0cos²θ

Where
I0 is the intensity when the incident E-vector is parallel to the transmission axis.

(Polaroids have long chains of hydrocarbons which become conducting at optical frequencies. When light falls perpendicularly on the sheet, the electric field parallel to the chains is absorbed but the field perpendicular to the chains gets transmitted. The direction perpendicular to the chains is called the transmissiona axis of the Polaroid.

(Chapter Light waves)

### Snell’s law

The law of refraction: the refracted ray lies in the plane of incidence.

Snell’ law is

n1 sin θ1 = n2 sin θ2

Where

n1 and n2 are the index of refraction of medium 1 and medium 2.
The index of refraction of a medium is the ratio between the speed of light c in vacuum and the speed of light v in that medium

n = c/v

θ1, θ2 are angles of incidence and angle of refraction in the medium 1 and medium 2.

(Chapter: Light waves)

### Laws of friction

1. If the bodies slip over each other, the force of friction is given by

fk = μ kN

Where
fk is the force of friction
N is the normal contact force
μ k is the coefficient of friction between the surfaces

2. The direction of kinetic friction on a body is opposite to the velocity of this body with respect to the body applying the force of friction.

3. If the bodies do not slip over each other, the force of friction is given by

fs = μ sN

where

fs = static force of friction
μ s = coefficient of static friction between the bodies
N = normal force between them

The direction and magnitude of static friction are such that the condition of no slipping between the bodies is ensured.

4. The frictional force fk or fs does not depend on the areas of contact. It depends on normal contact force only.

(Chapter: friction)

## Wednesday, July 9, 2008

### Joules Laws Heating Due to Current in a Resistor

1. The heat produced in a given resistor in a given time is proportional to the square of the current in it

2. The heat produced in a given resistor by a given current is proportional to the timwe for which the current exists in it.

The heat produced in a given resistor by a given current in a given time proportional to its resistance

(Topic: thermal and chemical effects of electric current)

## Monday, July 7, 2008

### MaxWell's Speed Distribution Law

MaxWell's Speed Distribution Law

It is an equation giving the distribution of molecules in different speeds in a gas at a temperature.

If dN represents the number of molecules with speeds between v and v+dv then

dN = 4πN[m/2πkT]3/2v²e-mv²/2kTdv

where
dN represents the number of molecules with speeds between v and v+dv
N = total number of molecules in the gas
m = mass of a molecule
T = absolute temperature of the gas
v = velocity of the molecules

The speed vp at which dN/dv is maximum is called the most probable speed.
Its value is given by

vp = √(2kT/m)

(Topic: Kinetic theory of gases)

### Laws of Thermodynamics

Zeroth law: If two bodies A and B are in thermal equilibrium and A and C are also in thermal equilibrium then B and C are also in thermal equilibrium.

First law of thermodynamics

∆U = ∆Q -∆W
Where
∆U = change in internal energy of a thermodynamic system
∆Q = Heat given to the system
∆W = work done by the system

Change in internal energy of a thermodynamic system is equal to the heat given to the system minus the work done by the system on surroundings or environment.

Second law of thermodynamics

Kelvin-Planck statement

It is not possible to design a heat engine which works in cyclic process and whose only result is to take heat from a body at a single temperature and convert it completely into mechanical work.

(Topic: Laws of Thermodynamics)