Diagonal Method of Squaring

DIAGONAL METHOD OF SQUARING

This is also an old Indian Method of Multiplying Numbers. The diagonal Method can be applied to find the square of any number irrespective of the number of digits.
let us learn the Diagonal Method of Squaring numbers as given below :-

WORKING RULES  :-

Here we find out the square of - 93

Step 1 -   First , draw a square . Then , divide it into sub - squares . The number of sub - squares will be equal to the number of digits of the given number.

Step 2 - Draw the diagonals of each square and write the digits of the number to be squared from left to right and top to bottom.

Step 3 - Multiply each digit from left to right of the square by each digit one by one from top to bottom of the column.

Step 4 - Write the product in the corresponding sub -squares.

Step 5 - If the number so obtained is a one digit number, write it below the diagonal.If it is a two digit numbers , write the tens digit above the diagonal and the ones digit below the diagonal.

Step 6  - Fill each empty places with the number 0 .

Steps 7 - Starting from the lowest diagonal , add the digits along the diagonals so obtained , underline the ones digit of the sum and take tens digit as carry (if any) and add to the sun in the next above diagonal.

Step 8 - Ones digits so underlined together will be the required square of the given number.

One more example  - Square of 345

Square of 345 is - 119025

Success is no accident . Put your heart , mind and soul into even smallest acts..:)

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Physics PART - 3

PHYSICS - PART - 3

Nicola Tesla (1856 - 1943 )was a key figure in the development of alternating current electricity,high-voltage transformers, and the transport of electrical power using AC transmission lines. Tesla's viewpoint was not in agreement with the idea of Edison , who committed himself to the use of direct current in power transmission .

 Tesla's approach won out !

Formulas : Alternative Current

1 - Instantaneous value of current → I = I₀ Sin ⍵t  = I₀ Sin(2πvt)    ( I is the instantaneous value of alternating current at an instant I and I₀ is its Peak value . ⍵ is the angular frequency , v is the frequency.)

2 - Instantaneous value of voltage → ɛ = ɛ₀ Sin ⍵t  = ɛ₀ Sin (2πvt)   ( ɛ is the instantaneous value of alternating voltage at instant t and ɛ₀ is its Peak value .)

3 - Root mean square value → ɛᵣₘₛ → ɛ₀ / √2   and  Iᵣₘₛ = I₀ / √2       (ɛᵣₘₛ and Iᵣₘₛ stand for the rms , virtual , effective values of ac voltage and current respectively .)

4 - Average OR Mean values → ɛₘ = 2ɛ₀ / π   and   Iₘ = 2I₀ / π       ( ɛₘ  and Iₘ  are the average OR mean values of ac voltage and current for half cycle respectively .)

5 - Inductive reactance  →  𝒳ⳑ = ⍵L = 2πvL       ( 𝒳ⳑ is the inductive reactance of an inductance L in an ac circuit of frequency v , In an ac circuit , containing inductance , current lags behind the voltage .by π / 2.)

6 - Capacitive reactance → 𝒳𝑐 = 1/⍵C  = 1 / 2πvC   ( 𝒳𝑐 is the capasitive reactance of a capacitance C in an ac circuit , containing inductance , current leads the voltage by π / 2 .)

7 - Impedance of series LCR circuit → Z = √ R² +  (𝒳ⳑ - 𝒳𝑐 )²   = √ R² + ( ⍵L - 1/⍵C )²     (Z is the impedance of series LCR circuit . Resistance R .)

8 - Phase angle  → tan ɸ = ( 𝒳ⳑ - 𝒳𝑐 )/ R  = (⍵L - 1 / ⍵C ) / R     ( ɸ is the phase angle between current and voltage . The sign of ɸ can be positive or negative depending upon whether 𝒳ⳑ is greater or less than 𝒳𝑐 .The phase angle is zero when 𝒳ⳑ = 𝒳𝑐 .)

9 - Net potential difference series LCR circuit → ɛ₀ = √ V²ʀ +  (Vⳑ - V𝑐)²   ( ɛ₀ is the net potential difference across series LCR circuit and Vʀ , Vⳑ and V𝑐 are the pds across R , L and C respectively.)

10 - Average power → Pₐᵥ = ɛᵣₘₛ Iᵣₘₛ cosɸ = I²ᵣₘₛ R    ( Pₐᵥ is the average power delivered by the generator in an LCR circuit . The term cosɸ is called the power factor.)

Pₐᵥ = ɛᵣₘₛ Iᵣₘₛ    ( In case of an Ideal inductor or a capacitor , ɸ is numerically equal to π / 2 and as such P = 0 . Thus , there is no power loss in ideal inductor or a capacitor .)

11 - Resonance frequency → vᵣ = 1 / 2π√LC    ( vᵣ is the resonance frequency of a LCR circuit.)

12 - Quality factor → Q = ⍵ᵣL /R  = 1 / ⍵ᵣCR  = 1/R √L/C    (Q is called the quality factor of an LCR circuit .)

13 - Band Width  → Q = ⍵ᵣ / 2Δ⍵    ( Δ⍵ = R / 2L  is called the band width of the circuit .)

14 - In case of ideal transformer → ɛₚ Iₚ = ɛₛ Iₛ
ɛₛ / ɛₚ = Iₚ / Iₛ = Nₛ / Nₚ
Rₛ / Rₚ = ( Nₛ / Nₚ )²            ( p and s stand for primary and secondary of the transformer respectively.)

Formulas : Electromagnetic Waves

Understanding about the production , propagation and absorption ,of electromagnetic waves has opened the door to modern methods of communication.

The major asymmetry arise due to the fact that there are no "magnetic monopoles " corresponding to "electric charges ". Dissatisfaction with such broken symmetries in nature has often led physicists to important discoveries and to profound insights into the nature of universe.We can see why physicists have not given up the search for magnetic monopoles.

1 - Displacement current → Iɗ = ∊₀ dɸₑ / dt = ∊₀ A dE/ dt = C dV/ dt    ( Iɗ is displacement current . dɸₑ / dt is the rate of change of electric flux , dE / dt is the rate of change of electric field and dV / dt is the rate of change of potential difference .)

2 - The sinusoidally varying electric and magnetic fields → E𝗒 = E₀ Sin ( kx - ⍵t )

B𝗓 = B₀ Sin ( kx - ⍵t )    ( E and B are the plane electromagnetic wave propagating along X - axis. E₀ and B₀ are the maximum values of E and B .

 k (propagation constant ) = 2π /λ

⍵ ( angular frequency ) = 2πv

3 - Speed of Electromagnetic wave in vacuum  → c = 1 /√∊₀𝜇₀ 

speed of electromagnetic wave in medium → v = 1 / √∊𝜇

4 - Instantaneous electric field energy density → uᴇ = 1/2 ∊₀ E²

Instantaneous magnetic field energy density → uʙ = B² / 2𝜇₀       (uᴇ = uʙ , E is the Electric field , B is magnetic field . ∊₀ and 𝜇₀ is electric and magnetic permittivity in vaccum.

5 - Total average energy density → uₐᵥ = 1/2 ∊₀E₀² = B²/2𝜇₀ = 1/2 (E₀B₀ / 𝜇₀c ) = (Eᵣₘₛ Bᵣₘₛ) / 𝜇₀c    ( total average density associated with electromagnetic wave. )

6 - Intensity of electromagnetic wave → I = uₐᵥc = 1/2 c ∊₀ E² = E₀² / 2𝜇₀c = cB₀²/ 2 𝜇₀ = 1/2 (E₀ B₀/  𝜇₀)  = Eᵣₘₛ Bᵣₘₛ / 𝜇₀
I = power/area = (energy/time) / area =( U /t ) / A = U / At   (U is the energy carried by the wave.)

The history of science is filled with many unforeseen surprises . nobody seriously would have expected young Michael Faraday to be anything but an errand boy or book binder , even if he managed to escape an early death from starvation or illness. 

The idea that he might be a scientist or educator was ridiculous and to predict that he would produce the most fundamental changes in the basic theories of Physics, to follow those oh Sir Issac Newton, would have been sheer madness .

Michael Faraday received no formal training in science and mathematics and showed no signs of genius or ability during his very limited formal education . He was the first to use the consept of a field and to picture field lines , which he referred to as "line of force".This approach was to flourish in the hand of James Maxwell , who possessed the mathematical education that Faraday lacked. 

Maxwell read Faraday's work  , as well as the more mathematical work of others . His synthesis resulted in the unification of electricity and magnetism , using the first field theory . Field theory has been an important part of theoretical physics ever since.


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Physics Part - 2

PART - 2

This session makes a great deal of formulae , connecting thread of ( PART - 1 ) . This part consist of laws of electric and magnetic field. If you want to be a rock and Roll star follow ME..:)

1 - Kirchhoff"s law → ΣI = 0  ( it follows from conservation of charge.) [ LOOP RULE]

Σε = ΣRI  ( it follows from law of conservation of energy )  [ JUNCTIONS LAW ]

2 - Wheatstone bridge → P/Q = R/S  ( is the condition for a balanced Wheatstone bridge .)

3 - Principal of a potentiometer → V ∝ l ( that is fall of potential across a wire is directly proportional to the length of the wire.)

4 - Comparison of emf of two cells →Ɛ₁ / Ɛ₂ = l₁/ l₂ ( Ɛ₁ and Ɛ₂ are the emfs of two cells and l₁ and l₂ are the corresponding balancing lengths of the potentiometer wire .)

5 - Internal resistance of a cell → r = [ (l₁ - l₂ ) / l₂ ] R  ( l₁ and l₂ are the balancing lengths without and with a resistance R connected to the cell.)

6 - Electric energy → VIt  ( V is the potential difference and t is the for which current I flows .)

7 - Heat →H = VIt/4.2 = I²Rt/4.2  ( Heat produced in calorie , when a current I flows through a resistor of resistance R.)

8 - Electric power → Pʀ = VI = I²R = V²/ R  ( Pʀ is the power dissipated in a resistor of resistance R when a potential difference V applied across its ends and as a result , a current I flows through it .)

9 - Power output of a source → Ps → ƐI   ( Ps is power output of a source of emf Ɛ and I is the current delivered by it .)

10 - Lorentz force → Fₘ = q ( v ⨯B )  ( where v and B is in vector form.)
Fₘ = qvB sinθ ( Fₘ is the magnetic Lorentz force acting on a charged particle of charge q and moving with a velocity v in a magnetic field B . θ is the angle between v and B.)

11 - Cyclotron radius → r = mv/qB  ( r is the radius of the circular path which a charged particle having mass m follows when moving with a velocity v initially at right angle to the magnetic field B . This radius is called Gyroradius or cyclotron radius.)

12 - Cyclotron frequency → v = qB / 2 𝝅m

13 - Kinetic energy → K = q²B²R² / 2m  ( K is the maximum kinetic energy of the charged particle of charge q and mass m emerging out of a cyclotron with dee radius R , B is the magnetic field perpendicular to the plane of the dees.)

14 - Force acting on a straight conductor → F = IlB  ( F is a force acting on a straight conductor of length l and carrying a current I when placed in a uniform external magnetic  field B . l is the direction of current .)

15 - Torque → ꞇ = NIABsin θ   ( ꞇ Acting on a coil of N turns of Area A and carrying a current I is when placed in a uniform magnetic field B  here A is perpendicular to the plane of the coil and θ is the angle between A and B .)
ꞇ = NIABcosα  ( α is the angle between B and the plane of the coil.)

16 - Torque → ꞇ = m x B   ( ꞇ is the torque acting on a current loop placed in a magnetic field B and m is the magnetic dipole moment of the current loop that is current carrying loop m = NIA.)

17 - Current sensitivity → CS = θ/ I = NAB/k
Voltage sensitivity of a galvanometer → VS = θ/V = NAB / kR = CS / R   ( k is called the torsional constant and R is the resistance of the galvanometer .)

18 - Shunt resistance → S = Iℊ G/ ( I - Iℊ) = G / ( n - 1)  ( S is the shunt resistance  require to convert a galvanometer of resistance G into a ammeter to measure a current I . Iℊ is the current that can safely flow through the galvanometer without damaging it or current which produces full scale deflection in the galvanometer .Iℊ also stands for the given current range and I is the required current range.)

19 - Biot - Savart Law → B = kₘ ( Il x r )/ r³  ( B is the magnetic field at any point due to a current element l carrying a current I . r is the vector from the current element to the point. kₘ is the magnetic constant. )

20 - Magnetic field → B = kₘ (2l/a) = μ₀ I/2πa  ( B is the magnetic field due to an infinitely long conductor at a point distance 'a' from it .)
B = kₘ (I/a) = μ₀I/ 4πa  ( When the point lies at a distance 'a' near one end of an infinitely long conductor .)

21 - Force per unit length between two straight parallel infinite conductor → f = kₘ (2 I₁I₂ ) / a = (μ₀I₁I₂)/ 2πa   ( The force is attractive if the currents are in the same direction and repulsive if they are in opposite directions.)

22 - Magnetic field at the center of a coil → B = kₘ (2πNI)/r = (μ₀NI)/2r      (Magnetic field at the center of a coil of radius r and N turns through which I current flows .)

23 - Magnetic field due to a coil at a point on its axis → B = kₘ (2πNI∮R² )/ (R² + x²)³/²
= (μ₀NIR²)/ 2(R² + x²)³/²   ( B is the magnetic field due to a coil N turns radius R and through which a current I flows at a point on its axis distance x from the center .)

24 - Magnetic field inside a long solenoid → B = μ₀ nI  ( B is the magnetic field inside a long solenoid carrying a current I at points near its center and n is the number of turns per unit length .)
B = (μ₀nI)/2  ( B is the magnetic field at a point situated at one of the ends of a long solenoid.)

25 - Magnetic field inside a tortoid → B = (μ₀NI )/ 2πr     ( B is the magnetic field inside a tortoid of a radius r having N turns .)

26 - Ampere's Circuital Law → ∮ B.dl = μ₀ I   ( The line integral in the equation is evaluated around a closed loop called an Amperean loop and I is the net current encircled by the loop.)

27 - Magnetic Flux → ɸʙ = BS   (ɸʙ is the magnetic flux linked with a surface of area S placed perpendicular to a uniform magnetic field B)

28 - Gauss's law of magnetism → ∮B.dS = 0

29 - Magnetic dipole moment → m = qₘ x 2a = 2qₘa   ( m is the magnetic dipole moment of a magnetic dipole whose each pole has a strength qₘ.)

30 - Magnetic field due to a dipole on its axial line → Bₐₓᵢₐₗ→ kₘ (2mr)/ (r² - a²)²
For a short dipole → Bₐₓᵢₐₗ = kₘ 2m/r³    ( Bₐₓᵢₐₗ is the magnetic field due to a dipole on its axial line , at a point distance r from its center.)

31 - Magnetic field due to a dipole on its equatorial line → Bₑ𝔮ᵤₐₜₒᵣᵢₐₗ → kₘ = m/(r² + a² )³/²
Bₑ𝔮ᵤₐₜₒᵣᵢₐₗ = kₘ m/r³  ( for a short dipole )

32 - Magnetic potential energy → Uʙ = - m . B = - mBcosθ  ( Uʙ is the magnetic potential energy of the dipole in an external uniform field B making an angle θ with B .)

33 - Time period → T = 2π✓I /mB   ( T is the time period of a freely suspended magnet of magnetic moment m and moment of inertia I in a magnetic field B .)

34 - Horizontal component → Bʜ = B cos 𝛿,       Bv =Bsin𝛿    and  B = ✓B²ʜ + B²v      ( B is the resultant magnetic field at a place on earth with dip 𝛿  and Bʜ , Bv are the horizontal and the vertical components of B .)

35 - Magnetic susceptibility → 𝒳ₘ = M/H   ( 𝒳ₘ is called the magnetic susceptibility , which is a dimensionless quantity and has no units .)

36 - Induced emf  →  ε = - dФʙ / dt     ( ε is the induced emf set up in a circuit when the rate of change of flux is dФʙ / dt .)

37 - Induced emf set up across the ends of a conductor →  ε = Blv   ( ε is the induced emf set up across the ends of a conductor of length l when moving in a magnetic field B with a velocity v and in a direction perpendicular to B.)

38 - Faraday's law in general form →  ε = ∮ E .dl   ( E is a non conservative , time varying electric field that is produced by the changing magnetic flux .)
The essence of the law is that a changing magnetic flux induces an electric field.
ε = 1/2 B⍵l²  ( ε is the emf induced in a rod of length l rotating with constant angular speed ⍵ in a uniform magnetic field .)
ε = NAB⍵sinθ  ( ε is the induced emf in the rotating armature of a dynamo whose total number of turns is N ,area is A , B is the magnetic induction ,⍵ is the angular velocity and θ is the angle between B and A .)

39 - Current → I = (V - ε ) /R    ( I is the current through the armature coil of a dc motor of resistance R . Here V is the applied voltage and ε is the back emf .)

40 - Efficiency of dc motor →    𝜼 = ε / V

41 - Self induction → L = Φʙ / I  ( L is the self induction of the coil and Фʙ is the magnetic flux linked with it when a current I flows through it.)

42 - Coefficient of self induction → ε = L ( dl/dt)    ( ε is the induced emf set up in a coil when the rate of change of current in the coil is dl/dt . L is the coefficient of self induction or self inductance or simply the inductance of coil . Inductance is the measure of the opposition of a device to a change in current.)

43 - Inductance of a solenoid → L = (μ₀N²A / l )   ( L is the inductance of a solenoid whose core is a vacuum ), N is the total number of turns , A is the cross sectional area  and l is the length of a solenoid .)

44 - Mutual induction → Фʙ = MI    ( M is the mutual induction of two coils , I being the current through one coil and Фʙ is the flux linked with the other coil.)

45 - Energy stored in the magnetic field of an inductor → Uʙ = 1/2 LI²

46 - Magnetic energy density → uʙ = B²/ 2μ₀  ( uʙ is the magnetic energy density at a point where the magnetic field is B . Thus the energy density is proportional to the square of the field at that point. )

Physics is about questioning , studying , probing nature .You probe and if you are lucky , you get strange...

well..!! wait for next session..till than bye bye..:)

Note : For any queries or suggestion please post a comment .

PHYSICS in our life..!!

PHYSICS➨PART- 1

Nothing happens until something moves

Well this session consists of  PHYSICS .  This part consist Physical constant and formulae . I tries to tell ,in such a way as to be understood by everyone.The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them.

Physical Constants

1 - Average acceleration of gravity (g) →                9.81 m/s²
2 - Gravitational constant (G) →                             6.67× 10 ⁻ ¹¹ Nm²/kg²
3 - Mass of electron (mₑ ) →                                     9.11 × 10 ⁻³¹ kg = 0.511 MeV/c²
4 - Mass of proton (mₚ ) →                                      1.673 × 10 ⁻²⁷ Kg = 1.007 u
                                                                                   = 938.3 MeV/c²
5 - Mass of neutron (mₙ ) →                                   1.657 × 10 ⁻²⁷ Kg = 1.009 u
                                                                                = 939.6 MeV/c²  
6 - Speed of light in vaccum ( c) →                       3.00 × 10 ⁸ m/s
7 - Universal gas constant ( R ) →                         8.31 J/mol K
8 - Boltzmann constant ( kʙ ) →                           1.38 × 10⁻²³ J/k
9 - Avogadro constant (Nᴀ) →                              6.02 × 10 ²³ / mol
10 - Permittivity constant ( ∈₀ ) →                       8.85 ×  10⁻¹²  C²/Nm² = 8.85 pF/m
11 - Coulomb constant or
      Electric constant ( kₑ ) →                                1/4 𝝅∈₀ = 8.99 × 10⁹ Nm²/C²
12 - Elementary charge ( e ) →                           1.60 × 10 ⁻¹⁹ c
13 - Permeability constant ( μ₀ ) →                      4𝝅× 10⁻⁷ Tm/A = 1.26 × 10 ⁻⁶ N/A²
14 - Magnetic constant ( kₘ = μ₀ /4𝝅 ) →             10⁻⁷ Tm/A
15 - Planck constant ( h ) →                                 6.63 × 10 ⁻³⁴ Js = 4.14 × 10 ⁻¹⁵ eVs
16 - Bohr radius ( r₀ ) →                                       5.29 × 10⁻¹¹ m = 0.529 Å
17 - Electron volt ( eV ) →                                    1.60 × 10⁻¹⁹ J
18 - Unified atomic mass unit ( amu) ( n) →       1.66 × 10⁻²⁷ Kg
19 - Rydberg constant ( R ) →                             1.097 × 10⁷m⁻¹
20 - Bohr magneton (μ ) →                                  9.274 × 10⁻²⁴ J/T

Formulae of Physical Quantities

1 - Quantisation of charge → q = ne   (q is the charge on a body and n is an integer , it can be positive or negative )

2 - Coulomb's law → F = kₑ q₁ q₂/r² ( F is the force in between two charges q₁and q₂ placed in a vacuum at a distance r )

3 - Relative permittivity  Or Dielectric constant → ∈ᵣ = F/Fₘ  ( ∈ᵣ is the relative permittivity of a medium . Fₘ is the force acting between two point charges q₁ and q₂ placed in a medium at the same distance r apart. )

4 - Absolute permittivity → ∈ᵣ = ∈ /∈₀  ( ∈ is absolutely permittivity of a medium )

5 - Electric field intensity → E = F / q₀  ( E is the electric field intensity and F is the force acting on a test charge q₀ placed in the field )

6 - Electric field due to an isolated point charge → E = kₑ q/r²  ( E is the electric field due to an isolated point charge q at a point distant r from q. )

7 - Electric intensity due to an infinitely long thin wire → E = 2kₑλ/x = λ/ 2𝜋∈₀x  ( E is the intensity due to an infinitely long thin wire at a distance x , λ being the linear charge density of the wire.)

8 - Electric intensity due to a conducting ring → E = kₑ qx/ (R²+x²)³/²  ( E is the electric intensity due to a conducting ring of radius R and carrying a charge q at a point lying on it's axis of symmetry at a distance x from it's center .)

9 - Electric dipole moment → p = q × 2a = 2qa ( p is the electric dipole moment of an electric dipole of each charge of magnitude q , placed a distance 2a apart. )

10 - Electric field intensity due to a dipole on its axial line → E (axial) → kₑ2pr/(r²-a²)²
E (axial) → kₑp/r³  ( E axial is the electric field intensity due to a dipole on the axial line at a point distance r from it's center , p being the dipole moment of the dipole.)

11 - Electric field intensity due to a dipole on its equatorial line → E (equatorial ) = kₑp/ (r²+a²)³/²
E (equatorial) = kₑp/r³ (for a short dipole)  (E equatorial is the electric field intensity due to a dipole on its equatorial line at a point distance r from its center , p being the dipole moment of the dipole.)

12 - Electric field intensity  ➝ E= σ/2∊₀ ( E is the electric field intensity due to an infinite uniformly charged plane sheet of surface charge density σ.)

E = σ /∊₀  ( E is the electric field intensity due to an infinite charged conducting plate of surface charge density σ. )

13 - Torque → ꞇ = p × E  Or  ꞇ = pEsin𝜃  (ꞇ is the torque acting on an electric dipole of dipole moment p lying at angle 𝜃 with the electric field E)

14 - Potential energy → U = - p.E ( p and E both are in vector form )= -pEcosθ       (U is the potential energy of an electric dipole of dipole moment p  lying at  an angle θ with the electric field E.)

15 - Work done → W = 2 pE  ( work done to reverse the dipole , that is turning its end for end.)

16 - Electric flux → Φᴇ = EA = q/∊₀   ( Φᴇ is the electric flux through a surface of area A, placed perpendicular to a uniform electric field E.)

17 - Potential difference → Δ V = Vʙ - Vᴀ = Wᴀʙ / q₀  ( ΔV is the potential difference between points A and B and Wᴀʙ is the work done in carrying a positive charge q₀ from A to B.)

18 - Potential gradient → E = - dV/dr  ( dV/dr is called the potential gradient )
E = V/r  ( E is a uniform electric field .)

19 - Potential energy → U = kₑ ( q₁q₂/x₁ + q₂q₃/x₂ + q₃q₁/x₃)   ( U is the potential energy of a configuration of three charges q₁,q₂,and q₃ placed in vacuum .x₁ is the distance between q₁ q₂ ; x₃ is the distance between q₂ q₃ ; x₃ is the distance between q₃ q₁.)

20 - Capacitance of a conductor → C= Q/V ( C is the capacitance of a conductor having charge Q and a potential V.)

21 - Capacitance of a parallel plate conductor → C = ∊₀A /d  ( C is the capacitance of a parallel plate conductor , A is the area of the plates and d is the separation between them.)
[ A capacitor consists of isolated conductors plates carrying equal and opposite charges +Q₁ , and - Q₂ .)

22 - Capacitance of a spherical capacitors →  C = ab/ kₑ(b - a) = 4𝝅∊₀ab/ (b - a)   (C is the capacitance of a spherical capacitor , a is the radius of the inner charged sphere and b that of the outer earthed shell.)

24 - Resultant capacitance in series combination → 1/Cₛ = 1/C₁ + 1/C₂ +1/C₃  (Cₛ is the resultant capacitance in series combination of capacitors of capacitance C₁, C₂, C₃.)

25 - Resultant capacitance in parallel combination → Cₚ = C₁ + C₂ + C₃  ( Cₚ is the resultant capacitance in parallel combination of capacitors of capacitance C₁ ,C₂ and C₃ )

26 - Electric potential energy stored in a capacitor → U = 1/2CV² = QV/2 = Q²/2C  (U is the electric potential energy stored in a capacitor of capacitance C , having charge Q and at a potential V.)

27 - Energy density → uₑ = 1/2∈₀ E²  ( uₑ is called the energy density that is electric potential energy per unit volume stored in an electric field E.)

28 - Loss of energy → U₁-U₂ = 1/2 (C₁C₂/C₁+C₂)(V₁-V₂)²   ( U₁-U₂ is the loss of energy on sharing the charges ; C₁ and C₂ are the capacitance of two conductors at potential V₁ and V₂.)

29 - Dielectric constant → ∈ᵣ = C/C₀  ( ∈ is the dielectric constant and C and C₀ denote the capacitance of a capacitor with and without dielectric.)

30 - Capacitance of a parallel plate capacitors → C = ∈₀A/ (d-t) + t/∈ᵣ  ( C is the capacitance of a parallel -plate capacitor with a dielectric slab of thickness t between its plates, each of area A and having a separation d.)
C = ∈₀A/d-t  ( C is the capacitance of the parallel - plate capacitor with a conducting slab of thickness t between its plates, each of area A and having a separation d.)

31 - Steady current → I = q/t
Instantaneous current I = dq /dt  ( I is the current due to flow of charge q Or dq through a conductor for a time t Or dt.)

32 - Ohm's Law → V = RI   ( V is the potential difference across the ends of a conductor through which a current I flows . Here R is the resistance of the conductor.)

33 - Conductance → G =1/R = 1/ V  (G is the conductance of a conductor.)

34 - Resistivity → ⍴ = RA/l     =  m/ne²ꞇ     (⍴ is the resistivity of the material .R is the resistance of a conductor of a lenght l , cross sectional area A . n is the number density of electrons , ꞇ is the relaxation time that is mean time between the collision of an electron with the ions of metal lattice , m is the mass of the electron and e its charge.)

35 - Current in terms of drift speed → I = nevdA   ( A is the cross sectional area of a conductor , n is the number density of electrons ( number of electrons per unit volume ) , e is the charge on the electron and vd is the drift speed of the electron when current I flows through the conductor.)

36 - Current density → J = I/A = n e vd  ( J is the current density.)

37 - Electrical conductivity →  σ = 1/⍴ = ne²ꞇ/m   ( σ is electrical conductivity)

38 - Resultant resistance → Rₛ = r₁ +r₂ +r₃   ( Rₛ is the resultant resistance of a number of resistors in series.)
1/Rₚ = 1/r₁ +1/r₂ +1/r₃   ( Rₚ is the resultant resistance of a number of resistors in parallel)

39 - emf of cell → ɛ = dW/dq   (E is the emf of cell and dW is the amount of work that the cell does to force positive charge dq from the negative terminal to the positive terminal.)

40 - Internal resistance → r =(ɛ-V/V)R   ( r is the internal resistance of a cell of emf ɛ and V is the potential difference across the terminals of the source when the current flows through an external resistance R.)

41 - Delivering current → V = ɛ - IR  ( when the cell is  delivering current , that is cell is discharging itself.)

42 - Receiving current → V = ɛ + IR   ( when the cell is receiving current , that is , the cell is being charged .)

43 - Series grouping of cell → I = nɛ / R+nr   ( A number (n) of cells are grouped in series to get maximum current I when the external resistance R is much more than the internal resistance r of each cell .)

44 - Parallel grouping of cells → I = ɛ /(R+r/m)   ( A number (m) of cells are grouped in parallel to get maximum current I when the external resistance R is much less than the internal resistance r of each cell.)


Note : For any queries and suggestion please comment. 

Floatation with Chemistry

Give reason : 

1 - Anhydrous HCl is a poor conductor while aqueous HCl is an excellent conductor.

2 - When the stopper of a bottle full of hydrogen chloride gas is opened there are fumes in the air.

3 - A solution of hydrogen chloride in water turns blue litmus red , and conduct electricity , while a solution of the same gas in toluene ;

(a) has no effect on litmus

(b) does not conduct electricity.

4 - Thick white fumes are formed when a glass rod dipped in ammonium hydroxide is bought near the mouth of a bottle full of HCl gas

5- Hydrogen chloride gas is not collected over water.
6 - Ammonium compounds do not occur as minerals.
7 - Ammonium nitrate is not used in the preparation of ammonia.
8 - Conc H2SO4 is a good drying agent , yet it is not used to dry NH3.
9 - Liquid ammonia is used as a refrigerant in ice plants.
10 - Aqueous solution of ammonia is used for removing grease stains from wollen clothes.
11 - Aqueous solution of ammonia gives a pungent smell.
12 - Concentrated Sulphuric acid kept in air tight bottles.
13 - only all glass apparatus should be used for the preparation of nitric acid by heating concentrated sulphuric acid and potassium nitrate.
14 - Nitric acid is kept in a reagent bottle for along time.
15 - Why is freshly prepared ferrous sulphate solution used for testing the nitrate radicals in the brown ring test.
16 - Pure nitric acid is colorless but the acid obtained in laboratory is slightly yellow.
17 - Sulphuric acid forms two types of salts with NaOH.
18 - Red brown vapours are produced when concentrated sulphuric acid is added to hydrogen bromide.
19 - A piece of wood becomes black when concentrated sulphuric acid is poured on it 
20 - Brisk effervescence is seen when oil of vitriol is added to sodim carbonate.
21 - Ethyne is more reactive than ethene.
22 - Ethene is more reactive than ethane.
23 - Hydrocarbons are excellent fuels.
24 - Why the impurity of arsenic oxide must be removed before passing the mixture of SO₂ and air through the catalytic chamber in the preparation of sulphuric acid.
25 - Why is pure acetic acid known as glacial acetic acid?

**ANSWERS**

Comprehensive list of basic MATH formulas ..!!!

ALGEBRIC IDENTITIES :

 (a+b+c)²= a²+b²+c²+2(ab+bc+ca)

1. (a+b)²= a²+2ab+b²

2. (a+b)²= (a-b)²+4ab

3. (a-b)²= a²-2ab+b²

4. (a-b)²= f(a+b)²-4ab

5. a² + b²= (a+b)² - 2ab.

6. a² + b²= (a-b)² + 2ab.

7. a²-b² =(a + b)(a - b)

8. 2(a² + b²) = (a+ b)² + (a - b)²

9. 4ab = (a + b)² -(a-b)²

10. ab ={(a+b)/2}²-{(a-b)/2}²

11. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

12. (a + b)³ = a³ + 3a²b + 3ab² + b³

13. (a + b)³ = a³ + b³ + 3ab(a + b)

14. (a-b)³=a³-3a²b+3ab²-b³

15. a³ + b³ = (a + b)(a² -ab + b²)

16. a³ + b³ = (a+ b)³ -3ab(a+ b)

17. a³ -b³ = (a -b)(a² + ab + b²)

18. a³ -b³ = (a-b)³ + 3ab(a-b)

Trigonometric Identities :

Sin0° =0

Sin30° = 1/2

Sin45° = 1/√2

Sin60° = √3/2

Sin90° = 1

Cos is opposite of Sin

tan0° = 0

tan30° = 1/√3

tan45° = 1

tan60° = √3

tan90° = ∞

Cot is opposite of tan

Sec0° = 1

Sec30° = 2/√3

Sec45° = √2

Sec60° = 2

Sec90° = ∞

Cosec is opposite of Sec

2(Sin a)(Cos b)=Sin(a+b)+Sin(a-b)

2(Cos a)(sin b)=Sin(a+b)-Sin(a-b)

2(Cos a)(Cos b)=Cos(a+b)+Cos(a-b)

2(Sin a)(sin b)=Cos(a-b)-Cos(a+b)

Sin(a+b)=(Sin a)(Cos b)+(Cos a)(sin b).

» Cos(a+b)=(Cos a)(Cos b) - (Sin a)(sin b).

» Sin(a-b)=(Sin a)(Cos b)-(Cos a)(sin b).

» Cos(a-b)=(Cos a)(Cos b)+(Sin a)(sin b).

» tan(a+b)= ((tan a) + (tan b))/ (1−(tan a)(tan b))

» tan(a−b)= ((tan a) − (tan b)) / (1+ (tan a)(tan b))

» Cot(a+b)= ((Cot a)(Cot b) −1) / ((Cot a) + (Cot b))

» Cot(a−b)= ((Cot a)(Cot b) + 1) / ((Cot b)− (Cot a))

» Sin(a+b)=(Sin a)(Cos b)+ (Cos a)(sin b).

» Cos(a+b)=(Cos a)(Cos b) +(Sin a)(sin b).

» Sin(a-b)=(Sin a)(Cos b)-(Cos a)(sin b).

» Cos(a-b)=(Cos a)(Cos b)+(Sin a)(sin b).

» tan(a+b)= ((tan a) + (tan b))/ (1−(tan a)(tan b))

» tan(a−b)= ((tan a) − (tan b)) / (1+ (tan a)(tan b))

» Cot(a+b)= ((Cot a)(Cot b) −1) / ((Cot a) + (Cot b))

» Cot(a−b)= ((Cot a)(Cot b) + 1) / ((Cot b) − (Cot a))

a/(Sin a) = b/(sin b) = c/Sinc = 2r

» a = b Cosc + c (Cos b)

» b = a Cosc + c (Cos a)

» c = a (Cos b) + b (Cos a)

» (Cos a) = (b² + c²− a²) / 2bc

» (Cos b) = (c² + a²− b²) / 2ca

» Cosc = (a² + b²− c²) / 2ca

» Δ = abc/4r

» SinΘ = 0 then,Θ = nΠ

» SinΘ = 1 then,Θ = (4n + 1)Π/2

» SinΘ =−1 then,Θ = (4n− 1)Π/2

» SinΘ = (Sin a) then,Θ = nΠ (−1)^na

1. Sin2a = 2(Sin a)(Cos a)

2. Cos2a = Cos²a − Sin²a

3. Cos2a = 2Cos²a − 1

4. Cos2a = 1 − 2Sin²a

5. 2Sin²a = 1 − Cos2a

6. 1 + Sin2a = ((Sin a) + (Cos a))²

7. 1 − Sin2a = ((Sin a) − (Cos a))²

8. tan2a = 2(tan a) / (1 − tan²a)

9. Sin2a = 2(tan a) / (1 + tan²a)

10. Cos2a = (1 − tan²a) / (1 + tan²a)

11. 4Sin³a = 3(Sin a) − Sin3a

12. 4Cos³a = 3(Cos a) + Cos3a

» Sin²Θ+Cos²Θ=1

» Sec²Θ-tan²Θ=1

» Cosec²Θ-Cot²Θ=1

» SinΘ=1/CosecΘ

» CosecΘ=1/SinΘ

» CosΘ=1/SecΘ

» SecΘ=1/CosΘ

» tanΘ=1/CotΘ 

» CotΘ=1/tanΘ

» tanΘ=SinΘ/CosΘ

Formulas of three dimensional solids :(Mensuration)

For a cuboid :  

1 - Volume (V) = l × b × h

2 - Total surface area = 2 (lb × bh × hl )

3 - Diagonal = √ l²× b²×h²

For a cube :

1 - Volume (V) = a³

2 - Total surface area = 6a²

3 - Diagonal = √3 a

For a solid right circular cylinder:

1- Volume (V) = π r²h

2 - Lateral surface area = 2 π r h

3 - Total surface area = 2πr (h + r)

For a hollow right circular cylinder :

1 - Thickness of the material = (R - r)

2 - Volume of the material = π(R² - r²)h

3 - External curved surface area = 2πRh

4 - Internal curved surface area = 2πrh

5 - Total surface area = {2π(R + r) h + 2π(R² - r²)

For a right circular cone:

1 - Slant height = √h²+r²

2 - Volume (V) = 1/3 πr²h

3 - Lateral surface area = πrl

4 - Total surface area = πrl + πr²

For a sphere :

1 - Volume (V) = 4/3 πr³

2 - Surface area = 4πr²

For a spherical shell :

1 - Thickness of the material = ( R - r)

2 - Volume of the material = 4/3 π (R³- r³)

3 - Exterior surface area = 4πR²

4 - Interior surface area = 4πr²

For a hemisphere :

1 - Volume (V) = 2/3πr³

2 - Curved surface area = 2πr²

3 - Total surface area = 3πr²

Distance Formula :

1 - The distance between two points (xₗ , yₗ ) and (x₂ , y₂)

= √(xₗ - x₂)² + (yₗ - y₂)²

= √(difference of x - coordinates )² + ( difference of y - coordinates)²

Middle point of a line segment :

The middle point of a line segment joining the points (xₗ , yₗ ) and (x₂ , y₂ ) has the coordinates

(xₗ +x₂ /2) ,( yₗ + y₂/2)

Section Formula :

If AB is divided at P(x , y) in the ratio m : n , then the coordinates of P are

x = mx₂ + nxₗ / m + n    ,   y = my₂ + nyₗ /m + n

Formulas of equation of straight line :

Equation  of a straight line in the straight intercept  form :

1 - tan𝛉 = mx + c    , where m is slope, which cut off intercept c on the y axis.

2 - The equation of a line passing through (x₁ , y₁ ) and having the slope m is y - y₁ = m(x - x₁)

3 - The equation of the straight line passing through (x₁ , y₁) and (x₂ , y₂ ) is y - y₁ = y₁ - y₂/x₁ - x₂ (x - x₁)

Kill the fear of chemistry

Kill the fear of chemistry 

Let prepare for "BOARD EXAMS"

Few of the posts I've made in the recent cover a part of chemistry . This part of chemistry will give you additional matter  for your Rocking Success in 10 th Grade.

Give reason :

1 - Why is the size of inert gases are greater than in corresponding period ?

2 - Group 17 elements are strong non-metals ,while group 1 elements are strong

      metal.

3 - Metallic character of elements decreases from left to right in a period while it 

     increases in moving down a group.

4 - Halogens have a high electron affinity.

5 - The reducing power of element increases down in group while decreases in a 

      period.

6 - Size of atoms progressively becomes smaller when we move from sodium (Na)

     to Chlorine(Cl) in the third period of the periodic table.

7 - The size of Cl ion is greater than the size of Cl atom.

8 - Argon atom is bigger than Cl atom.

9 - Ionisation potential of the element is increases across a period.

10 - Alkali metals are good reducing agent.
11 - Nitric acid is not used in the preparation of hydrogen.

12 - Why covalent compounds are volatile in nature ?

13 - Covalent compounds are insoluble in water where as elctrovalent compounds 

       are soluble.

14 - Electrovalent compounds are usually hard crystal yet brittle.

15 - Polar covalent compounds conduct electricity .

16 - Carbonic acid gives an acid salt but hydrochloric acid does not.

17 - Dil. HCl acid is stronger than highly concentrated acetic acid.

18 - H3PO3 is not a tribasic acid .

19 - Lead carbonate do not react with dilute HCl .

20 - Nitrogen dioxide is a double acid anhydride.
21 - It is necessary to find out the ratio of reactants required in the preparation of 
       sodium sulphate.
22 - Fused calcium chloride is used in the preparation of FeCl3?
23 - Anhydrous FeCl3 cannot be prepared by heating hydrated Iron (3) chloride.
24 - Why common salt get wet during the rainy season ?
25 -  Why " The number of atoms in a certain volume of hydrogen is twice the                        number of atoms in the same volume of helium at the same temperature 
       and pressure".
26 - why "when stating the volume of a gas the pressure and temperature should 
       also be given."
27 - Why inflating a baloon seems to violate Boyles Law.
28 - Why C - 12 was selected as a unit and masses of other atoms were compared 
       with it ?
29 - A solution of cane sugar does not conduct electricity , but a solution of soium              chloride is agood conductor.
30 - Hydrochloric acid is a good conductor of electricity.
32 - On electrolysis of dilute copper(II) sulphate solution , copper is deposited at the
       cathode but no hydrogen gas evolves there.
33 - When a dilute aqueous solution of sodium chloride is electrolysed between                   platinum electrodes , hydrogen gas is evolved at the cathode but metallic sodium is not deposited.
34 - Zinc can produce hydrogen on reacting with acids but copper can not.
35 - In the electrolysis of acidified water , dilute sulphuric acid is preferred to dilute nitric acid for acidification.
36 - Electrolysis of molten lead bromide is considered to be a reaction in which                  oxidation and reduction go side by side i.e. redox reaction.
37 - The blue color of aqueous copper sulphate fades when it is electrolysed using             platinum electrodes.
38 - Lead bromide undergoes electrolytic dissociation in the molten state but is a non
       electrolyte in the solid state.
39 - Aluminium is extracted from its oxide by electrolytic reduction and not by                    conventional reducing agent.
40 -  The ratio of hydrogen and oxygen formed at the cathode and the anode is 2 :1 
         by volume.
41 - Ammonia is unionised in the gaseous state but in the aqeuous solution , it is a 
       weak electrolyte.
42 - A graphite anode is preferred to other inert electrodes during electrolysis of                fused lead bromide.
43 - For electroplating with silver , silver nitrate is not used as electrolyte.
44 - Carbon tetrachloride is a liquid but does not conduct electricity.
45 - Alkali metals kept in Kerosene oil.
46 - The rate of reaction decreases on moving from potassium to Lead.
47 - Why does metal not occur free in nature.
48 - The highly reactive metals can be used as reducing agents.
49 - Aluminium reacts with acids sulphuric acid and hydrochloric acid but not nitric            acid.
50 - In the electrolytic reduction of alumina , the graphite anode is gradually                        consumed.
51 - Roasting is carried out on sulphide ores and not on carbonate ores.
52 - Carbon can reduce lead oxide but not aluminium oxide.
53 - Electrolytic reduction is done to obtain aluminium.
54 - Zinc is used to cover iron so as to prevent rusting of iron .Why?
55 - A neutral gas other than oxygen which is formed at the anode during electrolysis        of fused alumina. 
56 - Nitric acid should be stored in aluminium container.              

***ANSWERS***