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2021-11-09 15:51:23 #Physics #ImportantConcept #jee #neet

C-FRAME

If we attach an Inertial frame of Reference with the centre of mass of many particle system then centre of mass in that frame of reference would be at rest or, Vcm=0 , and such type of reference frames are known as centre of mass frame of reference.


Total Linear Momentum of a many particle system is zero in centre of mass frame of reference i.e., pcm=MVcm=0 since Vcm=0.

Therefore C-reference frames are also known as zero momentum reference frames.

Since in absence of any external force the centre of mass of any system moves with constant velocity in inertial frame of reference therefore for a many particle system C-frame of reference is an inertial frame of reference.

Reference frames connected to laboratory are known as L-frame of reference or laboratory frame of reference. ओपन / कमेंट

2021-11-04 11:32:37 Happy diwali ओपन / कमेंट

2021-10-28 14:36:55 JEE

https://vinaygond.gumroad.com/l/CaTYb ओपन / कमेंट

2021-10-24 12:07:43 Revision Notes on Vectors


The length or the magnitude of the vector = (a, b, c) is defined by w = √a2+b2+c2

A vector may be divided by its own length to convert it into a unit vector, i.e. ? = u / |u|. (The vectors have been denoted by bold letters.)

If the coordinates of point A are xA, yA, zA and those of point B are xB, yB, zB then the vector connecting point A to point B is given by the vector r, where r = (xB - xA)i + (yB – yA) j + (zB – zA)k , here i, j and k denote the unit vectors along x, y and z axis respectively.

Some key points of vectors:
1) The magnitude of a vector is a scalar quantity

2) Vectors can be multiplied by a scalar. The result is another vector.

3) Suppose c is a scalar and v = (a, b) is a vector, then the scalar multiplication is defined by cv = c (a, b) = (ca, cb). Hence each component of vector is multiplied by the scalar.

4) If two vectors are of the same dimension then they can be added or subtracted from each other. The result is gain a vector.

If u, v and w are three vectors and c, d are scalars then the following results of vector addition hold true:
1) u + v = v + u (the commutative law of addition)

2) u + 0 = u

3) u + (-u) = 0 (existence of additive inverses)

4) c (du) = (cd)u

5) (c + d)u = cu + d u

6) c(u + v) = cu + cv

7) 1u = u

8) u + (v + w) = (u + v) + w (the associative law of addition)

Some Basic Rules of Algebra of Vectors:
1) a.a = |a|2 = a2

2) a.b = b.a

3) a.0 = 0

4) a.b = (a cos q)b = (projection of a on b)b = (projection of b on a) a

5) a.(b + c) = a.b + a.c (This is also termed as the distributive law)

6) (la).(mb) = lm (a.b)

7) (a ± b)2 = (a ± b) . (a ± b) = a2 + b2 ± 2a.b

8) If a and b are non-zero, then the angle between them is given by cos θ = a.b/|a||b|

9) a x a = 0

10) a x b = - (b x a)

11) a x (b + c) = a x b + a x c

Any vector perpendicular to the plane of a and b is l(a x b) where l is a real number.

Unit vector perpendicular to a and b is ± (a x b)/ |a x b|

The position of dot and cross can be interchanged without altering the product. Hence it is also represented by [a b c]

1) [a b c] = [b c a] = [c a b]

2) [a b c] = - [b a c]

3) [ka b c] = k[a b c]

4) [a+b c d] = [a c d] + [b c d]

5) a x (b x c) = (a x b) x c, if some or all of a, b and c are zero vectors or a and c are collinear.

Methods to prove collinearity of vectors:
1) Two vectors a and b are said to be collinear if there exists k ? R such that a = kb.

2) If p x q = 0, then p and q are collinear.

3) Three points A(a), B(b) and C(c) are collinear if there exists k ? R such that AB = kBC i.e. b-a = k (c-b).

4) If (b-a) x (c-b) = 0, then A, B and C are collinear.

5) A(a), B(b) and C(c) are collinear if there exists scalars l, m and n (not all zero) such that la + mb+ nc = 0, where l + m + n = o

Three vectors p, q and r are coplanar if there exists l, m ? R such that r = lp + mq i.e., one can be expressed as a linear combination of the other two.

If [p q r] = 0, then p, q and r are coplanar.

Four points A(a), B(b), C(c) and D(d) lie in the same plane if there exist l, m ? R such that b-a = l(c-b) + m(d-c).

If [b-a c-b d-c] = 0 then A, B, C, D are coplanar.

Two lines in space can be parallel, intersecting or neither (called skew lines). Let r = a1 + μb1 and r = a2 + μb2 be two lines.

They intersect if (b1 x b2)(a2 - a1) = 0

The two lines are parallel if b1 and b2 are collinear.

The angle between two planes is the angle between their normal unit vectors i.e. cos q = n1 . n2

If a, b and c are three coplanar vectors, then the system of vectors a', b' and c' is said to be the reciprocal system of vectors if aa' = bb' = cc' = 1 where a' = (b xc) /[a b c] , b' = (c xa)/ [a b c] and c' = (a x b)/[a b c] Also, [a' b' c'] = 1/ [a b c]

Dot Product of two vectors a and b defined by a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is given by a1b1 + a2b2 + ..., + anbn . ओपन / कमेंट

2021-10-17 14:56:16 Notes on Atomic Physics


e/m of an electron (Thomson Method):-

(a) e/m of a particle is called the specific charge of the particle.

e/m = v/rB

Here, r is the radius of curvature, B is the strength of magnetic field, v is the velocity, e is the charge on cathode ray particle and m is the mass.

(b) v = E/B

Electric field:- E = V/d

Photo electric effect:- Photo-electric effect is the phenomenon of emission of electrons from the surfaces of certain substances, mainly metals, when light of shorter wavelength is incident upon them.

Effect of collector’s potential on photoelectric current:-

(a) Presence of current for zero value potential indicates that the electrons are ejected from the surface of emitter with some energy.

(b) A gradual change in the number of electrons reaching the collector due to change in its potential indicates that the electrons are ejected with a variety of velocities.

(c) Current is reduced to zero for some negative potential of collector indicating that there is some upper limit to the energy of electrons emitted.

(d) Current depends upon the intensity of incident light.

(e) Stopping potential is independent of the intensity of light.

Effect of intensity of light:- The photoelectric currentis directly proportional to theintensity of incident radiation.

Effect of frequency of light:-

(a) Stopping potentialdepends upon thefrequency of light. Greater the frequency of light greater is the stopping potential.

(b) Saturation current is independent of frequency.

(c) Threshold frequency is the minimum frequency, that capable of producing photoelectric effect.

Laws of Photoelectricity:-

(a) Photoelectric effect is an instantaneous process.

(b) Photoelectric current is directly proportional to the intensity of incident light and is independent of its frequency.

(c) The stopping potential and hence the maximum velocity of the electrons depends upon the frequency of incident light and is independent of its frequency.

(d) The emission of electrons stops below a certain minimum frequency known as threshold frequency.

Energy contained in bundle or packet:-

E = hf = hc/λ

Here h is the Planck’s constant and f is the frequency.

Work function:- It is defined as the minimum energy required to pull an electron out from the surface of metal. It is denoted by W0.

Einstein’s equation of photoelectric effect:-

(a) ½ mvmax2 = hf – W0

(b) ½ mvmax2 = hf – hf0 = h(f- f0) = h [c/λ – c/λ0]

(c) eV0 = hf - W0

(d)V0 = [(h/e)f] – [W0/e]

Here f0 is threshold frequency.

Threshold frequency (f0):- f0 = work function/h = W/h

Maximum kinetic energy of emitted photo electrons:-

?Kmax= ½ mvmax2 = eV0

Threshold wavelength:- λ0 = c/f0 = hc/hf0 = hc/W

Slope of V0~ v graph:- Slope= h/e

Rest mass of photon = 0, Charge = 0

Energy of photon:- E = hf = hc/λ

Momentum of photon:- p = E/c = h/λ = hf/c

Mass od photon:- m = E/c2 = h/cλ = hf/c2

For electron, λe = [12.27/√V]Å

For proton, λp = [0.286/√V]Å

For alpha particle, λα = [0.286/√V]Å

For particle at temperature T, λ = h/√3mKT (E = 3/2 KT)

The wavelength of electron accelerated by potential difference of V volts is:-

λe= [12.27/√V]Å

Number of photons:-

(a) Number of photons per sec per m2, np = Intensity/hf

(b) Number of photons incident per second, np = Power/hf

(c) Number of electrons emitted per second = (efficiency per surface)× (number of photons incident per second)

Compton wave length:-
(a) λc = h/m0c

Here h is the Planck’s constant, m0 is the rest mass of electron and c is the speed of light.

(b) Change in wavelength:- λ' – λ =λc (1-cos?)

de Broglie wavelength (λ):-λ = h/mv = h/√(2mE) = h/√(2meV)
In accordance to Bohr’s postulate of atomic structure, the angular momentum of an electron is an integral multiple of h/2π.

So, mvr = nh/2π

Bragg’s diffraction law:- 2dsinθ = nλ

Here λ is the wavelength of electron and d is distance between the planes.

Rutherford’s atomic model (α-particle scattering):-
(a) N(θ) ∝ cosec4(θ/2)

(b) Impact parameter, b = [(Ze2) (cot θ/2)]/[(4πε0)E]

Here, E = ½ mv2 = KE of theα particle. ओपन / कमेंट

2021-10-12 17:28:49 (c) Distance of closest approach, r0 = 2Ze2/(4πε0)E

Here E = ½ mv2 = KE of the α particle.

Bohr’s atomic model:-
(a) The central part of the atom called nucleus, contains whole of positive charge and almost whole of the mass of atom. Electrons revolve round the nucleus in fixed circular orbits.

(b) Electrons are capable of revolving only in certain fixed orbits, called stationary orbits or permitted orbits. In such orbits they do not radiate any energy.

(c) While revolving permitted orbit an electron possesses angular momentum L (= mvr) which is an integral multiple of h/2π.

L=mvr =n (h/2π)

Here n is an integer and h is the Planck’s constant.

(d) Electrons are capable of changing the orbits. On absorbing energy they move to a higher orbit while emission of energy takes place when electrons move to a lower orbit. If f is the frequency of radiant energy,

hf= W2-W1

Here W2 is the energy of electron in lower orbit and W1 is the energy of electron in higher orbit.

(e) All the laws of mechanics can be applied to electron revolving in a stable orbit while they are not applicable to an electron in transition.

Bohr’s Theory of Atom:-
(a) Orbital velocity of electron:- vn= 2πkZe2/nh

For a particular orbit (n= constant), orbital velocity of electron varies directly as the atomic number of the substance.

vn∝Z

(b) For a particular element (Z= constant), orbital velocity of the electron varies inversely as the order of the orbit.

vn∝1/n

(c) v = nh/2πmr

Relation between vn and v1:-vn = v1/n

Radius of electron:-

r= n2h2/4π2kmZe2

So, r∝n2

For, C.G.S system (k = 1), r = n2h2/4π2mZe2

S.I (k = 1/4πε0), r =(ε0/π) (n2h2/mZe2)

Kinetic energy of the electron:- It is the energy possessed by the electron by virtue of its motion in the orbit.
K.E = ½ mv2 = ½ k (Ze2/r)

Potential energy:- It is the energypossessed by the electronby virtue of its position near the nucleus.
P.E = -k (Ze2/r )

Total energy:-
W= K.E + P.E

W=- ½ k (Ze2/r) = -k2 2π2Z2me4/n2h2

For, C.G.S (k = 1), W = - [2π2Z2me4/n2h2]

For, S.I. ( k = 1/4πε0), W = - (1/8ε02) [Z2me4/n2h2]

Since, W∝1/n2, a higher orbit electron possesses a lesser negative energy (greater energy) than that of a lower orbit electron.

Frequency, wavelength and wave number of radiation:-
Frequency, f = k2[2π2Z2me4/h3] [1/n12 – 1/n22]

Wave number of radiation,

Here R is the Rydberg’s constant and its value is,

R= k2 [2π2Z2me4/ch3]

Bohr’s theory of hydrogen atom (Z=1):-
(a) Radius of orbit:-
r= n2h4/4π2me2 (C.G.S)
r= (ε0/π) (n2h2/me2) (S.I)

(b) Energy of electron:-
W= 2π2me4/n2h2 (C.G.S)
W =(1/8ε0)[me4/n2h2]

(c) Frequency, wavelength and wave number of radiation:-
C.G.S:- k =1 and Z=1
Frequency= f=2π2me4/h3 [1/n12 – 1/n22]
Wave number = 1/λ = 2π2me4/ch3 [1/n12 – 1/n22]

S.I:- k =1/4πε0 and Z=1

Frequency= f = (1/8ε0) (me4/h3)[1/n12 – 1/n22]
Wave number = 1/λ = (1/8ε02) (me4/ch3)[1/n12 – 1/n22]
Rydberg’s constant:- R=k2 =2π2z2 me4/ch3

For hydrogen atom, Z = 1, R = RH = k2 (2π2 me4/ch3).
For C.G.S system (k=1), RH = 2π2 me4/ch3
For S.I system (k=1/4πε0), RH = (1/8ε02) (me4/ch3)
Wave number, 1/λ = RH [1/n12 – 1/n22]

Hydrogen Spectrum:-
(a) For Lyman series:- 1/λ = R [1– 1/n2], n = 2,3,4…..∞
(b) For Balmer series:- 1/λ = R [1/22 – 1/n2], n =3,4,5…..∞
(c) For Paschen series:-1/λ = R [1/32 – 1/n2], n =4,5,6…..∞
(d) For Brackett series:-1/λ = R [1/42 – 1/n2], n =5,6,7…..∞
(e) P-fund series:-1/λ = R [1/52 – 1/n2], n =6,7,8…..∞

Series limits (λmin):-
(a) Lyman:- λmin = 912 Å
(b) Balmer:-λmin = 3645 Å
(c) Paschen:- λmin = 8201 Å

Energy levels of hydrogen atom:-
W = -k22π2me4/n2h2
For, n=1, W1 = -13.6 eV
For the first excited state, n=2, W2 =W1/4 = (-13.6/4) eV = -3.4 eV
For the second excited state, n=3, W3 =W1/9 = (-13.6/9) eV = -1.51 eV
Similarly, for other excited states, W4 = -0.85 eV and W5 = -0.54 eV
Number of emission lines from excited state:-n = n(n-1)/2

Ionization energy:-

- E1 = +(13.6Z2)eV

(a) For H-atom, I.E = 13.6 eV
(b) For He+ ion, I.E = 54.4 eV
(c) For Li++ ion, I.E = 122.4 eV

Ionization potential:-
(a) For H-atom, I.P = 13.6 eV
(b) For He+ ion, I.P = 54.42 eV ओपन / कमेंट

2021-10-01 13:33:08 Electric Charges and Fields

1. Electric Charge Charge is the property associated with matter due to which it produces and experiences electric and magnetic effect.

2. Conductors and Insulators Those substances which readily allow the passage of electricity through them are called conductors, e.g. metals, the earth and those substances which offer high resistance to the passage of electricity are called insulators, e.g. plastic rod and nylon.

3. Transference of electrons is the cause of frictional electricity.

4. Additivity of Charges- Charge are scalars and they add up like real numbers. It means if a system consists of n charges q1, q2, q3 , … ,qn, then total charge of the system will be q1 +q2 + … +qn.

5. Conservation of Charge The total charge of an isolated system is always conserved, i.e. initial and final charge of the system will be same.

6. Quantisation of Charge -Charge exists in discrete amount rather than continuous value and hence, quantised.

Mathematically, charge on an object, q=±ne
where, n is an integer and e is electronic charge. When any physical quantity exists in discrete packets rather than in continuous amount, the quantity is said to be quantised. Hence, charge is quantised.

7. Units of Charge
(i) SI unit coulomb (C)
(ii) CGS system
(a) electrostatic unit, esu of charge or stat-coulomb (stat-C)
(b) electromagnetic unit, emu of charge or ab-C (ab-coulomb)
1 ab-C = 10 C, 1 C = 3 x 109 stat-C ओपन / कमेंट

2021-09-30 14:35:34 Force and Pressure

When an object is either pushed or a pulled is known as a force.
Motion, generated in an object, is because of an action of a force.
The applied force makes the table move in a given direction.
The strength of a force is commonly expressed by the magnitude.
Force also has direction; likewise, if the magnitude or direction changes, it directly affects the force.
If the force is applied in the direction opposite to the direction of motion, then it results in a decrease in the speed of the object.
If an object is in motion, then external force may change in the state or direction of motion of that object.
The state of motion of an object is explained by its speed and the direction of motion.
The state of ‘rest’ of an object is considered to be the zero speed, as
An object cannot move by itself.
An object cannot change its speed by itself.
An object cannot change its direction by itself.
An object cannot change by itself.
A force may make an object move from rest.
A force may change the speed of a moving object.
A force may change the direction of a moving object.
A force may change the shape of an object.

The force caused by the action of muscles is known as the muscular force.
Some force, decreases the speed of a moving object, is known as ‘friction.’ E.g. moving wheel on road; once the source of force stops working, then wheel stops because of friction.
The force applied by a charged body on another charged or uncharged body is known as ‘electrostatic force.’
Objects or things that fall towards the earth, as earth pulls it towards itself; this force is known as the force of gravity or gravity.
The force of gravity is applicable on all objects. In fact, every object in this universe, irrespective of its size and shape, exerts some force on every other object. It happens only because of the ‘gravitational force.’

Pressure
The force, applied on a unit area of a surface is known as pressure
Pressure= force / area on which it acts.

If the area is smaller, then the pressure on a surface would be greater; e.g. this is the reason that the area of one end of a nail is pointed to exert sufficient pressure and other end is bigger.
This envelop of air is known as the atmosphere that extends up to many kilometers above the surface of the earth.
The pressure exerted by the air is known as atmospheric pressure.
The pressure inside our bodies is exactly equal to the atmospheric pressure and annuls the pressure acting from outside.
Liquids and gases also exert pressure on the walls of their respective containers. ओपन / कमेंट

2021-09-28 13:29:52 Revision Notes on Vectors


The length or the magnitude of the vector = (a, b, c) is defined by w = √a2+b2+c2

A vector may be divided by its own length to convert it into a unit vector, i.e. ? = u / |u|. (The vectors have been denoted by bold letters.)

If the coordinates of point A are xA, yA, zA and those of point B are xB, yB, zB then the vector connecting point A to point B is given by the vector r, where r = (xB - xA)i + (yB – yA) j + (zB – zA)k , here i, j and k denote the unit vectors along x, y and z axis respectively.

Some key points of vectors:
1) The magnitude of a vector is a scalar quantity

2) Vectors can be multiplied by a scalar. The result is another vector.

3) Suppose c is a scalar and v = (a, b) is a vector, then the scalar multiplication is defined by cv = c (a, b) = (ca, cb). Hence each component of vector is multiplied by the scalar.

4) If two vectors are of the same dimension then they can be added or subtracted from each other. The result is gain a vector.

If u, v and w are three vectors and c, d are scalars then the following results of vector addition hold true:
1) u + v = v + u (the commutative law of addition)

2) u + 0 = u

3) u + (-u) = 0 (existence of additive inverses)

4) c (du) = (cd)u

5) (c + d)u = cu + d u

6) c(u + v) = cu + cv

7) 1u = u

8) u + (v + w) = (u + v) + w (the associative law of addition)

Some Basic Rules of Algebra of Vectors:
1) a.a = |a|2 = a2

2) a.b = b.a

3) a.0 = 0

4) a.b = (a cos q)b = (projection of a on b)b = (projection of b on a) a

5) a.(b + c) = a.b + a.c (This is also termed as the distributive law)

6) (la).(mb) = lm (a.b)

7) (a ± b)2 = (a ± b) . (a ± b) = a2 + b2 ± 2a.b

8) If a and b are non-zero, then the angle between them is given by cos θ = a.b/|a||b|

9) a x a = 0

10) a x b = - (b x a)

11) a x (b + c) = a x b + a x c

Any vector perpendicular to the plane of a and b is l(a x b) where l is a real number.

Unit vector perpendicular to a and b is ± (a x b)/ |a x b|

The position of dot and cross can be interchanged without altering the product. Hence it is also represented by [a b c]

1) [a b c] = [b c a] = [c a b]

2) [a b c] = - [b a c]

3) [ka b c] = k[a b c]

4) [a+b c d] = [a c d] + [b c d]

5) a x (b x c) = (a x b) x c, if some or all of a, b and c are zero vectors or a and c are collinear.

Methods to prove collinearity of vectors:
1) Two vectors a and b are said to be collinear if there exists k ? R such that a = kb.

2) If p x q = 0, then p and q are collinear.

3) Three points A(a), B(b) and C(c) are collinear if there exists k ? R such that AB = kBC i.e. b-a = k (c-b).

4) If (b-a) x (c-b) = 0, then A, B and C are collinear.

5) A(a), B(b) and C(c) are collinear if there exists scalars l, m and n (not all zero) such that la + mb+ nc = 0, where l + m + n = o

Three vectors p, q and r are coplanar if there exists l, m ? R such that r = lp + mq i.e., one can be expressed as a linear combination of the other two.

If [p q r] = 0, then p, q and r are coplanar.

Four points A(a), B(b), C(c) and D(d) lie in the same plane if there exist l, m ? R such that b-a = l(c-b) + m(d-c).

If [b-a c-b d-c] = 0 then A, B, C, D are coplanar.

Two lines in space can be parallel, intersecting or neither (called skew lines). Let r = a1 + μb1 and r = a2 + μb2 be two lines.

They intersect if (b1 x b2)(a2 - a1) = 0

The two lines are parallel if b1 and b2 are collinear.

The angle between two planes is the angle between their normal unit vectors i.e. cos q = n1 . n2

If a, b and c are three coplanar vectors, then the system of vectors a', b' and c' is said to be the reciprocal system of vectors if aa' = bb' = cc' = 1 where a' = (b xc) /[a b c] , b' = (c xa)/ [a b c] and c' = (a x b)/[a b c] Also, [a' b' c'] = 1/ [a b c]

Dot Product of two vectors a and b defined by a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is given by a1b1 + a2b2 + ..., + anbn . ओपन / कमेंट