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Tuesday, November 26, 2019

Electric Current in Atoms - Bohr Magneton ; magnetic moment of electron in orbit

The revolution of electron in its orbit around nucleus resembles a magnetic dipole and the magnetic moment due to this orbital motion of electron is

𝜇ₑ₁ = - (e/2m) x angular momentum

angular momentum = mr²w

The minus sign indicates that dipole moment points in direction opposite to vector representing angular momentum.

The ratio of magnetic dipole moment of the electron due to its orbital motion and angular momentum of orbital motion is called "orbital gyro magnetic ratio" represented by '𝛾'.

𝛾 = (magnetic moment/angular momentum) = e/2m

The strength of magnetic dipole is given by

 ðœ‡â‚‘₁ = -𝜇B.l;  

𝜇B - Bohr Magneton = (eh/4πm) = 9.27 x 10⁻²⁴ Amp.m²

Therefore, '𝜇B'  represents magnetic moment of an elementary permanent magnetic dipole.

As we know that for a 'l' value there exists a quantum number 'ml' such that it takes +l to -l values hence for a d-electron for eg:

corresponding possible magnetic moment along direction of field are 2𝜇B, 𝜇B, 0, -𝜇B, -2𝜇B

 Therefore  𝜇ₑ₁ = -𝜇B.ml

Monday, November 25, 2019

Characteristics of electrical conduction in Metals

The general characteristics of electrical conduction in metals are summarized as follows:

1) The electrical current density in the steady state is proportional to electric field strength
     (Ohm's law).

2)  For pure specimens, the electric conductivity (σ) and the thermal conductivity (σ') vary with temperature as follows:

      σ∝T⁻¹ and σ' =const (for T > θD); θD is characteristic Debye temperature.

       so that  σ' / σT is independent of temperature (Weidmann-Franz law)

For T < θD;

      σ∝T⁻⁵ and σ' = T⁻² where 'θD' is characteristic Debye Temperature. 

      The relation  ρT⁵ is known as Bloch-Gruneisen T⁵ law.

 3) For metals that exhibit the phenomenon of superconductivity, their resistivity disappears at temperature above 0Kand below critical temperature for superconducting phase transition (critical temp=4.15K) for mercury.

4) For metals containing small amounts of impurities, the electrical resistivity(ρ) may be written as 
                                                          ρ = ρ₀ + ρᴘ(T)
where 'ρ₀' is a constant that increases with increasing impurity content and ρ(T) is temperature dependent part of resistivity. This is known as Mattheissen's rule.

5) For most metals, the electrical resistivity decreases with increasing pressure.

6) The resistivity of alloys that exhibit order-disorder transitions shows pronounced minima corresponding to ordered phases.

  

Sunday, November 24, 2019

Discovery of Artificial Disintegration

The artificial transmutation of one element into another is first accomplished by Rutherford in 1919.


The chamber 'c' was filled with a gas such as Nitrogen and Alpha particles from a radioactive source at 'A' were absorbed in the gas. A sheet of silver foil 'F', itself thick enough to absorb the alpha particles was placed over an opening in the side of chamber. A zinc sulphide screen 'S' was placed outside this opening and a microscope 'M' was used for observing any scintillatons - occuring on the screen 'S'. Scintillations were observed when the chamber was filled with Nitrogen, but when the Nitrogen was replaced by Oxygen or Carbondioxide no scintillations were observed.

Rutherford concluded that the scintillations were produced by high energy particles which were ejected from Nitrogen nuclei as a result of bombardment of these nuclei by alpha particles.

Magnetic deflection experiments indicated that these particles were Hydrogen nuclei or Protons.

Later experiments by Rutherford and Chadwick showed that these ejected Protons had Ranges upto 40cm in air.

Other light elements in the Range from Boron to Potassium were also disintegrated by bombardment with alpha particles.

The disintegration of Nuclei has also been studied with  Wilson cloud chamber. One of the first of these investigations was that of Blackett, who photographed the tracks of alpha particles in a Wilson cloud chamber containing 90% Nitrogen and 10% Oxygen. The majority of tracks photographed were straight tracks typical of alpha particle tracks.

Many of the tracks were observed to be forked tracks, indicating that an inelastic collision had taken place between an alpha particle and a Nitrogen Nucleus.

Measurement of the tracks showed that momentum of system was conserved but that the sum of kinetic energies of particles after impact was less than kinetic energy of alpha particle before impact.

On the basis of theory of nucleus advanced by Bohr, the disintegration of Nitrogen by bombardment with alpha particles may be thought as consisting of two separate parts.

The first is the capture of the alpha particle by Nitrogen nucleus which resulted in the formation of a new compound nucleus.

The second is the breaking up of compound nucleus into two particles, one of which is a Proton.

These two processes can be represented by means of a nuclear reaction equation analogous  to one representing a chemical reaction.

The nuclear reaction equation for this process is

₂He⁴ + ₇N¹⁴ --------->  (₉F¹⁸✷) ------->  ₈O¹⁷ ⁺ ₁H¹ + Q

Q is energy evolved or absorbed during nuclear reaction

Q --->  -Ve --->  energy has been absorbed (endothermic)
Q --->  +Ve ---> energy has been evolved (exothermic)
Q ---> nuclear reaction energy or disintegration energy

If sum of masses of the final particles exceeds that of initial particles, 'Q' must be negative; the energy absorbed in such a nuclear reaction must have been obtained from kinetic energies of the particle.

If 'E1' is kinetic energy of alpha particle just before capture, 'E2' the kinetic energy of Proton, 'E3' the kinetic energy of product nucleus,

Q = E2+E3-E1

In those cases in which Q is positive the sum of kinetic energies of final particles will be greater than kinetic energy of incident alpha particle.




Discovery of Meson

Yukawa predicted that it is due to the exchange of a massive particle between the nucleons leading to a short range force.

A result of much calculation is that the Range of a force is of same order of magnitude as compton wave length of exchanged particle. By analogy the nuclear force has a Range of about 1.4 x 10⁻¹³ cm.

A particle for which ℏ/mc = 1.4 x 10⁻¹³ cm will have its rest mass energy equal to 150 MeV or about 275 times the mass of electron.

The name Mesotron was given to this exchanged particle whose mass is intermediate between that   of electron and Proton. The modern name is Meson.

In 1937, a particle believed to be of the type was discovered by "S H Neddermeyer" and "C D Anderson" and independently by "J C Street" and "E C Stevenson" in cloud chamber studies of cosmic rays.

Estimates of the mass of this Meson were made from measurements of curvature of its track in a magnetic field which yielded values for mass of Meson in neighbourhood of 200 electron masses. Both positive and negative particles were observed.

WB Fretter (1946) made some very careful measurements of masses of mu particles, using two cloud chambers, one above the other. They were expanded simultaneously when ever a penetrating particle passed through them. This was accomplished by placing the Geiger Counters above each chamber, the two sets of actuating the expansion mechanism whenever an ionizing particle passed through  them as shown in below Fig.





The upper cloud chamber was placed in a magnetic induction of 5300 Gauss so that momentum of particle could be measured. The lower cloud chamber had a set of lead plates 0.5 inch thick and placed 1.5 inch apart so that Range in lead of particles could be measured. Out of 2100 tracks observed, 26 were suitable for measurement, their mass determination is yielded a value of 202Me.

The present accepted value is 207mₑ.

Later Occhialini and Powell and D M Perkins using a special nuclear emulsion photographic plates exposed at high altitudes, observed that some of Mesons stopped in photographic emulsions and produced so called stars - that is, nuclear disintegration with the emission of slow protons or alpha particles.

The photographs showed the curved track of heavy Meson which is named '𝚷' Meson; when captured by a nucleus in the emulsion, the resulting nuclear disintegration produces a star in which 3 charged particles are emitted.

The kinetic energy of muon emitted in the decay of a Pi Meson is always same and is equal to about 4 MeV.

𝚷⁺  ------------>  𝛍⁺ + 𝝂
𝚷⁻  ------------>  𝛍⁻ + 𝝂'    ;   ð‚' is anti neutrino     







Saturday, November 23, 2019

Properties of Stationary Waves

When two simple harmonic waves of same amplitude, frequency and time period travel in opposite directions in a straight line, the resultant wave obtained is called a stationary or a standing wave.

Properties of stationary waves:

1) In these waves, nodes and anti nodes are formed alternately.
   Nodes are positions where particles are at their mean positions having maximum strain.
   Anti nodes are positions where the particles vibrate with maximum amplitude having minimum strain.

2) The medium is split into segments and all particles of same segment vibrate in phase. The particles in one segment have a phase difference of '𝜫 ' with the particles in neighboring segment.

3) Condensations and rarefractions do not travel forward as in progressive wave but they appear and disappear alternately at same place.

4) As condensations and rarefractions do not travel forward there is no transfer of energy.

5) The distance between two adjacent nodes is 'ƛ/2' and also the distance between two adjacent antinodes is 'ƛ/4'. Between the two nodes there is anti node and vice versa.

6) The general appearance of wave can be represented by a sine curve but it reduces to straight line twice in each time period.

Saturday, October 5, 2019

REFLECTION, REFRACTION AT PLANE SURFACES

LAWS OF REFLECTION

 The angle of incidence is equal to angle of reflection.
 The incident ray, Normal and Reflected ray ray all lie in one plane. 


PROPERTIES OF IMAGE FORMED BY PLANE MIRROR
  1.  The image formed by a plane mirror is "virtual", "erect" and laterally reversed.
  2.  The size of image is equal to size of object.
  3.  The image is as far behind the mirror as the source is in front of it.
  4.  When the plane mirror is rotated through certain angle, the reflected ray turns through double the angle.
  5.  When two plane mirrors are kept facing each other at an angle '𝛳 ' and an object is placed between them, multiple images of the object are formed as a result of multiple successive reflections.
            a) If (360/𝛳) is "even", then no. of images is given by n = (360/𝛳)-1

            b)  If (360/𝛳) is "odd", then following two situations arise 
     
                   i) If object lies symmetrically, then n = (360/𝛳)-1
                  ii)  If object lies unsymmetrically, then n = 360/𝛳

            c) When two plane mirrors are placed parallel to each other, then  
                 n = (360/0) = ∞ (infinite no. of images)

Note:- 

I) The point object for a mirror is a point from which the rays incident on mirror actually diverge or towards which the incident rays appear to converge.

II) An optical image is a point where rays of light either intersect or appear to do so.


REFRACTION OF LIGHT

The Refracted ray bends towards the Normal when the second medium is denser than first medium and vice versa.

The deviation 'D' suffered by refracted ray is given by D =  i-r

LAWS OF REFRACTION

1. The Incident ray, the Refracted ray and the Normal to surface separating two media lie in one plane.

2. Snells Law: For any media, the ratio of sine of angle of incidence to sine of angle of refraction is a constant for a light beam of particular wavelength.

sini/sinr = 𝜇2/𝜇1 = constant

Refractive index 𝜇 = velocity of light in vacuum / velocity of light in medium


Saturday, May 25, 2019

Nature of orbits of satellites for different speeds

Let

'V' be velocity with which a body is projected from Earth.
Vs be minimum velocity of object to orbit around earth
Ve be escape velocity from surface of earth

then if,

i)  V < Vs ---  body falls to ground
ii) V = Vs --- body rotates round earth in circular orbit closer to surface of Earth
iii) Vs < V < Ve --- body revolves in elliptical orbit
iv)  V = Ve ----------body just  escapes from gravitational field
v)  V > Ve  --------- body moves in interstellar space with velocity equal to √V² -Ve²
vi)  V<Ve  ---------- Total energy of body is negative
vii)  V =Ve ---------- Total energy of body is zero

Satellites - Important points to be noted

1.  Orbital velocity of satellite is independent of mass of the satellite but depends on mass of planet and radius of orbit.

2. A satellite orbiting around a planet will have both Potential energy and Kinetic energy. Here Potential energy is negative and Kinetic energy is positive.

3. Total energy of satellite is negative.

4. With the increase of height of orbit from surface of planet, for a satellite

              a) Potential energy increases (from more negative to less negative)
              b) Kinetic energy decreases
              c) Orbital velocity decreases
              d) Total energy increases
              e) Period of revolution increases

5. A satellite orbiting very close to surface of Earth is known as its surface satellite. Orbital velocity for such a satellite is V = √gR = 8 Km.S⁻¹.

6. Relative velocity of parking satellite with respective to Earth is zero.

7. Orbital linear velocity is about 3 Km.Sec⁻¹.

8. A satellite cannot be coast in a stable orbit in a plane not passing through the Earth's center.

9. If two satellite move around the Earth in its equitorial plane such that one moves from West to East and other from East to West and other from East to West, the time period of revolution of first satellite will be more compared to other.

10. If a rocket launched in equitorial plane from West to East, advantage is up to 0.47 Km.Sec⁻¹  in the launching speed.  

11. If the Kinetic energy of an orbiting satellite is E, its Potential Energy will be -2E and total energy will be -E.

12. If a body is in a satellite which does not produce its own gravity, its true weight in that satellite W' is given by

W'/W =mg'/mg  ; W' = W/(1+[h/R])²

W - Weight of body on Earth
 h  - Height of orbit of satellite
 R - Radius of Earth

so true weight is lesser than its weight on Earth.  

13.  Apparent weight of a body in a satellite is zero and is independent of radius of orbit .

  

Wednesday, May 22, 2019

FRICTION - Important Points to be remembered

1. Friction is tangential force between the contact surfaces of two bodies.

2. Friction is due to Electromagnetic Forces between the surfaces in contact.

3. Friction is due to molecular interaction at the surfaces in contact.Friction is due to adhesive forces between molecules of two surfaces in contact.

4. Friction depends on nature of surfaces in contact and on the impurities present on these surfaces.

5. Normal Force: When two bodies are in contact or when one body is placed over another body, the contact force which either body exerts on other normal to contact surface is called Normal Force or Normal Reaction.

6. Limiting Friction is least force necessary to set a body into motion.

7. Static friction is a self adjusting force.

8. Kinetic Friction is not a self adjusting friction.

9. Generally coefficient of static friction is less than 1 but in some cases it may exceed 1.

10. Frictional force is a "Non-Conservative" force.

11. If a body of mass 'm' is on the floor of a lift which is moving with uniform acceleration 'a', Normal force on body or its apparent weight is

N = mg ±ma = m(g±a)

a) If the lift moves up, then N = m(g+a)
b) If the lift moves down, then N = m(g-a)
c) If the lift falls freely, then N=0
d) If the lift moves with uniform velocity, then  a =0, and N=mg

12. When a person falls on a rough road, the frictional force exerted by road on him is along his direction of motion.

13. The angle made by resultant of Normal force and Limiting friction with Normal force is called angle of friction. The tangent of this angle gives coefficient of static friction.

14. A good lubricant must be highly viscous and low volatile in nature.