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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.

Wednesday, August 22, 2018

Crystal Growth Methods - Brief Explanation

Various types of crystal growth methods are

Growth from Water Solution
Growth from Flux
Hydro Thermal Growth
Electrode Deposition
Gel Growth

Growth from Water Solution


This technique is used for soluble crystals like sugar, salt crystals for example NaCl, KCl, KBr are used. Their growth rates are very small. They have 5 mole percent solubility.

Nucleation is one such process. Liquid containing crystal solution solution having low viscosity is taken into a beaker. Crystal which has to be grown is taken in very small size which is called as seed crystal. We have to hang this crystal in the liquid in beaker. The molecules join crystal to form the crystal big in size.


Growth from Flux

This method is used for crystals which are not dissoluble. This technique uses oxide crystals/metal crystals. Crystals like quartz having high melting point of 1400 oC which is attained at higher energy are grown using this technique. For this they are combined with other crystals called as flux whose benefit is to reduce the melting point of crystals to form.

Advantages of this technique

a) Growth is at temperature well below the melting point
b) High quality crystals can be obtained
c) Doping with suitable materials could be done
d) Solid solution can be grown easily

Hydro thermal growth

This method is used for crystals whose melting point is very high. For instance, Al2O3 cannot be soluble in water. Normally Al2O3 dissolves in water at critical temperature of 353 ⁰C.

So when pressure is exerted on crystal then melting point of material decreases (of about 50000 pounds per sq. inch). So special devices such as autoclaves are used for this purpose.

conditions of growth in hydro thermal process:

i) The Temperature
ii) The Pressure
iii) The temperature difference between top and bottom ends of autoclave



Sunday, July 1, 2018

Charactersitics of electron in one dimensional periodic potential

CRYSTAL MOMENTUM

For free electron, the quantity ℏk represents true momentum of electron as follows:

E= (ℏk)²/2m = (1/2m)*(ℏ)²*(k)² = (1/2m)*(h/2𝝅)²*(2𝝅/ƛ)² = (1/2m)*(h/ƛ)² = P²/2m

Therefore, the dynamical behavior of free electron can be represented by true momentum.

But when we consider an electron in periodic potential, ℏk doesn't represent true momentum. ℏk does not represent true momentum. The energy doesn't vary with 'k' as in previous case.

The true instantaneous momentum of an electron in presence of lattice potential is not a constant of motion and cannot be calculated by quantum mechanical method we take average value.
So in order to describe the dynamical behavior of electron in periodic potential we introduce a new type of momentum called as Crystal Momentum.

When we deal with interactions of electrons with lattice, we use conservation of crystal momentum and not of true momentum.

The crystal momentum is perfectly well defined constant for a state of given energy.


VELOCITY

The quantum mechanical part describes that the velocity of electron in a one dimensional lattice will be equal to Group Velocity of waves representing the electron.

v = (dw/dk) ...................................................................................................(1)

where 'w' is angular frequency of debroglie waves.

Eqn(1) depends on actual E-K curve.

(dE/dk) = ℏ (dw/dk) ; v=(1/ℏ)(dE/dk) ..........................................................(2)

for free electrons, substituting E= ℏk, v=p/m

giving linear variation of 'v' with 'k'.

In band theory of solids, however, 'E' is not proportional to k².

The variation of 'E' with 'k' is as sown in fig:




using this type of variation of 'E' with 'k' as shown in fig below.




We observe that at bottom (k=0) of energy band, the velocity is zero and as the value of 'k' increases ('E' increases) the velocity increases reaching its maximum at k=kₒ, where kₒ corresponds to "point of inflection" on E-K curve. Beyond this point the velocity begins to decrease and finally assumes zero value at k=𝝅/a, which is top of band. These are entirely new features which do not appear at all in behavior of free electrons.

EFFECTIVE MASS OF ELECTRON

It is known for long time that an electron has well defined mass and when accelerated by an electric field, it obeys Newtonian Mechanics. What happens when electron is to be accelerated inside a crystal? How will it react to electric field.

The mass of an electron inside the crystal appears, in general, different from free electron mass and is usually referred to as "effective mass".

The velocity of an electron in one dimensional lattice is given by

v = (2𝝅/ℏ)(dE/dk)..................................................................................................(3)

a= dv/dt = (2𝝅/h)(d²E/dk²)*(dk/dt) .......................................................................(4)

so we have to find value of dk/dt.

Let an electron is subjected to influence of an electric field of  strength 'E' for a time dt. If velocity of electron is v, the distance travelled in time dt=vdt

Therefore Work done, dE=(e𝜀)*v*dt

we know

v = (2𝝅/h)(dE/dk) ; therefore  dE= (e𝜀)*(2𝝅/h)*(dE/dk)*dt

(dk/dt) =  2𝝅e𝜀/h .....................................................................................................(5)

substituting (5) in (4), we get

a =  (2𝝅/h)² *(e𝜀)*(d²E/dk²)  ...................................................................................(6)

For free particle, a= m(dv/dt) = eE;

a=e*E/m  ..................................................................................................................(7)

comparing (6) & (7) , both forms are identical, we introduce a new mass known as effective mass given by

m* = (h/4𝝅) * (d²E/dk²)⁻¹  ......................................................................................(8)

For free electron,

m* = m

  • Effective mass can also be determined using "Cyclotron Resonance Experiment".
From experimental values of effective mass, we can conclude that
  •  Effective mass need not always be greater than 'm'. It can be smaller than 'm'.
  • It can be negative.
Variation of m* with k:  




Physically speaking near the bottom of band the effective mass m* has a constant value which is positive because the quadratic eqn [E ∝k²] is  satisfied near the bottom of band.

But as 'k' increases m* is no longer a constant, being now a function of k, because quadratic relation is no longer valid.

The degree of freedom of an electron is defined by a factor


fk = (m/m*) = (m/ℏ2)*(d²E/dk²)

fk is measure of extent to which an electron in state 'k' is free.