Wave Optics : Part 2

  • In Young’s Experiment two parallel and very close slits S1and S2 (illuminated by other another narrow slit) behaves like two coherent sources and produces a pattern of dark and bright bands (interference fringes) on a screen. For a point P on the screen


    Where d is the separation distance between two slits, D1 is the distance between the slits and the screen and y1 is the distance of point P from the central fringe.

  • For constructive interference (bright band) , the path difference must be an integral multiple of wavelength λ i.e.,

    y1d/D1 = n λ or y1=nD1λ/d

  • The separation distance Δy1 between adjacent bright or dark fringes is

    Δy1 = D1λ/d

    Using this relation we can calculate wavelength λ.

  • The colors shown by thin films are due to interference between two beams , one reflected from the top surface of the film and other from the bottom. The path difference between the two may give constructive interference for one color and destructive interference for another. Hence the reflected light is colored.

  • Term diffraction refers to light spreading out from narrow holes and slits, and bending around corners and obstacles.

  • The single slit diffraction pattern shows the central maximum (θ=0) at angular separation θ=±n λ (n≠0) and secondary maxima at θ=±(n+1/2) λ (n≠0).

  • Different parts of the wave front at the slit acts as secondary sources ; diffraction pattern is the result of interference of waves from these sources.

  • An aperture of size a sends diffracted light into an angle ≈ λ/a.

  • Doppler effect is the shift in frequency of light when there is a relative motion between the source and the observer. It is given by

    Δν/ν ≈ vr/c for v/c << 1

    Where vr is the radial component of relative velocity v. This effect can be used to measure the speed of an approaching or receding object.

  • Polarization specifies the manner in which electric field E oscillates in the plane transverse to direction of propagation of light. If E oscillates back and forth in a straight line , the wave is said to be linearly polarized. If the direction of E changes irregularly then the wave is un-polarized.

Cylendrical capacitor

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Wave velocity in a continuous system

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Wave velocity in a continuous system

  • Any system whose particle motion are governed by classical wave equation is a system in which harmonic waves of any wavelength can travel with the speed v

  • The value of v depends on the elastic and inertial properties of the system under consideration.

(1) Transverse wave on a stretched string

  • Displacement of the string is governed by the equation

    Where T is the tension and µ is the linear density (mass per unit length of the string)
  • Velocity of wave on the string is
    v is the velocity of the wave.
  • Medium through which waves travel will offer impedance to these waves.
  • If the medium is loss less i.e., it does not have any resistive or dissipative components, the impedance is solely determined by its inertia and elasticity.
  • Characteristic impedance of string is determined by
  • Since v is determined by the inertia and elasticity this shows that impedance is also governed by these two properties of the medium.
  • For loss-less medium impedance is real quantity and it is complex if the medium is dissipative.

(2) Longitudinal waves in uniform rod

  • Equation for longitudinal vibrations of a uniform rod is

  • where ξ (x,t )→displacement
    Y is young’s modulus of the rod
    ρ is the density
  • Velocity of longitudinal wave in rod is

(3) Electromagnetic waves in space

  • When electric and magnetic field vary in time they produce EM waves.
  • An oscillating charge has an oscillating electric and magnetic fields around it and hence produces EM waves.
  • Example: - (1) Electrons falling from higher to lower energy orbit radiates EM waves of particular wavelength and frequency. (2) The motion of electrons in an antenna radiates EM waves by a process called Bramstrhlung.
  • Propagation of EM waves in a medium is also due to inertial and elastic properties of the medium.
  • Every medium (including vacuum) has inductive properties described by magnetic permeability µ of the medium.
  • This property provides magnetic inertia of the medium.
  • Elasticity of the medium is provided by the capacitive property called electrical permittivity ε of the medium.
  • Permeability µ stores magnetic energy and the permittivity ε stores the electric field energy.
  • This EM energy propagates in the medium in the form of EM waves.
  • Electric and magnetic fields are connected by Maxwell’s Equations (dielectric medium)
    ∇×H =ε (∂E )/∂t
    ∇×E = - μ (∂H⃗)/∂t
    ∇∙H =0
  • Here in above equations E ⃗ is electric field , H ⃗ is the magnetic field and ρ is charge density

Color code of carbon resistance

• Commercially resistors of different type and values are available in the market but in electronic circuits carbon resistors are more frequently used
• In carbon resistors value of resistance is indicated by four colored bands marked on its surface as shown below in figure

• The first three bands a,b.c determine the value of the resistance and fourth band d gives the tolerance of the resistance
• The color of the first and second band respectively gives the first and second significant figure of the resistance and third band c gives the power of the ten by which two significant digits are multiplied for obtaining the value of the resistance
• value of different colors for making bands in carbon resistors are given below in the table

Color Figure(first and second band) Multiplier(for third band) tolerance
• For example in a given resistor let first strip be brown ,second strip be red and third be orange and fourth be gold then resistance of the resistor would be 12 x 103 (± 5%

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Coulumb's law

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The full Syllabus for Part B of Paper I and Part B of Paper II.

The syllabus for Part A of Paper II comprises Sections I-VI.

I. Mathematical Methods of Physics

Dimensional analysis; Vector algebra and vector calculus; Linear algebra, matrices, Cayley Hamilton theorem, eigenvalue problems; Linear differential equations; Special functions (Hermite, Bessel, Laguerre and Legendre); Fourier series, Fourier and Laplace transforms; Elements of complex analysis: Laurent series-poles, residues and evaluation of integrals; Elementary ideas about tensors; Introductory group theory, SU(2), O(3); Elements of computational techniques: roots of functions, interpolation, extrapolation, integration by trapezoid and Simpson’s rule, solution of first order differential equations using Runge-Kutta method; Finite difference methods; Elementary probability theory, random variables, binomial, Poisson and normal distributions.

II. Classical Mechanics

Newton’s laws; Phase space dynamics, stability analysis; Central-force motion; Two-body collisions, scattering in laboratory and centre-of-mass frames; Rigid body dynamics, moment of inertia tensor, non-inertial frames and pseudoforces; Variational principle, Lagrangian and Hamiltonian formalisms and equations of motion; Poisson brackets and canonical transformations; Symmetry, invariance and conservation laws, cyclic coordinates; Periodic motion, small oscillations and normal modes; Special theory of relativity, Lorentz transformations, relativistic kinematics and mass–energy equivalence.

III. Electromagnetic Theory

Electrostatics: Gauss’ Law and its applications; Laplace and Poisson equations, boundary value problems; Magnetostatics: Biot-Savart law, Ampere's theorem, electromagnetic induction; Maxwell's equations in free space and linear isotropic media; boundary conditions on fields at interfaces; Scalar and vector potentials; Gauge invariance; Electromagnetic waves in free space, dielectrics, and conductors; Reflection and refraction, polarization, Fresnel’s Law, interference, coherence, and diffraction; Dispersion relations in plasma; Lorentz invariance of Maxwell’s equations; Transmission lines and wave guides; Dynamics of charged particles in static and uniform electromagnetic fields; Radiation from moving charges, dipoles and retarded potentials.

IV. Quantum Mechanics

Wave-particle duality; Wave functions in coordinate and momentum representations; Commutators and Heisenberg's uncertainty principle; Matrix representation; Dirac’s bra and ket notation; Schroedinger equation (time-dependent and time-independent); Eigenvalue problems such as particle-in-a-box, harmonic oscillator, etc.; Tunneling through a barrier; Motion in a central potential; Orbital angular momentum, Angular momentum algebra, spin; Addition of angular momenta; Hydrogen atom, spin-orbit coupling, fine structure; Time-independent perturbation theory and applications; Variational method; WKB approximation;

Time dependent perturbation theory and Fermi's Golden Rule; Selection rules; Semi-classical theory of radiation; Elementary theory of scattering, phase shifts, partial waves, Born approximation; Identical particles, Pauli's exclusion principle, spin-statistics connection; Relativistic quantum mechanics: Klein Gordon and Dirac equations.

V. Thermodynamic and Statistical Physics

Laws of thermodynamics and their consequences; Thermodynamic potentials, Maxwell relations; Chemical potential, phase equilibria; Phase space, micro- and macrostates; Microcanonical, canonical and grand-canonical ensembles and partition functions; Free Energy and connection with thermodynamic quantities; First- and second-order phase transitions; Classical and quantum statistics, ideal Fermi and Bose gases; Principle of detailed balance; Blackbody radiation and Planck's distribution law; Bose-Einstein condensation; Random walk and Brownian motion; Introduction to nonequilibrium processes; Diffusion equation.

VI. Electronics

Semiconductor device physics, including diodes, junctions, transistors, field effect devices, homo and heterojunction devices, device structure, device characteristics, frequency dependence and applications; Optoelectronic devices, including solar cells, photodetectors, and LEDs; High-frequency devices, including generators and detectors; Operational amplifiers and their applications; Digital techniques and applications (registers, counters, comparators and similar circuits); A/D and D/A converters; Microprocessor and microcontroller basics.

VII. Experimental Techniques and data analysis

Data interpretation and analysis; Precision and accuracy, error analysis, propagation of errors, least squares fitting, linear and nonlinear curve fitting, chi-square test; Transducers (temperature, pressure/vacuum, magnetic field, vibration, optical, and particle detectors), measurement and control; Signal conditioning and recovery, impedance matching, amplification (Op-amp based, instrumentation amp, feedback), filtering and noise reduction, shielding and grounding; Fourier transforms; lock-in detector, box-car integrator, modulation techniques.

Applications of the above experimental and analytical techniques to typical undergraduate and graduate level laboratory experiments.

VIII. Atomic & Molecular Physics

Quantum states of an electron in an atom; Electron spin; Stern-Gerlach experiment; Spectrum of Hydrogen, helium and alkali atoms; Relativistic corrections for energy levels of hydrogen; Hyperfine structure and isotopic shift; width of spectral lines; LS & JJ coupling; Zeeman, Paschen Back & Stark effect; X-ray spectroscopy; Electron spin resonance, Nuclear magnetic resonance, chemical shift; Rotational, vibrational, electronic, and Raman spectra of diatomic molecules; Frank – Condon principle and selection rules; Spontaneous and stimulated emission, Einstein A & B coefficients; Lasers, optical pumping, population inversion, rate equation; Modes of resonators and coherence length.

IX. Condensed Matter Physics

Bravais lattices; Reciprocal lattice, diffraction and the structure factor; Bonding of solids; Elastic properties, phonons, lattice specific heat; Free electron theory and electronic specific heat; Response and relaxation phenomena; Drude model of electrical and thermal

conductivity; Hall effect and thermoelectric power; Diamagnetism, paramagnetism, and ferromagnetism; Electron motion in a periodic potential, band theory of metals, insulators and semiconductors; Superconductivity, type – I and type - II superconductors, Josephson junctions; Defects and dislocations; Ordered phases of matter, translational and orientational order, kinds of liquid crystalline order; Conducting polymers; Quasicrystals.

X. Nuclear and Particle Physics

Basic nuclear properties: size, shape, charge distribution, spin and parity; Binding energy, semi-empirical mass formula; Liquid drop model; Fission and fusion; Nature of the nuclear force, form of nucleon-nucleon potential; Charge-independence and charge-symmetry of nuclear forces; Isospin; Deuteron problem; Evidence of shell structure, single- particle shell model, its validity and limitations; Rotational spectra; Elementary ideas of alpha, beta and gamma decays and their selection rules; Nuclear reactions, reaction mechanisms, compound nuclei and direct reactions; Classification of fundamental forces; Elementary particles (quarks, baryons, mesons, leptons); Spin and parity assignments, isospin, strangeness; Gell-Mann-Nishijima formula; C, P, and T invariance and applications of symmetry arguments to particle reactions, parity non-conservation in weak interaction; Relativistic kinematics.

Blog for graduate level physics: Waves in continous medium

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Waves in continous medium

  • There are essentially two ways of transporting energy from one place to another (a) Actual transport of matter for example a fired bullet and (b) Waves carry energy but there is no transport of matter for example sound waves carry energy so thay can move diphagram of the ear.
  • Here we will consider the oscillations of open or unbounded systems i.e., systems having no outer boundaries.
  • If such system is disturbed , waves travel in the system with a speed determined by the properties of the system.
  • Waves are not reflected back in such a system.
  • The waves generated by driving force are called travelling waves ; these waves travel from the point where the driving force produces the disturbance.
  • If the driving force produces a harmonic disturbance the travelling wave it produces are called harmonic travelling waves.
  • In the steady state, all moving parts of the system oscillates with simple harmonic motion at the driving frequency.
  • Waves where the displacements or oscillations are transverse (i.e., perpandicular) to the direction of wave propagation is called transverse wave.
  • The wavelength (denoted by λ) of the wave is defined as the distance, measured along the direction of the propagation of the wave, between two nearest points which are in the same state of viberation.
  • Wavelength λ is just the distance travelled by the wave during one time period T of particle oscillation. Thus wave velocity
    where ν=1/T - is the frequency of the wave.
  • This relation between wave velocity, frequency and wavelength also holds for longitudinal waves in which the displacements or oscillation in the medium are parallel to the direction of wave propagation.
  • Waves in spring and sound waves are longitudinal waves.
  • Wavelength for longitudinal waves is the distance between two successive compressions or rarefactions.
  • Sound waves are also compressional.
  • Assumptions that are made while obtaining wave equation are:-
    1. Amplitude A of particle oscillations does not change in course of the propagation of wave.
    2. The medium is isotopic and homogeneous so that velocity of wave does not chance from place to  place
  • Displacement of particle at x at any time t is
    Ψ(x,t) = A sin{2π(t-x/v)/T)}
  • The function Ψ(x,t) repeats itself in a distance λ . Wavelength of a wave is also known as spatial periodicity of the wave.
  • The wave is thus doubly periodic. It has temporal periodicity T and spatial periodicity λ.
  • Let us define quantities
    k=2π/λ and
    then wave function can be written as
    Ψ(x,t) = A sin{ωt-kx}
    where quantity k is known as wave number of the wave and ω is called angular frequency of particle oscillations in wave.
  • Harmonic wave travelling in
    + x direction    :     Ψ(x,t) = A sin{ωt-kx}
    - x direction    :     Ψ(x,t) = A sin{ωt+kx}
    above equations can also be equally well described by cosine function.
  • Classicsl wave equation is

  • Important inferences from above wave equation
    1. Whenever second order time derivative of any physical quantity is related to second order space derivative as in above equation , a wave of some sort must travell in the medium.
    2. Velocity of that wave is given by the square root of the coefficent of second order space derivative.
  • Individual derivatives which makes up the medium do not propagate through the medium with the wave; they merely oscillates ( transversly or longitudinally) about there equilibrium positions.
  • It is their phase relationship which we observe as wave.
  • Wave velocity is also called phase velocity with which crest or troughs in case of transverse wave and compressions or rarefactions in case of longitudinal waves travell through the medium.
  • The phase velocity is given by
  • Ψ(x,t)=f(vt-x) is the solution of the above given wave equation.

Wave Optics : Part 1

1. A wavefront is the locus of points having same phase of oscillation.
2. Rays are lines perpandicular to the wavefront, which shows the direction of propagation of energy.
3. The time taken for light to travel from one wavefront to another is same along any ray.
4. Huygen's construction is based on the principle that every point of a wavefront is the source of secondary wavefront that is the surface tangent to all secondary wavefronts gives rise to a new wavefront.
5. The law of refraction (i=r) and the Snell's law of refraction
sini /sinr =v1/v2=n2/n1 = n21
can be derived using the wave theory. Here v1 and v2 are the speed of light in media 1 and 2 wiyh refractive index n1 and n2 respectively.
6. The frequency ν remains same when light travels from one medium to another. The speed of the wave is given by
where λ is the wavelength of the wave and T is the period of oscillation.
7. Emission , absorption and scattering are the three proscesses by which matter interacts with radiation.
8. In emission , an accelerated charge radiates an looses energy.
9. In absorption the charge gains energy at the expence of the EM wave.
10. In scattering the charge accelerated by incident EM wave radiated in all direction.
11. Two sources of light are coherent if they have same frequency and stable phase difference.
12. In case of coherent sources of light the total intensity I  is not just the sum of individual intensities I1 and I2 due to two sources but also includes an interference term that is
I = I1 + I2 + 2kE1.E2
where E1 and E2 are the electric fields at a point due to the sources.
13. The interference term averaged over many cycles is zero if (i) the source of light have different frequencies or (ii) the source have the same frequency but not stable phase difference. For such incoherent sources I = I1 + I2

Overview of electrostatics and electricity

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Overview of electrostatics and electricity:
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Comparison between linear motion of a particle and rotational motion of a rigid body

Concept of force

  • Concept of force is central to all of physics whether it is classical physics,nuclear physics,quantum physics or any other form of physics
  • So what is force? when we push or pull anybody we are said to exert force on the body
  • Push or pull applied on a body does not exactly define the force in general.We can define force as an influence causing a body at rest or moving with constant velocity to undergo an accleration
  • There are many ways in which one body can exert force on another body
    Few examples are given below
    (a)Stretched springs exerts force on the bodies attached to its ends
    (b)Compressed air in a container exerts force on the walls of the container
    (c) Force can be used to deform a flexible object
    There are lots of examples you could find looking around yourself
  • Force of gravitational attraction exerted by earth is a kind of force that acts on every physical body on the earth and is called the weight of the body
  • Mechanical and gravitation forces are not the only forces present infact all the forces in Universe are based on four fundamental forces
    (i) Strong and weak forces: These are forces at very short distance (10-05 m) and are responsible for interaction between neutrons and proton in atomic nucleus
    (ii) Electromagnetic forces: EM force acts between electric charges
    (iii) Gravitational force -it acts between the masses
  • In mechanics we will only study about the mechanical and gravitational forces
  • Force is a vector quatity and it needs both the magnitude as well as direction for its complete description
  • SI unit of force is Newton (N) and CGS unit is dyne where
    1 dyne= 1005 N

Uncertainity Principle

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Uncertainity Principle: "Uncertainity principle says that 'If a measurement of position is made with accuracy Δx and if the measurement of momentum is made simultane..."
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Uncertainity Principle

Uncertainity principle says that "If a measurement of position is made with accuracy Δx and if the measurement of momentum is made simultaneously with accuracy Δp , then the product of two errors can never be smaller than a number of order h
ΔpΔx≥(∼h)                                  (1)
Similarly if the energy of the syatem is measured to accuracy ΔE , then time to which this measurement refers must have a minimum uncertainity given by
ΔEΔt≥(∼h)                                    (2)
In generalised sence we can say that if  Δq is the error in the measurement of any co-ordinate and Δp is the error in its canonically conjugate momentum then we have,
ΔpΔq≥(∼h)                                  (3)
Consider the relation between the range of position Δx and range of wave number Δk appearing in a wave packet then
ΔxΔk≥1                                                     (4)
and this is a general property not restricted to quantum mechanics. Uncertainity principle is obtaines when the following quantum mechanical interpretation of quantities appearing in above equation are taken into account.
(1) The de-Brogli equation p=hk creates a relationship between wave number and momentum , which is not present in classical mechanics.
(2) Whenever either the momentum or the position of an electron is measured , the result is always some definite number. A classical wave packet always covers a range of positions and range of wave numbers.
Δx is a measure of minimum uncertainity or lack of complete determination of the position that can be ascribed to the electron. and Δk is the measure of minimum uncertainity or lack of complete determination of the momentum that can be ascribed to it.

Relation of spreading wave packet to uncertainity principle
Narrower the wave packet to begin with , the more rapidly it spreads. Because of the confinement of the packet within the region Δx0 the fourier analysis contains many waves of length of order of Δx0 , hence momenta p≅h/Δx0
Although average velocity of the packet is equal to the group velocity , there is still a strong chance that the actual velocity will fluctuate about this average by the same amount. The distance covered by the particle is not completely determined but it may vary as much as
The spread of the wave packet may therefore be regarded as one of the manifestations of the lack of complete determination of initial velocity necesarily associated with the narrow wave packet.

Relation of stability of atom to uncertainity principle
From uncertainity principle if an electron is localized it must have on an average a high momentum and have high kinetic energy as it takes energy to localize a particle. According to uncertainity principle it takes a momentum  Δp≅h/Δx and an energy nearly equal to h2/2m(Δx)2 to keep an electron localised within a region Δx. Momentum creates a pressure which tends to oppose localization of the electron. In an atom the pressure is opposed by the force attracting the electron back to the nucleus. Thus the electron will come to equilibrium when the attractive forces balances the effective pressure and, this way , the mean radius of the lowest quantum state is determined. This point of balance can be found from the condition that total energy must be minimum. Thus we have
W≅  (h2/2m(Δx)2) - (e2/Δx)
Differentiating both the sides w.r.t. Δx and making ∂W/∂(Δx) = 0 we get
THis result is just the radius of first Bohr orbit although not exact but qualitative.. The limitation of the localizability of the electron is inherent in the wave-particle nature of matter. In order to have an electron in very small space , we must have very high fourier components in its wave function and therefore the possibility of very high moments. There is no way to force an electron to occpy a well defined position and still remain at rest.

SHM in short

  • Simple harmonic motion is simplest form of oscillatory motion
  • SHM is a kind of motion in which the restoring force is propotional to the displacement from the mean position and opposes its increase.Mathematically restoring force is
    K=Force constant
    x=displacement of the system from its mean or equilibrium position
    Diffrential Equation of SHM is
    d2x/dt2 + ω2x=0

  • Solutions of this equation can both be sine or cosine functions .We conveniently choose
    where A,ω and φ all are constants
  • Quantity A is known as amplitude of SHM which is the magnitude of maximum value of displacement on either sides from the equilibrium position
  • Time period (T) of SHM the time during which oscillation repeats itself i.e, repeats its one cycle of motion and it is given by
    where ω is the angular frequency
  • Frequency of the SHM is the number of the complete oscillation per unit time i.e, frequency is reciprocal of the time period
    Thus angular frequncy
  • Total energy remains constant in a SHM.So you can find the energy at any position and differentiate it to find the out the frequency
  • Problem of SHM are basically to find out the timeperiod.So the concenteration should be on getting the net restoring force
  • The basic approach to solve such problem is
    1. Consider the system is displaced from equilibrium position
    2. Now consider all the forces acting on the system in displaced position
    3. find the restore force which comes out to be in the form

Vector Algebra 2(quick recap)

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Vector Algebra 1(quick recap)

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Blog for graduate level physics: Force on a conductor

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Force on a conductor

We have already learned in our previous discussion that field inside a conductor is zero and the field immidiately outside is
En=n(σ/ε0)                               (1)
where n is the unit normal vector to the surface of the conductor. We also know that any charge a conductor may carry is distributed on the surface of the conductor.
In presence of an electric field this surface charge will experience a force. If we consider a small area element ΔS of the surface of the conductor then force acting on area element is given by
ΔF=(σΔS).E0       (2)
where σ is the surface charge density of the conductor , (σΔS) is the amount of charge on the area element ΔS and E0 is the field in the region where charge element (σΔS) is located.
Now there are two fields present  Eσ and E0 and the resultant field both inside and outside the conductor near area element  ΔS would be equal to the superposition of both the fields  Eσ and E0 . Figure below shows the directions of both the fields inside and outside the conductor

Now field E0 has same value both inside and outside the conductor and surface element ΔS suffers discontinuty because of the charge on the surface and this makes field  Eσ on either side pointing away from the surfaceas shown in the figure given above. Since E=0 inside the conductor
Since direction of  Eσ and E0 are opposite to each other and outside the conductor near its surface
Thus , E0 =E/2                      (3)
Equation (2) thus becomes,regardless of the of ΔF=½(σΔS).E                  (4)
From equation 4 , force acting per unit area of the surface of the conductor is
f=½σ.E                                       (5)
Here  is the  Eσ electric field intensity created by charge on area element ΔS at the point very close to this area element. In this region this area element behaves as infinite uniformly charged sheet hence we have,
Eσ=σ/2ε0                                                  (6)
which is in accordance with equation 1. Hence from equation 5
f2/2ε0 = (ε0E2/2)n                (7)
This quantity f is known as surface density of force. From equation 7 we can conclude that regardless of the sign of σ and hence direction of E , f is always directed in outward direction of the conductor.

Thomson Effect

  • Thomson effect is related to the emf that develops between two parts of the single metal when they are at different temperature

  • Thus thomson effect is the absorption or evolution of heat along a conductor when current passes through it when one end of the conductor is hot and another is cold

  • If two parts of the metal are at small temperature difference dT ,themn the electric potential difference is proportional to dT dV α dT
    where σ is the constant of proportinality and is known as thomson coefficent

  • Peltier coefficent and thomson coeficent are related to thermopower according to following relations
    and σ =-T(ds/dT)=-T(d2E/dT2)

  • We have seen that all the three effects are defined in terms of three coefficent namely seaback,peltier and thomson coefficent but the basic quantity is thermo-power which is the rate of change of thermo-emf with temperature
  • Kinetic Energy

    For full notes on Work , Energy and Power visit Physicscatalyst.com
    • Kinetic energy is the energy possesed by the body by virtue of its motion
    • Body moving with greater velocity would posses greater K.E in comparison of the body moving with slower velocity
    • Consider a body of mass m moving under the influenece of constant force F.From newton's second law of motion
      Where a is the acceleration of the body
    • If due to this acceleration a,velocity of the body increases from v1 to v2 during the displacement d then from equation of motion with constant acceleration we have
      v22 -v12=2ad or
      a=v22 -v12/2d Using this acceleration in Newton's second law of motion
      we have
      F=m(v22 -v12)/2d
      Fd=m(v22 -v12)/2
      Fd=mv22/2 -mv12/2           (7)
      We know that Fd is the workdone by the force F in moving body through distance d
    • In equation(7),quantity on the right hand side mv2/2 is called the kinetic energy of the body
    • Finally we can define KE of the body as one half of the product of mass of the body and the square of its speed
    • Thus we see that quantity (mv2/2) arises purely becuase of the motion of the body
    • In equation 7 quantity
      is the final KE of the body and
      is the initial KE of the body .Thus equation 7 becomes
      W=K2-K1=ΔK           (9)
    • Where ΔK is the change in KE.Hence from equation (9) ,we see that workdone by a force on a body is equal to the change in kinetic energy of the body
    • Kinetic energy like work is a scalar quantity
    • Unit of KE is same as that of work i.e Joule
    • If there are number of forces acting on a body then we can find the resultant force ,which is the vector sum of all the forces and then find the workdone on the body
    • Again equation (9) is a generalized result relating change in KE of the object and the net workdone on it.This equation can be summerized as
      Kf=Ki+W           (10)
      which says that kinetic energy after net workdone is equal to the KE before net work plus network done.Above statement is also known as work-kinetic energy theorem of particles
    • Work energy theorem holds for both positive and negative workdone.if the workdone is positive then final KE increases by amount of the work and if workdone is negative then final KE decreases by the amount of workdone

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