Wave Optics

 1. The locus of all particles of the medium vibrating in the same phase at a  given instant is known as wavefront. Depending on the shape of sources of light, wavefront can be of three types.

2. Spherical wavefront: When the source of light is a point source, the wavefront is spherical.

3. Cylindrical wavefront: When the source of light is linear, the wavefront is cylindrical.

4. Plane wavefront: When the point source or linear source of light is at a very large distance, a small portion of the spherical or cylindrical wavefront appears to be plane. Such a wavefront is known as a plane wavefront.

5. Huygens principle: According to Huygens principle, (a) Every point on a given wavefront (primary wavefront) acts as a fresh source of new disturbance, called secondary wavelets. (b) The secondary wavelets spread out in all directions with the speed of light in the medium. A surface touching these secondary wavelets tangentially in the forward direction at any instant gives the new (secondary) wavefront at that instant.

6. Huygen's principle can be used to verify laws of reflection and refraction.

7. The sources of light, which emit continuously light waves of the same wavelength (monochromatic light), same frequency, and in the same phase or have a constant phase difference with time are known as coherent sources. Two sources of light that do not emit light waves with a constant phase difference are called incoherent sources. 

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Dual Nature of Matter and Radiation

 1.  Quantum mechanics is a mathematical model that describes the behavior of the particles on an atomic and subatomic scale.

2. According to Planck's quantum theory, the Energy of light comes in form of packets (it is not a material particle). These packets are called Quanta or Photons.

3. Energy of the photon depends on frequency. Frequency does not change with changing medium. Energy is given by \[ E = hf = (6.636 \times 10^{-34})f\] 

4. Energy of the photon having wavelength $ \lambda $ is given by \[ E = \frac{hc}{\lambda} = \frac{12400}{\lambda (\ in \ Angstrom)}eV  \] and momentum of photon is given by \[  p = \frac{h}{\lambda}  \] where $ h $ is plack's constant and $ c $ is speed of light.

5. Photons are electrically neutral particles. These are not deflected by the electric and magnetic fields.

6. Photons can collide with material particles like electrons. During the collision, total Energy and total momentum remain constant i.e. collision is elastic.   

7. Rest mass of the photon is zero. This means that photon does not exist at rest. Equivalent mass of the photon is given by, \[ m = \frac{hf}{c^2} \]

8. Intensity is defined as energy radiated per unit time per unit area. i.e. \[ I = \frac{E}{tA}= \frac{P}{A} \] where P is power. SI unit of intensity is  $  \frac{W}{m^2} $.

9. A source is at power P and emitting radiation energy of wavelength $ \lambda $ then number of photons emitted by the source per second is given by, \[ n = \frac{P}{E}= \frac{P \lambda}{hc} \] 

10. Photon Flux is the number of photons incident on a surface normally per second per unit area. \[ Photon \ Flux, \phi = \frac{Intensity}{Energy \ of \ a \ photon} = \frac{I\lambda}{hc} \]

11. When radiation is incident on the surface then it will apply some force on the surface.  If radiation falls on the surface at some angle and is reflected by the surface, the average force is given by, \[  F = \frac{2IAcos^2\theta}{c} \]Radiation pressure is given by, \[ P = \frac{2Icos^2\theta}{c} \]

Study of Photoelectric Effect

12. Minimum energy required to escape an electron from the surface is known as Work Function $ \phi $.  The minimum frequency of incident light that is just capable of ejecting electrons from metal is called the threshold frequency.

        Metal            Work Function (eV)

        Cesium                    1.9

        Potassium                2.2

        Sodium                    2.3

        Lithium                    2.5

        Calcium                   3.2

        Copper                     4.5

        Silver                       4.7

        Platinum                  5.6

13. When electromagnetic radiation of suitable wavelength is incident on the metal surface such that electrons emitted from the surface, this phenomenon is known as Photoelectric emission.

14. When energy is given in form of heat to the metal surface such that electrons are emitted from the surface, this phenomenon is known as Thermionic emission.

15. When the strong field is applied in such a way that electrons get accelerated and overcome the potential barrier, this phenomenon is known as Field emission.

16. The phenomenon of the photoelectric effect was discovered by Heinrich Hertz in 1887. While performing an experiment for the production of electromagnetic waves by means of spark discharge. Hertz observed that sparks occurred more rapidly in the air gap of his transmitter when ultraviolet radiations were directed at one of the metal plates. Hertz could not explain his observations.

17. Phillip Lenard observed that when ultraviolet radiations were made incident on the emitter plate of an evacuated glass tube enclosing two metal plates (called electrodes), current flows in the circuit, but as soon as ultraviolet radiation falling on the emitter plate was stopped, the current flow stopped. These observations indicate that when ultraviolet radiations fall on the emitter (cathode) plate, the electrons are ejected from it, which are attracted towards the anode plate. The electrons flow through the evacuated glass tube, complete the circuit and current begins to flow in the circuit.

18. Hallwachs studied further by taking a plate and an electroscope. The zinc plate was connected to an electroscope. He observed that: (i) When an uncharged zinc plate was irradiated by ultraviolet light, the zinc plate acquired a positive charge. (ii) When a positively charged zinc plate is illuminated by ultraviolet light, the positive charge of the plate was increased. (iii) When a negatively charged zinc plate was irradiated by ultraviolet light, the zinc plate lost its charge. All these observations show that when ultraviolet light falls on zinc plate, the negatively charged particles (electrons) are emitted.

19. Einstein's Photoelectric equation is \[ hf = \phi + KE \implies \frac{1}{2}mv^2 = hf - \phi \]where f is incident frequency and v is maximum velocity of the electron.

20. Photoelectric current is directly proportional to the intensity of the incident light keeping frequency and potential the same. On increasing the intensity, photoelectric current will increase and vice-versa. It does not depend on the incident energy.

21.  If the collector is given negative potential with respect to the emitter, then at some potential electrons will not reach the collector. Due to this, there will be no current in the circuit. If the potential of the collector is further increased in the negative, no current will be in the circuit. This potential at which the photoelectric current is zero is known as Stopping Potential. \[ eV_0 = K.E. = \frac{1}{2}mv^2 \]Stopping potential is independent of the intensity of the incident light. It depends only on the incident energy. 

22. Kinetic energy of the electrons depends only on the incident energy, not on the intensity of the incident light.

23. There is no time lag between the incidence of light and the emission of photoelectrons.

24. De Broglie gave the wavelength associated with moving object, which is given by, \[ \lambda = \frac{h}{mv} = \frac{h}{p}  \]where m is mass of the object and v is the speed of the object. 

25. If E is the kinetic energy of the electron, then the De-Broglie wavelength of the electron will be \[ \lambda = \frac{h}{\sqrt{2mE}} \] 

26. If electron is accelerated by potential V then De-Broglie wavelength of electron will be \[ \lambda = \frac{h}{\sqrt{2meV}} \]

27. Davisson and Germer Experiment gives the experimental evidence for the wave nature of the electrons.

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Physics Syllabus class 11

CBSE SYLLABUS (2018-19)

Unit I: Physical World and Measurement

Chapter–1: Physical World Physics-scope and excitement; nature of physical laws; Physics, technology and society. 

Chapter–2: Units and Measurements Need for measurement: Units of measurement; systems of units; SI units, fundamental and derived units. Length, mass and time measurements; accuracy and precision of measuring instruments; errors in measurement; significant figures. Dimensions of physical quantities, dimensional analysis and its applications. 

Unit II: Kinematics 

Chapter–3: Motion in a Straight Line Frame of reference, Motion in a straight line: Position-time graph, speed and velocity. Elementary concepts of differentiation and integration for describing motion, uniform and non-uniform motion, average speed and instantaneous velocity, uniformly accelerated motion, velocity - time and position-time graphs. Relations for uniformly accelerated motion (graphical treatment). 

Chapter–4: Motion in a Plane Scalar and vector quantities; position and displacement vectors, general vectors and their notations; equality of vectors, multiplication of vectors by a real number; addition and subtraction of vectors, relative velocity, Unit vector; resolution of a vector in a plane, rectangular components, Scalar and Vector product of vectors. Motion in a plane, cases of uniform velocity and uniform acceleration-projectile motion, uniform circular motion. 

Unit III: Laws of Motion 

Chapter–5: Laws of Motion Intuitive concept of force, Inertia, Newton's first law of motion; momentum and Newton's second law of motion; impulse; Newton's third law of motion. Law of conservation of linear momentum and its applications. Equilibrium of concurrent forces, Static and kinetic friction, laws of friction, rolling friction, lubrication. Dynamics of uniform circular motion: Centripetal force, examples of circular motion (vehicle on a level circular road, vehicle on a banked road). 

Unit IV: Work, Energy, and Power

Chapter–6: Work, Engery and Power Work done by a constant force and a variable force; kinetic energy, work-energy theorem, power. Notion of potential energy, potential energy of a spring, conservative forces: conservation of mechanical energy (kinetic and potential energies); non-conservative forces: motion in a vertical circle; elastic and inelastic collisions in one and two dimensions. 

Unit V: Motion of System of Particles and Rigid Body

Chapter–7: System of Particles and Rotational Motion Centre of mass of a two-particle system, momentum conservation and center of mass motion. Centre of mass of a rigid body; centre of mass of a uniform rod. Moment of a force, torque, angular momentum, law of conservation of angular momentum and its applications. Equilibrium of rigid bodies, rigid body rotation and equations of rotational motion, comparison of linear and rotational motions. Moment of inertia, radius of gyration, values of moments of inertia for simple geometrical objects (no derivation). Statement of parallel and perpendicular axes theorems and their applications. 

Unit VI: Gravitation  

Chapter–8: Gravitation Kepler's laws of planetary motion, universal law of gravitation. Acceleration due to gravity and its variation with altitude and depth. Gravitational potential energy and gravitational potential, escape velocity, orbital velocity of a satellite, Geo-stationary satellites. 

Unit VII: Properties of Bulk Matter 

Chapter–9: Mechanical Properties of Solids Elastic behaviour, Stress-strain relationship, Hooke's law, Young's modulus, bulk modulus, shear modulus of rigidity, Poisson's ratio; elastic energy. 

Chapter–10: Mechanical Properties of Fluids Pressure due to a fluid column; Pascal's law and its applications (hydraulic lift and hydraulic brakes), effect of gravity on fluid pressure. Viscosity, Stokes' law, terminal velocity, streamline and turbulent flow, critical velocity, Bernoulli's theorem and its applications. Surface energy and surface tension, angle of contact, excess of pressure across a curved surface, application of surface tension ideas to drops, bubbles and capillary rise. 

Chapter–11: Thermal Properties of Matter Heat, temperature, thermal expansion; thermal expansion of solids, liquids and gases, anomalous expansion of water; specific heat capacity; Cp, Cv - calorimetry; change of state - latent heat capacity. Heat transfer-conduction, convection and radiation, thermal conductivity, qualitative ideas of Blackbody radiation, Wein's displacement Law, Stefan's law, Greenhouse effect. 

Unit VIII: Thermodynamics

Chapter–12: Thermodynamics Thermal equilibrium and definition of temperature (zeroth law of thermodynamics), heat, work, and internal energy. First law of thermodynamics, isothermal and adiabatic processes. Second law of thermodynamics: reversible and irreversible processes, Heat engine and refrigerator. 

Unit IX: Behaviour of Perfect Gases and Kinetic Theory of Gases  

Chapter–13: Kinetic Theory Equation of state of a perfect gas, work done in compressing a gas. Kinetic theory of gases - assumptions, concept of pressure. Kinetic interpretation of temperature; rms speed of gas molecules; degrees of freedom, law of equi-partition of energy (statement only) and application to specific heat capacities of gases; concept of mean free path, Avogadro's number. 

Unit X: Mechanical Waves and Ray Optics  

Chapter–14: Oscillations and Waves Periodic motion - time period, frequency, displacement as a function of time, periodic functions. Simple harmonic motion (S.H.M) and its equation; phase; oscillations of a loaded springrestoring force and force constant; energy in S.H.M. Kinetic and potential energies; simple pendulum derivation of expression for its time period. Free, forced and damped oscillations (qualitative ideas only), resonance. Wave motion: Transverse and longitudinal waves, speed of wave motion, displacement relation for a progressive wave, principle of superposition of waves, reflection of waves, standing waves in strings and organ pipes, fundamental mode and harmonics, Beats, Doppler effect. 

Chapter–15: RAY OPTICS: Reflection of light, spherical mirrors, mirror formula, refraction of light, total internal reflection and its applications, optical fibres, refraction at spherical surfaces, lenses, thin lens formula, lensmaker's formula, magnification, power of a lens, combination of thin lenses in contact, refraction and dispersion of light through a prism. Scattering of light - blue colour of sky and reddish apprearance of the sun at sunrise and sunset. Optical instruments: Microscopes and astronomical telescopes (reflecting and refracting) and their magnifying powers.

**Syllabus taken from cbse website.

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Unit and Dimensions: Problems

1. In SI system, the unit of Temperature is

    (a) Degree Celcius

    (b) Degree Centigrade

    (c) Degree Kelvin

    (d) Degree Fahrenheit 

[d]

2. Which of the following is a unit of distance?

     (a) Metre

     (b) Astronomical unit

     (c) Light year

     (d)  All of the above

[d]

3. What is the dimension of surface tension:

     (a) $[M^2L^2T^{-2} $

     (b) $[M^2LT^{-2}]$

     (c) $ [ML^0T^{-2}]$

     (d) None of these

[c]

4. Which of the following have the same dimensions?

     (a) Stress

     (b) Bulk modulus

     (c) Thrust

     (d) Energy Density

[a and b]

5. If C and R denote the capacitance and resistance, then the dimension of RC is:

     (a) $[M^0L^0T ]$

     (b) $[ML^0T] $

     (c)  $[MLT]$

     (d)  $[M^0L^0T^2]$

[a]

6. Write any two physical quantities which have the dimension of Energy? [Torque, Work]

7. Is it possible to add any two physical quantities? [No]

8. Force on a sphere of radius 'a' moving in the medium with velocity 'v' is given by $F = 6\pi \eta av $. Find the dimension of $ \eta $. [$ML^{-1}T^{-1}$]

9. What is the dimensional formula for Planck's constant? [$ML^2T^{-1}$]

10. In the formula, $X = 3YZ^2$, X and Z have the dimensions of capacitance and magnetic induction respectively. Find the dimension of Y in MKSQ system. [$M^{-3}L^{-2}T^{4}Q^{4}$]

11. The equation of state for a real gas is given by $ (p+\frac{a}{V^2})(v-b) = RT $. Find the dimensions of 'a' and 'b'. [$[ML^5T^{-2}]$, $[L^3]$]

12. Find the dimension of $\frac{1}{2}\epsilon_0 E^2$. [$ML^{-1}T^{-2}$]

13. A quantity X is given by $ \epsilon_0 L\frac{\Delta V}{\Delta t} $ where $\epsilon_0 $ is the permittivity of the free space, L is the length,  $\Delta$V is the potential difference and ${\Delta t}$ is a time interval. The dimensional formula for X is the same as that of  (a) Resistance (b) Charge (c) Voltage (d) Current. [d]

14. Pressure depends on distance as $ p = \frac{\alpha}{\beta}e^{-\frac{\alpha z}{k \theta}} $, where $\alpha$, $\beta$ are constant, z is the distance, k is Boltzman's constant and $\theta$ is temperature. Find the dimension of $\beta$. [$L^{2}$]

15. Which of the following pair (s) has the same dimension?

     (a) Torque and work

     (b) Angular momentum and Work

     (c) Energy and Young's modulus

     (d) Light-year and Wavelength 

     (e) Reynold number and co-efficient of friction

     (f) Curie and Frequency of the light wave

     (g) Latent heat and gravitational potential

     (h) Planck's constant and torque

[a,d,e,f,g]

16. Let $ [\epsilon_0]$ denote the dimensional formula of permittivity of the vacuum, and $[\mu_0]$ that of permeability of the vacuum. Find their dimensional formula in term of mass M, length L, time T, and electric current I. [$\epsilon_0 = [M^{-1}L^{-3}T^4I^2]$, $\mu_0= [MLT^{-2}I^{-2}]$]

17. Planck's constant h, speed of light c, and gravitational constant G are used to form a unit of length L and a unit of mass M. Then find the correct option (s) is (are): 

   (a) $M \propto \sqrt{c}$ 

   (b) $M \propto \sqrt{G}$ 

   (c) $L \propto \sqrt{h}$ 

   (d) $L \propto \sqrt{G}$

[a,c,d]

18. In term of potential difference V, electric current I, permittivity $\epsilon_0$, permeability $\mu_0$ and speed of light c, the dimensionally correct equation(s) is(are):

    (a) $\mu_0 I^2 = \epsilon_0V^2$ 

    (b) $ \mu_0 I = \epsilon_0V$ 

    (c) $ I = \epsilon_0 cV$ 

    (d) $ \mu_0 c I = \epsilon_0 V$ 

[a,c]

19. A length-scale (l) depends on the permittivity ($\epsilon $) of a dielectric material. Boltzmann constant ($k_B$), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and charge (q) carried by each of the particles. Which of the following expression(s) for l is/are dimensionally correct? 

    (a) $l = \sqrt{\frac{nq^2}{\epsilon k_B T}}$ 

    (b) $l = \sqrt{\frac{\epsilon k_B T}{nq^2}} $ 

    (c) $ l = \sqrt{\frac{q^2}{\epsilon n^{2/3}k_B T}}$ 

    (d) $l = \sqrt{\frac{q^2}{\epsilon n^{1/3}k_B T}}$

[b,d]

20. Give the MKS unit of each of the following: 

    (a) Young's Modulus

    (b) Magnetic Induction

    (c) Power of lens

[$N/m^2$, Tesla, Dioptre]

21. A gas bubble, from an explosion underwater, oscillate with a period T proportional to $p^ad^bE^c$, where  'P' is the static pressure, 'd' is the density of the water, and 'E' is the total energy of the explosion. Find the values of a, b, and c. [$a = -\frac{5}{6},b=\frac{1}{2},c=\frac{1}{3}$]

22. Write the dimensions of the following in terms of mass, time, length, and charge

    (a) Magnetic flux 

    (b) Rigidity modulus

[$[ML^2T^{-1}Q^{-1}], [ML^{-1}T^{-2}]$]

23. Match the following with their dimensions where Q is for charge:

Column I	Column II
(A) Angular momentum	(a) $[ML^2T^{-2}]$
(B) Latent Heat	(b) $[ML^2Q^{-2}]$
(C) Torque	(c) $[ML^2T^{-1}]$
(D) Capacitance	(d) $[ML^3T^{-1}Q^{-2}]$
(E) Inductance	(e) $[M^{-1}L^{-2}T^{2}Q^2]$
(F) Resistivity	(f) $[L^2T^{-2}]$

[$(A) \to (c),(B) \to (f) ,(C)\to (a), (D)\to (e), (E)\to (b),(F)\to (d)$]

24. Match column I with column II:

Column I	Column II
(A) Capacitance	(i)                  Ohm-second
(B) Inductance	(ii)                Coulomb2-joule-1
(C) Magnetic Induction	(iii)               Coulomb (volt)-1
	(iv)               Newton (amp-metre)-1
	(v)                Volt-second (ampere)-1

[$(A) \to (ii),(iii), (B)\to (i),(v), (C)\to (iv)$]

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Mathematics for Physics II

Geometry

1. Two triangles are similar when the ratio of sides is same and angles are same. If two triangles have same configuration i.e. same sides and same angles, then triangles are congruent.

2. Vertical opposite angles are equal.

3. Alternate angles are equal.

4. General Equation of the line is $ y = mx + c$ where m is the slope and c is the intercept.

5. Slope of any line passing through points $ (x_1, y_1) $ and $ (x_2, y_2) $ is $ m = \frac{y_2-y_1}{x_2-x_1} $. Slope is inclination of the line with positive x-axis. If $ \theta $ is the angle made by the line with positive x-axis then slope will be $ m = tan\theta $. For horizontal line, slope is 0 and for vertical line, slope is $ \infty $. 

6. Equation of line passing through points $ (x_1, y_1) $ and $ (x_2, y_2) $ is $ (y-y_1) = \frac{y_2-y_1}{x_2-x_1} (x-x_1) $

7. If we are given two lines, then lines $y=m_1x+c_1$ and $y=m_2x+c_2$ are parallel only when these have the same slopes, i.e. $m_1 = m_2 $. 

8. Two lines $y=m_1x+c_1$ and $y=m_2x+c_2$ are perpendicular when the product of the slopes of two lines is -1, i.e. $m_1m_2 = -1$.

9. Equation of the circle is of form, $ ax^2 + ay^2 + 2bx + 2cy + d = 0 $.

10. Equation of the parabola is either of the form $ (y-c)^2=4a(x-b) $, or $ (x-c)^2 = 4a(y-b) $ or $ y = ax^2 - bx $ or $ x = ay^2-by $.

11. Equation of ellipse is $ \frac{x^2}{a^2}+ \frac{y^2}{b^2}  = 1 $.

12. Equation of hyperbola is $\frac{x^2}{a^2} - \frac{y^2}{b^2}  = 1 $. 

Calculus

1. Function is defined as a operation in which element of the set first are related to elements of second set by some relation. 

Domain of the function is the set all elements (values) which function can take. Range is set of values which function can give. 

Let any function $ y = f(x) $. Let us consider that curve $ y=f(x) $ passes through $(x,y) $. If we need to find the tangent at this point, then slope of the curve will be given by $ m = \frac{dy}{dx}=\frac{d f(x)}{dx} = f'(x) $ where $ \frac{dy}{dx}$ is derivative of y with respect to x.

Derivative of some functions are given below:

1. $ y = constant \implies \frac{dy}{dx}=0$

2. $ y = x^n  \implies \frac{dy}{dx} = nx^{n-1} $ 

3. $ y = sinx \implies \frac{dy}{dx}=cosx$

4. $ y = cosx \implies \frac{dy}{dx}= -sinx $

5. $ y = tanx \implies \frac{dy}{dx}= sec^2x $

6. $ y = lnx \implies \frac{dy}{dx}=\frac{1}{x} $

Some integral formulas:

1. $ \int x^n dx = \frac{x^{n+1}}{n+1} + C $ 

2. $ \int sinx dx = -cosx + C$

3. $ \int cosx dx = sinx + C $

4. $ \int tanx dx = log|sec x| + C$

Problem for Practice:

1. Find the equation of the line which is parallel to the given line $ y = 6x + 4 $ and passes through point (4, 6).

2. Find the  equation of the curve for which every point of the curve is at same distance from the point (3,2).

3. Differentiate the following with respect to x:

 (a) $ y = x^2 + 4x $ 

 (b) $z = 5x^3+10$

 (c) $y = sin4x + log|x|$

 (d) $y = sin^2x$

 (e)$ y = cos5x + tan2x + log|sin x| $

4. Integrate the following functions:

 (a) $y = sinx$

 (b)$y=5x^2 + 4x$

 (c) $y = sin^2x$

 (d) $ y = cos5x $

 (e)$y = logx $

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