Chapter 1. Rotational dynamics
Q. 1. State (1 mark) (July '22) and prove the principle (or law) of conservation of angular momentum.
(3 marks)
Ans. Principle (or law) of conservation of angular momentum: The angular momentum of a body is conserved if the resultant external torque on the body is zero.
Proof: Consider a moving particle of mass m whose position vector with respect to the origin at any instant is r
Then, at this instant, the linear velocity of this particle is \vec{v}=\frac{\vec{dr}}{dt}, its linear momentum is \vec{p}=m\vec{v} and its angular momentum about an axis through the origin is \vec{l}=\vec{r}\times\vec{p}.
Suppose its angular momentum \overline{I} 7changes with time due to a exerted on the particle. torque
The time rate of change of its angular momentum, \frac{\vec{dt}}{dt}=\frac{d}{dt}(\vec{r}\times\vec{p})=\frac{\vec{r}}{dt}\times\vec{p}+\vec{r}\times\frac{\vec{dp}}{dt} =xm+rxF =xF =\vec{\tau} where \frac{dp}{dt}=\vec{F}, the net force on the particle. Hence, \vec{r}=0,\frac{\vec{dl}}{dt}=0
(\cdot\cdot\vec{v}\times\vec{v}=0)
\vec{I}= constant, i.e.,\vec{l}i conserved. This proves the principle (or law) of conservation of angular momentum.
For a rigid body or a system of particles, 7 = Στ, and L = L, giving
dl dt Again for t=0, L= constant.
Chapter 2. Mechanical properties of fluids
(1 mark)
Q. 2. State Pascal's law.
Ans. Pascal's law: A change in the pressure applied to an enclosed fluid at rest is transmitted undiminished to every point of the fluid and to the walls of the container, provided the effect of gravity can be ignored.
Q. 3. State Laplace's law for a spherical membrane. (1 mark)
Ans. Laplace's law for a spherical membrane: The excess pressure 2T inside a small spherical liquid drop of radius R, is p-po= where p is R the pressure inside the drop, po is the pressure outside the drop and 7' is the surface tension of the liquid.
Q. 4. State and explain Newton's law of viscosity. (3 marks)
Ans. Newton's law of viscosity: In a steady flow of a fluid past a solid surface, a velocity profile is set up such that the viscous drag per unit area on a layer is directly proportional to the velocity gradient.
Explanation: When a fluid flows past a solid surface in a streamline
flow or when a solid body moves through a fluid, the force of fluid friction opposing the motion is called the viscous drag. The magnitude of the viscous drag of a fluid is given by Newton's law of viscosity.
do If is the velocity gradient, the viscous drag per unit area on a dy layer,
Fdv
A dy
F
A
do
dy
where the constant of proportionality, n, is called the coefficient of viscosity of the fluid.
(1 mark)
(3 marks)
Q. 5. State Stokes' law.
Derive Stokes' law using dimensional analysis.
Ans. Stokes' law: If a fluid flows past a sphere or a sphere moves through a fluid, for small enough relative speed v_{0} for which the flow is streamline, the viscous force on the sphere is directly proportional to the coefficient of viscosity of the fluid n, the radius of the sphere and the free-stream velocity \vec{v}_{n}
Derivation: Since the viscous force ƒ depends on \eta, r and LD_{0} , we can write f=-b\cdot\eta^{q}\cdot r^{\beta}\cdot v_{0}^{7} where b is a dimensionless proportionality constant. Therefore, [f]=[\eta]^{2}\cdot[r]^{\beta}\cdot[v_{0}]^{7} With [n] = ML-1T-1, [r]=L [f]=ML^{1}T^{-1} and [v_{0}]=LT^{-1} we get ML^{1}T^{-2}=(ML^{-1}T^{-1})^{\kappa}.(L)^{\beta}.(LT^{-1}) L^{1}T^{-2}=M^{\alpha}\cdot L^{-x+\beta+7}.T^{-x-\gamma} x=1.-x+\beta+\gamma=1 and -x-\gamma=-2 On solving, we get \gamma=-x+2=-1+2=1 Thus, By=1 f=-b\eta rv_{c} Inserting the value of b from theory and experiments, \vec{f}=-6\pi\eta~r\vec{v} This is Stokes' law. VM School \beta=1+\alpha-\gamma=1
Homogeneity of the above dimensional equation requires that Digest
Q. 6. State Bernoulli's principle.
(1 mark)
Ans. Bernoulli's principle: Where the velocity of an ideal fluid in streamline flow is high, the pressure is low, and where the velocity of a fluid is low, the pressure is high. OR
At every point in the streamline flow of an ideal (i.e., nonviscous and incompressible) fluid, the sum of the pressure energy, kinetic energy and potential energy of a given mass of the fluid is constant at every point.
(1 mark)
Q. 7. State the law of efflux.
Ans. Law of efflux (Torricelli's theorem): The speed of efflux for an open tank through an orifice at a depth h below the liquid surface is equal to the speed acquired by a body falling freely from rest through a vertical distance h.
Chapter 3. Kinetic theory of gases and Radiation
11 maak 106/4
Q. 8. State the law of equipartition of energy.
Ans. Law of equipartition of energy: For a gas equilibrium at absolute temperature 7, the average energy of 44 associated with each quadratic term (each degree of freedom), is where kg is the Boltzmann constant. 2
OR
The energy of the molecules of a gas, in thermal equilibrium at a thermodynamic temperature T'and containing large number of molecules, is equally divided among their available degrees of freedom, with the energy per molecule for each degree of freedom equal to Boltzmann constant. 1 2 kat. where knis Dit hest
Q. 9. State and prove Kirchhoff's law of heat (31 marks) radiation
Ans. Kirchhoff's law of neat radiation : Ata given temperature, the
ratio of the emissive power to the coefficient of absorption of a body is equal to the emissive power of a perfect blackbody at the same temperature for all wavelengths. S OR
For a body emitting and absorbing thermal radiation in thermal equilibrium, the emissivity is equal to its absorptivity.
Theoretical proof: Consider the following thought experiment: An ordinary body A and a perfect blackbody B are enclosed in an athermanous enclosure.
According to Prevost's theory of heat exchanges, there is a continuous exchange of radiant energy between each body and its surroundings. Hence, the two bodies, after some time, will attain the same temperature as that of the enclosure.
Let a and e be the coefficients of absorption and emission respectively, of body A. Let R and R, be the emissive powers of bodies A and B, respectively. Suppose that Q is the quantity of radiant energy incident on each body per unit time per unit surface area of the body.
Body A will absorb the quantity aQ per unit time per unit surface area
107/4
and radiate the quantity R per unit time per unit surface area. Since there is no change in its temperature, we must have aQ = R As body B is a perfect blackbody, it will absorb the quantity Q per unit
44
time per unit surface area and radiate the quantity R, per unit time per unit
surface area.
Since there is no change in its temperature, we must have Q=R From Eqs. (1) and (2), we get a = RR QR From Eq. (3), we get, R = R a This is Kirchhoff's law of heat radiation. By definition of coefficient of emission, R R e= From Eqs. (3) and (4), we get a S.M School Digest e This is another form of Kirchhoff's law of heat radiation.
(2)
(4)
Q. 10. State Wien's displacement law.
What is its significance?
(1 mark) (Sept. '21)
(1 mark)
Ans. Wien's displacement law: For a blackbody at an absolute temperature T, the product of T and the wavelength 4 corresponding to the maximum radiation of energy is a constant.
OR
As the temperature of a blackbody rises, the maximum of the spectral energy distribution curve is displaced towards the short-wavelength end
of the spectrum such that T= constant, where is the wavelength corresponding to the maximum radiation of energy and T is the absolute temperature.
Significance:
(1) It can be used to estimate the surface temperature of stars.
(2) It explains the common observation of the change of colour of a solid on heating-from dull red (longer wavelengths) to yellow (smaller wavelengths) to white (all wavelengths in the visible region).
Q. 11. State (1 mark) (Sept. '21) and explain the Stefan-Boltzmann
law. (I mark)
Ans. The Stefan-Boltzmann law: The rate of emission of radiant energy per unit area or the power radiated per unit area of a perfect blackbody is directly proportional to the fourth power of its absolute temperature. OR The quantity of radiant energy emitted by a perfect blackbody per unit time per unit surface area of the body is directly proportional to the fourth power of its absolute temperature.
The power per unit area radiated from the surface of a blackbody at Digest
Explanation:
an absolute temperature T is its emissive power or radiant power R temperature. According to the Stefan-Boltzmann law, ROC T ..RGT where the constanta is called Stefan's constant. sch Cowen at that
N.S.M
If A is the surface area of the radiated per unit time, is ΑσΤ. power, i.e., energy
Chapter 4. Thermodynamics
Q. 12. State the zeroth law of thermodynamics. (July '22) (1 mark)
Ans. Zeroth law of thermodynamics: If two systems are each in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.
Q. 13. State the first law of thermodynamics. (1 mark)
Ans. First law of thermodynamics: The change in the internal energy of a system (AU) is the difference between the heat supplied to the system (Q) and the work done by the system on its surroundings (W).
Mathematically, AUQ-W, where all quantities are expressed in the same units.
Q. 14. State the two forms of the second law of thermodynamics.
(2 marks)
Ans. Second law of thermodynamics:
(1) Kelvin-Planck statement: It is impossible to extract an amount of heat from a hot reservoir and use it all to do work W. Some amount of heat Q must be exhausted (given out) to a cold reservoir. This prohibits the possibility of a perfect heat engine.
This statement is also called the first form or the engine law or the engine statement of the second law of thermodynamics.
(2) Clausius statement: It is not possible for heat to flow from a colder body to a warmer body without any work having been done to accomplish this, i.e., there is no perfect refrigerator.
This statement is called the second form of the second law of thermodynamics.
Chapter 5. Oscillations
Digest
(1
Ans. The period of a simple pendulum at a given where I is the length of the simple pendulum and g is the acceleration due to gravity at that place. From the above expression, the laws of simple pendulum are as follows: choul
Q. 15. State the laws of a simple pendulum.
L
g
(1) Law of length: The period of a simple pendulum at a given place (g constant) is directly proportional to the square root of its length.
(2) Law of acceleration due to gravity: The period of a simple pendulum of a given length (L constant) is inversely proportional to the square root of the acceleration due to gravity.
. Toc 1
√g
(3) Law of mass: The period of a simple pendulum does not depend on the mass or material of the bob of the pendulum.
(4) Law of isochronism: The period of a simple pendulum does not depend on the amplitude of oscillations, provided that the amplitude is small.
Chapter 6. Superposition of waves
(3 marks)
Q. 16. State the laws of vibrating strings.
Ans. The fundamental frequency of vibration of a stretched string or wire is given by
2L
where L is the vibrating length, m the mass per unit length of the string and 7 the tension in the string. From the above expression, we can state the following three laws of vibrating strings:
(1) Law of length: The fundamental frequency of vibrations of
a stretched string is inversely proportional to its vibrating length, if the tension and mass per unit length are kept constant. (2) Law of tension: The fundamental frequency of vibrations of a
stretched string is directly proportional to the square root of the applied tension, if the length and mass per unit length are kept constant.
(3) Law of mass: The fundamental frequency of vibrations of stretched string is inversely proportional to the square root of its mass unit length, if the length and tension are kept constant. per
Q. 17. Give a brief account of Huygens' wave theory of light. Soos meest
Chapter 7. Wave optics
(2 marks)
Ans. Huygens' wave theory of
light (1678): cht (1678).
(1) Light emitted by a source propagates in the form of waves. Huygens' original theory assumed them to be longitudinal waves.
(2) In a homogeneous isotropic medium, light from a point source spreads by spherical waves.
(3) It was thought that a wave motion needed a medium for its propagation. Hence, the theory postulated a medium called luminiferous ether that exists everywhere, in vacuum as well as in transparent bodies. Ether had to be assigned some extraordinary properties a high modulus of elasticity to account for the high speed of light, zero density so that it offers no resistance to planetary motions, and perfect transparency.
(4) The different colours of light are due to the different wavelengths.
Q. 18. State the merits and demerits of Huygens' theory.
Ans. Merits of Huygens' theory:
(2 marks)
(1) Huygens' wave theory satisfactorily explains reflection and refraction as well as their simultaneity.
(2) In explaining refraction, the theory concludes that the speed of light in a denser medium is less than that in a rarer medium, which agreed with later experimental findings.
(3) The theory was later used by Young (1800-04), Fraunhofer and Fresnel (1814) to satisfactorily explain interference, diffraction and rectilinear propagation of light. The phenomenon of polarization could also be explained by considering the light waves to be transverse.
Demerits of the theory:
(1) It was found much later that the hypothetical medium, lumini- ferous ether, has no experimental basis. In 1905, Einstein discarded the idea
of ether completely.
(2) Phenomena like absorption and emission of light by atoms and molecules, photoelectric effect, Raman effect, Compton effect, etc., be explained on the basis of the wave theory.
Q. 19. State Huygens' principle.
Ans. Huygens' principle: Every point on a wavefront acts as a secondary source of light and sends out secondary wavelets in all directions. The secondary wavelets travel with the speed of light in the medium and are effective only in the forward direction. At any instant, the forward-going envelope or the surface of tangency to these these wavelets gives the position of the new wavefront at that instant.
Q. 20. State Brewster's law.
Ans. Brewster's law: The tangent of the polarizing angle is equal to the refractive index of the reflecting medium with respect to the surrounding
(1 mark)
(12). If 08 is the polarizing angle,
tan 08=2=2
Here is the absolute refractive index of the surrounding and 2 is that of the reflecting medium.
The angle og is also called the Brewster angle.
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NAVNEET 21 M. L. Q. SETS PHYSICS-STD. XII
Chapter 8. Electrostatics
Q. 21. State Gauss's law in electrostatics.
Ans. Gauss's law: The net flux through a closed surface in free
(1 mark)
(Sept. '21) (1 mark) Digest ) (1 mark)
Chapter 8. Electrostatics
Q. 21. State Gauss's law in electrostatics.
(1 mark)
Ans. Gauss's law: The net flux through a closed surface in free space is related to the net charge qenct that is enclosed by that surface and is given by
where && is the permittivity of free space and the electric flux Eds is integrated over the entire area of the surface.
Chapter 9. Current electricity
Q. 22. State Kirchhoff's laws for electrical circuit.
(July '22) (2 marks)
State the sign conventions used in these laws. (1) marks)
Are the laws applicable to both AC and DC networks? (mark)
Ans. Kirchhoff's circuital laws when steady currents are passing in an
electrical network are as follows:
Kirchhoff's first law (or current law or junction law) : algebraic sum of the currents at any junction is zero.
chool Digest
where I is the current in the ith conductor meeting junction.
Sign convention: A current entering the junction is taken while a current leaving the junction is taken as negative. as positive
Kirchhoff's second law (or voltage law or loop law): Around a closed loop of an electrical network, the algebraic sum of the emfs and the potential differences across all the circuit elements in that loop is zero. ΣΕ +EIR = 0
Sign convention: (a) In going round a loop, if we traverse a resistor in the direction of the current through it, the potential difference (p.d.) across the resistor is taken as negative. The p.d. is taken as positive if the direction in which we traverse the resistance is opposite to the current through it. (b) The emf of a cell is taken as positive when we traverse the cell from its negative terminal to the positive terminal while it is taken as negative when traversed in the opposite direction.
Kirchhoff's laws are applicable to both AC and DC circuits (networks). For AC circuits, instantaneous currents and voltages are taken in the summations.
Chapter 10. Magnetic fields due to electric current
Q. 23. State the Biot-Savart law (Laplace law) for the magnetic induction produced by a current element. Express it in vector form.
(2 marks)
Ans. Consider a very short segment of length dl of a wire carrying a current I. The product I dl is called a current element, the direction of the vector dl is along the wire in the direction of the current.
Biot-Savart law (Laplace law): The magnitude of the incremental magnetic induction dB produced by a current element I dl at a distance r from it is directly proportional to the magnitude Idl of the current element, the sine of the angle between the current element Idl and the unit vector r directed from the current element toward the point in question, and inversely proportional to the square of the distance of the point from the current element; the magnetic induction is directed perpendicular to both Idl and i as per the cross product rule. Idl sin 0
dB oc
. dB =
µo Idl sin 0
4π
2
B = (14) S.M School
(1)
... (2)
In vector form, dB = where = and the constant 4o is the permeability of free space. Equations (1) and (2) are called the Biot-Savart law.
Q. 24. State Ampère's circuital law.
(1 mark)
Ans. Ampère's circuital law: In free space, the line integral of magnetic induction around a closed path in a magnetic field is equal too times the net steady current enclosed by the path, where µo is the permeability of free space.
Chapter 11. Magnetic materials
Q. 25. Discuss Curie's law of paramagnetism.
(2 marks)
Ans. Curie's law: The magnetization of a paramagnetic material is directly proportional to the external magnetic field and inversely proportional to the absolute temperature of the material.
If a paramagnetic material at an absolute temperature T is placed in an external magnetic field of induction Be the magnitude of its magnetization M₂ oc Bext T M=C Bext T
where the proportionality constant C is called the Curie constant.
Q. 26. Explain ferromagnetism on the basis of the domain theory.
(3 marks)
Ans. Atoms of a ferromagnetic material have a permanent non-zero magnetic dipole moment, arising mainly from spin magnetic moments of the electrons.
According to the domain theory, a ferromagnetic material is composed of small regions called domains.
Weak B
Strong B
Stronger B
Chool Digest
(a)
(b)
(c)
Domains in a single crystal of iron
A domain is an extremely small region (e.g., a size of about 10 cm³) containing a large number of atoms (something like 1015 atoms as in common iron). Within each domain, the atomic magnetic moments of nearest-neighbour atoms interact strongly through exchange interaction (a quantum mechanical phenomenon) and align themselves parallel to each other even in the absence of an external magnetic field. A domain is, therefore, spontaneously magnetized to saturation.
QUESTION SET 4: LAWS AND THEORIES
113
In an unmagnetized material, however, the directions of magnetization of the different domains are so oriented that the net magnetization is zero.
When an external magnetic field is applied, the resultant magnetization
(d)
In an unmagnetized material, however, the directions of magnetization of the different domains are so oriented that the net magnetization is zero.
When an external magnetic field is applied, the resultant magnetization of the specimen increases. This is achieved in either of two ways: Either a domain that is favourably oriented grows in size at the expenses of a less favourably oriented domain, or the direction of magnetization of an entire
domain changes and tends to align along the external magnetic field. When a weak magnetic field is applied, favourably oriented domains grow in size by domain boundary displacement, Fig. (b). In strong fields, the domains change their magnetization by rotation, Figs. (c) and (d).
After the external field is removed, it may be energetically favourable for a domain's direction of magnetization to persist. Then, the specimen has a permanent magnetic dipole moment. This phenomenon, called magnetic remanence, i.e., magnetic retentivity, is the basis of the existence of permanent magnets.
Chapter 12. Electromagnetic induction
Q. 27. State Faraday's laws of electromagnetic induction. (1 mark)
Ans. Faraday's laws of electromagnetic induction:
First law: Whenever there is a change in the with a circuit, an emf is induced in the circuit. magnetic flux associated
Second law : The magnitude of the induced emf is directly proportional to the time rate of change of magnetic flux through the circuit.
Q. 28. State and explain Lenz's law of electromagnetic induction in the light of the principle of conservation of energy. (3 marks) OR
State Lenz's law of electromagnetic induction.
energy.
(mark) Explain how it is a consequence of the law of conservation of
(3 marks)
Ans. Lenz's law: The direction of the induced current is such as to oppose the change that produces it.
The change that induces a current may be (i) the motion of a conductor in a magnetic field or (ii) the change of the magnetic flux through a stationary circuit.
Chapter 14. Dual nature of radiation and matter
Q. 29. State the de Broglie hypothesis (1 mark) (July '22) and the corresponding equation. (2 marks)
115
QUESTION SET 4: LAWS AND THEORIES
Ans. De Broglie hypothesis: In 1924, Louis de Broglie proposed that the wave-particle duality may not be unique to light but a universal characteristic of nature, so that a particle of matter in motion also has a wave or periodicity associated with it which becomes evident when the magnitude of Planck's constant h cannot be ignored.
De Broglie equation: A particle of mass m moving with a speed v will, under suitable experimental conditions, exhibit the characteristics of a wave of wavelength & given by h mv h P where p = mo = magnitude of the momentum of the particle. This relation is called the de Broglie equation, and the wavelength associated with a particle momentum is called its de Broglie wavelength. The corresponding waves are termed as matter waves. = =
Chapter 15. Structure of atoms and nuclei
Q. 30. State the postulates of Bohr's atomic model. State the first and second postulates of Bohr's atomic model. (3 marks) gest
(July '22) (2 marks)
Ans. The postulates of Bohr's atomic model (for the hydrogen atom):
(1) The electron revolves with a constant speed in a circular orbit around the nucleus. The necessary centripetal force is the Coulomb force of attraction of the positive nuclear charge on the negatively charged electron.
(2) The electron can revolve without radiating energy only in certain orbits, called allowed or stable orbits, in which the angular momentum of the electron is equal to an integral multiple of h/2n, where h is Planck's constant.
(3) Energy is radiated by the electron only when it jumps from one of its orbits to another orbit having lower energy. The energy of the quantum of electromagnetic radiation, i.e., the photon, emitted is equal to the energy
difference of the two states
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NAVNEET 21 M. L. Q. SETS PHYSICS-STD. XII
Q. 31. State the equations corresponding to Bohr's atomic model.
(3 marks)
Ans. Let m_{e} be the mass ande the charge on an atomic electron in the ath circular orbit of radius r,, of the atom and Ze the charge on the nucleus, where n and Z are positive integers. Let the orbital speed of the electron in the ath orbit be \upsilon_{n} Then,
centripetal force on the electron m Coulomb force on the electron \frac{Ze^{2}}{4\pi\epsilon_{0}r^{2}}
where E_{0} is the permittivity of free space. For hydrogen atom, Z=1. By Bohr's postulate of quantization of angular momentum of the electron, m_{e}\mathcal{\nu}_{n}r_{n}=n\frac{h}{2\pi}
(2)
where h is Planck's constant and 1n=1,2, By Bohr's radiation postulate, the atom radiates energy in the form of a photon of frequency v and energy hv, only when the electron makes a transition from a higher orbit (energy state) to a lower orbit (energy state). If E, and E, are the energies of the electron in the initial (higher) energy state and final (lower) energy state, E_{i}-E_{f}=hv
MS Canad Prosigumon -
(3)
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