HSC Physics SAMPLE PAPER-2 – Maharashtra Board Solutions

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HSC Physics SAMPLE PAPER-2

Questions

Section A

Select and write the correct answers to the following questions:

(i) The Kinetic energy of a body is

and its

. Then its angular momentum will be:
(a)

(b)

(c)

(d)

(ii) The ratio of emissive power of perfectly blackbody at and is:
(a)
(b)
(c)
(d)

(iii) A standing wave is produced on a string fixed at one end with the other end free. The length of the string:

(a) Must be an odd integral multiple of

(b) Must be an odd integral multiple of

(d) Must be an even integral multiple of

(iv) Kirchhoff’s first law, i.e, at a junction, deals with the conservation of.
(a) Charge
(b) Energy
(c) Momentum
(d) Mass

(v) The internal energy of one mole of argon at is —
(a)
(b)
(c)
(d)

(vi) In a certain unit, the radius of gyration of a uniform disc about its central and transverse axis is . Its radius of gyration about a tangent in its plane (in the same unit) must be:
(a)
(b) 2.5
(c)
(d)

(vii) A charged particle is in moving having initial velocity when it enter into a region of uniform magnetic field perpendicular to . Because of the magnetic force the kinetic energy fo the particle will:

(a) Remain unchanged

(b) Get reduced

(d) be reduced to zero (viii) What is the energy required to build up a current of in an inductor of ?
(a)
(b)
(c)
(d)

(ix) In series LCR circuit, at resonance, phase difference between current and emf of source is
(a)
(b)
(c)
(d) 0 (zero) rad

(x) An electron, a proton, an -particle and a hydrogen atom are moving with the same kinetic energy. The associated de Broglie wavelength will be longest for:
(a) Electron
(b) Proton
(c) -particle
(d) Hydrogen atom

19. Answer the following questions:

(i) What should be the order of the size of an obstacle of aperture to produce diffraction of light?

(ii) The emissive power of a sphere of area is . What is the amount of heat radiated by the spherical surface is 20 second?

(iii) Give an example of some familiar process in which no heat is added to or removed from a system but the temperature of the system changes.

(iv) A plane wave front of light of wavelength is incident on two stats on a screen perpendicular to the direction of light rags. If the total separation of 10 bright fringes on a screen away is . Find distance between the slits.

(v) What do you mean by electromagnetic induction?

(vi) Explain why the inductance of two coils connected in parallel is less than the inductance of either coil

(vii) What do you understand by the term wave-particle duality? Where does it apply?

(viii) The expression defines the momentum of photon. Can this expression be used for momentum of an electron or proton?

20. Section B

21. Attempt any Eight of the following questions:

On what factors does the frequency of a conical pendulum depends? Is it independent of some factors?
The total energy of a body of mass performing S.H.M. is . Find its speed while crossing the centre of the path.
What is a wavefront? How is it related to rays of light?
A capacitor is charged by a supply. It is then disconnected from the supply and is connected to another uncharged capacitor. How much electrostatic energy of the first capacitor is lost in the form of heat and electromagnetic radiation?
State the uses of a potentiometer.
Calculate the value of magnetic field at a distance of 2 from a very long straight wire carrying a current of . (Given: ).
An iron rad is placed par all is to magnetic field of intensity . The magnetic flux through the rod is wb and its cross sectional area is . Calculate the magnetic permeability of the rod.
An aircraft of wing span of flies horizontally in earth’s magnetic field of at a speed of 400 . Calculate the emf generated between the tips of the wings of the aircraft.
The safest way to protect yourself from lightening is to be inside a car. Justify.
What do you understand by the term wave-particle duality? Where does it apply?
State the difficulties faced by Rutherford’s atomic model.
The common-base DC current gain of a transistor is 0.967. If the emitter current is . What is the value of base current?

22. Section C

23. Attempt any Eight of the following questions:

Deceive an expression that relates the angular momentum with the angular velocity of a rotating right body.
A rectangular wire frame of size is dipped in a soap solution and taken out. A soap film is formed, if the size of the film is changed to , calculate the work done in the process. The surface tension of soap film is .
State the law of equipartition of energy and hence calculate motor specific heat of mono- and di-atomic gases at constant volume and constant pressure.
A gas contained in a cylinder fitted with a frictionless piston expands against a constant external pressure of from a volume of 5 litres to a volume of 10 litres. In doing so it absorbs of thermal energy from its surroundings. Determine the change in internal energy of system.
Obtain the expression for the period of a magnet vibrating in a uniform magnetic field and performing S.H.M.
A wave of frequency is travelling with a speed of .

(i) What is the phase difference between two displacements at a certain point at times apart? (ii) What will be the smallest distance between two points which are out of phase at an instant of time?

Describe Young’s double slit interference experiment and derive conditions for occurrence of dark and bright fringes on the screen. Define fringe width and derive a formula for it.
A potentiometer wire has a length of and resistance of . It is connected in series with the cell of emf 4 volt and internal resistance Calculate the potential drop per centimetre of the wire.
An electron is moving with a speed of in a magnetic field of perpendicular to its path. What will be the radius of the path? What will be frequency and the energy in keV? [Given: mass of electron , charge ,
Obtain and expression for orbital magnetic moment of an electron rotating about the nucleus in an atom.
A plane coil of 10 turns is tightly wound around a solenoid of diameter having 400 turns per centimetre. The relative permeability of the core is 800 . Calculate the mutual inductance.
Why is the emitter, the base and the collector of a BJT doped differently?

24. Section D

25. Attempt any Three of the following questions:

(i) State and explain Lenz’s law in the light of principle of conservation of energy.

(ii) How long will it take for a radioactive sample to reduce to of its actual activity. (Half life of the sample is 5.3 years)

(i) State the characteristics of progressive waves.

(ii) One hundred twenty-five small liquid drops, each carrying a charge of and each of diameter 0.1 form a bigger drop. Calculate the potential at the surface of the bigger drop.
29. (i) Give an example of some familiar process in which heat is added to an object, without changing its temperature.

(ii) Find kinetic energy of 5 litre of a gas at S.T.P. given standard pressure is .

Determine the series limit of Balmer, Paschen and Pfund series, given the limit for Lyman series is .
Derive expression for excess pressure inside a drop of liquid?

26. [.].] Answer Key

27. Section A

(i) (b)

(ii) (b)

(iii) (a) Must be an odd integral multiple of

(iv) (a) Charge

(v) (b)

(vi) (b) 2.5

(vii) (a) Remain unchanged

(viii) (a)

(ix) (d) 0 (zero) rad

(x) (a) Electron

(i) For pronounced diffraction, the size of an obstacle or aperture should be of the order of the wavelength of light or greater.

(ii) Given:

This is the required quantity.

(iii) Adiabatic compression is the process in which no heat is transferred to or from the system but the temperature of the system changes. When we compress gas in an adiabatic process the volume of the gas will decrease and the temperature of the gas rises as it is compressed which we have seen the warming of a bicycle pump. Conversely, the temperature falls when the gas expands but the heat remains constant throughout the process.

(iv) Given,

Distance between 10 bright fringes

Fringe

Width,

but

(v) The phenomenon of production of emf in a conductor or circuit by a changing magnetic flux through the circuit is called electromagnetic induction.

(vi) Assuming that their mutual inductance can be ignored, the equivalent inductance of a parallel combination of two coils is given by

Hence, the equivalent inductance is less than the inductance of either coil.

(vii) Depending upon experimental conditions of the structure of matter, electromagnetic radiation and material particles exhibit wave nature or particle nature. This is known as wave-particle duality.

It applies to all phenomena. The wave nature and particle nature are liked by the de Broglie relation , where is the wavelength of matter waves, also called de Broglie waves I Schrodinger waves, is the magnitude of the momentum of a particle or quantum of radiation and is the universal constant called Planck’s constant.

(viii) The definition of momentum of photon is given by de-Broglie hypothesis is applicable only to those particles whose rest mass is zero and travel with speed of light.

As electron and proton have finite rest mass and their speed is not equal to speed of light, this expression of momentum, cannot be used for electron and proton.

28. Section B

The frequency of a conical pendulum of string length and semi-vertical angle is

where is the acceleration due to gravity as the place. From the above expression, we can see that

(i)

(ii)

(iii)

(If increases, decreases and increases) (iv) The frequency is independent of the mass of the bob.

Given: Mass , Energy

The speed of the body while crossing the centre of the path (mean position) is and the total energy is kinetic energy.

Wavefront or wave surface: The locus of all points where waves starting simultaneously from a source reach at the same instant of time and hence the particles at the points oscillate with the same phase is called a wavefront or wave surface.

Consider a point source of light 0 in a homogeneous isotropic medium in which the speed of light is . The source emits light in all directions. In time t, the disturbance (light energy) from the source, covers a distance in all directions, i.e, it reaches out to all points which are at a distance from the point source. The locus of these points which are in the same phase is the surface of a sphere with centre 0 and radius . It is a spherical wavefront. In a given medium, a set of straight lines can be drawn which are perpendicular to the wavefront. According to Huygens, these straight lines are the rays of light. Thus, rays are always normal to the wavefront. In the case of a spherical wavefront, the rays are radial. If a wavefront has travelled a large distance away from the source, a small portion of this wavefront appears to be plane. This part is a plane wavefront.

Given:

The electrostatic energy in the capacitor

The charge on this capacitor,

When two capacitors of capacitances and are connected in parallel, the equivalent capacitance

By conservation of charge,

The energy of the system

The energy lost .

The applications (uses) of the potentiometer:

(i) Voltage divider: The potentiometer can be used as a voltage divider to change the output voltage of a voltage supply.

(ii) Audio control: Sliding potentiometers are commonly used in modern low-power audio systems as audio control devices.

Both sliding (faders) and rotary potentiometers (knobs) are regularly used for frequency attenuation, loudness control and for controlling different characteristics of audio signals.

(iii) Potentiometer as a sensor: If the slider of the potentiometer is connected to the moving part of a machine, it can work as a motion sensor. A small displacement of the moving part causes a change in potential which is further amplified using an amplifier circuit. The potential difference is calibrated in terms of displacement of the moving part. (iv) To measure the emf (for this, the emf of the standard cell and potential gradient must be known).

(v) To compare the emfs to two cells.

(vi) To determine the internal resistance of a cell.

Given:

The magnetic induction,

Given:

The magnitude of the induced emf,

A car is an almost deal faraday cage, when a car is truck by lightning, the charge flows on the outside surface of the car to the ground but electric field inside remains zero. This leaves the passenger inside unharmed.
Depending upon experimental conditions or the structure of matter, electromagnetic radiation and material particles exhibit wave nature or particle nature. This is known as wave-particle duality.

It applies to all phenomena. The wave nature and particle nature are liked by the de Broglie relation , where is the wavelength of matter waves, also called de Broglie waves I Schrödinger waves, is the magnitude of the momentum of a particle or quantum of radiation and is the universal constant called Planck’s constant.

(i) According to Rutherford, the electrons revolve in circular orbits around the atomic nucleus. The circular motion is an accelerated motion. According to the classical electromagnetic theory. an accelerated charge continuously radiates energy. Therefore, an electron during its orbital motion should go on radiating energy. Due to the loss of energy, the radius of its orbit should go on decreasing. Therefore, the electron should move along a spiral path and finally fall into the nucleus in a very short time, of the order of in the case of a hydrogen atom. Thus, the atom should be unstable. We exist because atoms are stable.

(ii) If the electron moves along such a spiral path, the radius of its orbit would continuously decrease. As a result, the speed and frequency of revolution of the electron would go on increasing. The electron. therefore, would emit radiation of continuously changing frequency and hence give rise to a continuous spectrum. However, the atomic spectrum is a line spectrum.

Given: Current gain

Emitter current

To find: The value of base current of the transistor.

The common gain DC current us is given by

The base current of the transistor is given by the formula

The value of base current of the transistor is .

29. Section C

15.

A rigid object rotating with a constant angular speed ‘ ‘ about an axis perpendicular to the plane of paper. Consider the object to be consisting of particles of masses at respective perpendicular distances from axis of rotation.

All particles perform with different linear speeds ,

Angular momentum of each particle

Similarly for all particles

Total angular momentum is the sum of individual angular momentum.

Where

Given: ,

As the film has two surfaces, the work done is

Law of equipartition of energy states that for a dynamical system in thermal equilibrium the total energy of the system is shared equally by the degrees of freedom. The energy associated with each degree of freedom per molecule is . where is the Boltzmann’s constant.

For example, for a monoatomic molecule, each molecule has 3 degrees of freedom. Accordingly to kinetic theory of gases, the mean kinetic energy of a molecule is .
Specific heat capacity of monatomic gas: The molecules of a monatomic gas have 3 degrees of freedom.

The average energy of a molecule at temperature is .

The total internal energy of a mole is T where is the Avogadro number.

The molar specific heat at constant volume is

For an ideal gas,

where is molar specific heat at constant pressure.

Thus,

Specific heat capacity of diatomic gas: The molecules of a monatomic gas have 5 degrees of freedom, 3 translation and 2 rotational.

The average energy of a molecule at temperature is .

The total internal energy of a mole is .

The molar specific heat at constant volume is For an ideal gas,

For an ideal gas,

where is the molar specific heat at constant pressure.

Thus,

A soft or non-rigid diatomic molecule has, in addition, one frequency of vibration which contributes two quadratic terms to the energy. Hence, the energy per molecule of a soft diatomic molecule is

Therefore, the energy per mole of a soft diatomic molecule is

In this case,

and

Given:

liters

The work done by the system (gas in this case) on its surroundings,

The change in the internal energy of the system,

The minus sign shows that there is a decrease in the internal energy of the system.

The expression is given as

Explanation: The time period of oscillation of a magnet in a uniform magnetic field ‘ ‘ is given by

Formula:

where

Time-period of an oscillation body about a fixed point can be defined as the time taken by the body to complete one vibration around that particular point is called time period.

Given:

(i) , the path difference is the distance covered

Phase difference Path difference

(ii)

Path difference Phase difference

Description of Young’s doubles-slit interference experiment

(i) A plane wavefront is obtained by placing a linear source of monochromatic light at the focus of a convex lens. It is then made to pass through an opaque screen having two narrow and similar slits and and are equidistant from so that the wavefronts starting simultaneously from and reaching and at the same time are in phase. A screen is placed at some distance from as shown in the following figure.

Young’s double-slit experiment

(ii) and act as secondary sources. The crests/ troughs of the secondary wavelets superpose and interfere constructively along straight lines joining the black dots shown in the above figure. The point where these lines meet the screen have high intensity and its bright.

(iii) Similarly, there are points shown with red dots where the crest of one wave coincides with the trough of the other. The corresponding points on the screen are dark due to destructive interference. These dark and bright regions are called fringes or bands and the whole pattern is called an interference pattern.

Condition for occurrence of dark and bright fringes on the screen: Consider Young’s double-slit experimental set up. Two narrow coherent light sources are obtained by wavefront splitting as monochromatic light of wavelength emerges out of two narrow and closely spaced, parallel slits and of equal widths. The separation is very small. The interference pattern is observed on a screen placed parallel to the plane of and at considerable distance. from the slits. is the perpendicular bisector of a segment

Geometry of the double-slit experiment: Consider, a point on the screen at a distance from . The two light waves from and reach along paths and , respectively. If the path difference between and is an integral multiple of , the two waves arriving there will interfere constructively producing a bright fringe at . On the contrary, if the path difference
between and is a half-integral multiple of , there will be destructive interference and a dark fringe will be produced at .

From above figure,

and

In practice, and

Path difference,

The expression for the fringe width (or band width) : The distance between consecutive bright (or dark) fringes is called the fringe width (or bandwidth) W. Point P will be bright (maximum intensity), if the path difference,

Point will be dark (minimum intensity equal to zero), if

Thus, for bright fringes (or bands),

and for dark fringes (or bands).

These conditions show that the bright and dark fringes (or bands) occur alternately and equally spaced. For point , the path difference . Hence, point will be bright. It corresponds to the centre of the central bright fringe (or band). On both sides of , the interference pattern consists of alternate dark and bright fringes (or band) parallel to the slit.

Let and , be the distances of the th and bright fringes from the central bright fringe.

The distance between consecutive bright fringes

Hence, the fringe width,

(for bright fringes)

Alternatively, let and be the distances of the and dark fringes respectively from the central bright fringe.

and .

and

The distance between consecutive dark fringes.

Equations (7) and (11) show that the fringe width is the same for bright and dark fringes.

Given:

The potential drop per centimeter of the wire is .

Given:

The radius of the circular path,

The frequency of revolution,

Since the magnetic force does not change the kinetic energy of the charge,

In the Bohr model of a hydrogen atom, the electron of charge performs a uniform circular motion around the positively charged nucleus. Let and be the orbital radius, speed and period of motion of the electron. Then,

Therefore, the orbital magnetic moment associated with this orbital current loop has a magnitude,

Therefore, the magnetic dipole moment associated with this electronic current loop has a magnitude

Multiplying and dividing the right-hand side of the above expression by the electron mass

where is the magnitude of the orbital angular momentum of the electron is opposite to .

which is the required expression.

According to Bohr’s second postulate of stationary orbits in this theory of hydrogen atom, the angular momentum of the electron in the th stationary orbit is equal to , where is the Planck constant and is a positive integer. Thus, for an orbital electron,

Substituting for in equation (4).

For .

The quantity is a fundamental constant called the Bohr magneton, (or ) .

Given,

A BJT being a bipolar device, both electrons and holes participate in the conduction process. Under the forwardbiased condition, the majority carriers injected from the emitter into the base constitute the largest current component in a BJT. For these carriers to diffuse across the base region with negligible recombination and reach the collector junction, these must overwhelm the majority carriers of the opposite charge in the base. The total emitter current has two components, that due to majority carriers in the emitter and that due to minority carriers diffused from the base into the emitter. The ratio of the current component due to the injected majority carriers from the emitter to the total emitter current is a measure of the emitter efficiency. To improve the emitter efficiency and the common-base current gain (a), it can be shown that the emitter should be much heavily doped than the base

Also, the base width is a function of the base-collector voltage. A low doping level of the collector increases the size of the depletion region. This increases the maximum collector-base voltage and reduces the base width. Further, the large depletion region at the collector-base junction-extending mainly into the collector-corresponds to a smallest electric field and avoids avalanche breakdown of the reverse-biased collector-base junction.

30. Section D

(i) 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:

(a) The motion of a conductor in a magnetic field or

(b) The change of the magnetic flux through a stationary circuit.

Explanation: Consider Faraday’s magnet and coil experiment. If the bar magnet is moved towards the coil with its -pole facing the coil, as in the shown first figure, the number of magnetic lines of induction (pointing to the left) through the coil increases. The induced current in the coil sets up a magnetic field of its own pointing to the right (as given by Amperes right-hand rule) to oppose the growing flux due to the magnet. Hence, to move the magnet towards the coil against this repulsive flux of the induced current, we must do work. The work done shows up as electric energy in the coil.

When the magnet is withdrawn, with its -pole still facing the coil, the number of magnetic lines of induction (pointing left) through the coil decreases. The induced current reverses its direction to supplement the decreasing flux with its own, as shown in the second figure. Facing the coil along with the magnet, the induced current is in the clockwise sense. The electric energy in the coil comes from the work done to withdraw the magnet, now against an attractive force. Thus, we see that Lenz’s law is a consequence of the law of conservation of energy.

(ii) Here

years, ?

We have,

(i) Progressive wave: A progressive wave is defined as the onward transmission of the vibratory motion of a body in an elastic medium from one particle to the successive particle.

31. Characteristics of a progressive wave:

Energy is transmitted from particle to particle without the physical transfer of matter.
The particles of the medium vibrate periodically about their equilibrium positions.
In the absence of dissipative forces, every particle vibrates with the same amplitude and frequency but differs in phase from its adjacent particles. Every particle lags behind in its state of motion compared to the one before it.
Wave motion is doubly periodic, i.e., it is periodic in time and periodic in space.
The velocity of propagation through a medium depends upon the properties of the medium.
A transverse wave can propagate only through solids but not through liquids and gases while a longitudinal wave can propagate through any material medium.
Progressive waves are of two types: Transverse and longitudinal. In a transverse mechanical wave, the individual particles of the medium vibrate perpendicular to the direction of propagation of the wave. The progressively changing phase of the successive particles results in the formation of alternate crests and troughs that are periodic in space and time. In an em wave, the electric and magnetic fields oscillate in mutually perpendicular directions, perpendicular to the direction of propagation. In a longitudinal mechanical wave, the individual particles of the medium vibrate along the line of propagation of the wave. The progressively changing phase of the successive particles results in the formation of typical alternate regions of compressions and rarefactions that are periodic in space and time. Periodic compression and rarefactions result in periodic pressure and density variations in the medium. There is no longitudinal em wave.

(ii)

The radius of each small drop.

The volume of the larger drop being equal to the volume of the smaller drops, the radius of the layer drop is

The charge on the larger drop,

The electric potential of the surface of the larger
drop,

(i) (a) Melting of ice

(b) Boiling of water

(ii) Given: litres

This is the required energy.

Given:

For hydrogen spectrum,

as and

From Eqs. (1) and (2), we get.

This is the series limit of the Balmer series.

From Eqs. (1) and (3), we get

This is the series limit of the Paschen series.

From Eqs. (1) and (4), we get

Consider a liquid drop of radius and surface tension .

Due to surface tension, the molcules on the surface film experience the net force in the inward direction normal to the surafce. Therefore there is more pressure inside than outside.

Let be the pressure inside the liquid drop and be the pressure outside the drop.

Therefore excess of pressure inside the liquid drop is,

Due to excess pressure inside the liquid drop the surface of the drop will experience the net force in outward direction due to which the drop will expand.

Let the free surface displace by dR under isothermal conditions.

Therefore excess of pressure does the work in displacing the surface and that work will be stored in the form of potential energy.

The work done by an excess of pressure in displacing the surface is,