**SAMPLE PAPER-4**

Physics

**49. Questions**

**50. Section A**

**51. Select and write the correct answers to the following questions:**

(i) Two capillary tubes of radii

(a)

(b)

(c)

(d)

(ii) For polyatomic molecule having ‘ ‘ vibrational modes, the ratio of two specific heats, is,

(a)

(b)

(c)

(d)

(iii) The tension in a piano wire is increased by . Its frequency becomes times the original frequency.

(a) 0.8

(b) 1.12

(c) 1.25

(d) 1.56

(iv) A parallel plate capacitor is charged and then isolated. The effect of increasing the plate separation on charge, potential, capacitance respectively are:

(a) Constant, decreases, decreases

(b) Increases, decreases, decreases

(c) Constant, decreases, increases

(d) Constant, increases, decreases

(v) When the balance point is obtained in the potentiometer, a current is drawn from:

(a) Both the cells and auxiliary battery

(b) Cell only

(c) Auxiliary battery only

(d) Neither cell nor auxiliary battery

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

(a) Remain unchanged

(b) Get reduced

(c) Increase

(d) Be reduced to zero

(vii) A conducting thick copper rod of length carries a current of and is located on the Earth’s equator. There the magnetic flux lines of the Earth’s magnetic field are horizontal, with the field of , south to north. The magnitude and direction of the force on the rod, when it is oriented so that current flows from west to east, are:

(a) , downward

(b) , downward

(c) , upward

(d) , upward

(viii) A conductor rod of length ( ) is moving with velocity (v) in a direction normal to a uniform magnetic field (B). What will be the magnitude of induced emf produced between the ends of the moving conductor?

(a) BLv

(b)

(c)

(d)

(ix) In a circuit L, C and are connected in series with an alternating voltage of frequency . The current leads the voltage by 450 . The value of is:

(a)

(b)

(c)

(d)

(x) In the spectrum of hydrogen atom which transition will yield longest wavelength?

(a) to

(b) to

(c) to

(d) to

- Answer the following questions:

(i) Do we need a banked road for a two-wheeler?

(ii) How much amount of work is done in forming a soap bubble of radius ?

(iii) A gas receives an amount of heat equal to and performs of work. What is the change in internal energy of gas?

(iv) A gas contained in a cylinder surrounded by a thick layer of insulating material is quickly compressed. Has work been done?

(v) A metal plate is introduced between the plates of a charged parallel plate capacitor. What is its effect on the capacitance of the capacitor?

(vi) If the difference between the radii of the two spheres of a spherical capacitor is increased, state whether the capacitance will increase or decrease.

(vii) A voltmeter has a resistance What will be its reading. when it is connected across a cell of emf having internal resistance ?

(viii) On what factor does the potential gradient of the wire depends?

**52. Section B**

**53. Attempt any Eight of the following questions:**

- Why is it useful to define radius of gyration?
- A drop of water of radius breaks into number of droplets each of radius . How many droplets will be formed?
- Define athermanous substances and diathermanous substances.
- One mole of an ideal gas is initially kept in a cylinder with a movable frictionless and massless piston at pressure of and temperature . It is then expanded till its volume is doubled. How much work is done if the expansion is isobaric?
- State the laws of simple pendulum.
- What are harmonics and overtones?
- Why do we need filters in a power supply?
- When a plate of magnetic material of volume is located in magnetising field of , then a magnetic moment of is induced in it.

Find magnetic induction in rod.

- Explain why the inductance of two coils connected in parallel is less than the inductance of either coil.
- The total impedance of a circuit decreases when a capacitor is added in series with and . Explain why?
- Explain the inverse linear dependence of stopping potential on the incident wavelength in a photoelectric effect experiment.
- The energy of a photon is . Find its frequency.

**54. Section C**

**55. Attempt any Eight of the following questions:**

- Why two or more mercury drops form a single drop when brought in contact with each other?
- A resistor held in running water carries electric current. Treat the resistor as the system:

(i) Does heat flow into the resistor?

(ii) Is there a flow of heat into the water?

(iii) Is any work done?

(iv) Assuming the state of resistance to remain unchanged, apply the first low of thermodynamics to this process.

- A particle performing linear S.H.M. of period seconds about the mean position is observed to have a speed of , when at a distance (metre) from . If the particle is moving away from at that instant, find the time required by the particle, to travel a further distance b.
- Explain how vibrating strings can be verified using a sonometer.
- Derive the laws of reflection of light using Huygen’s principle.
- 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?
- Describe how a potentiometer is used to compare the emfs of two cells by combination method.
- Obtain and expression for orbital magnetic moment of an electron rotating about the nucleus in an atom.
- A capacitor is connected to a source. Find the capacitive reactance and the current (rms and peak) in the circuit. If the frequency is doubled, what will happen to the capacitive reactance and the current
- The threshold wavelength of tungsten is .

(i) Explain why no photoelectrons are emitted when the wavelength is more than .

(ii) What will be the maximum kinetic energy of electrons ejected in each of the following cases:

(a) If ultraviolet radiation of wavelength and

(b) Radiation of frequency is made incident on the tungsten surface.

- What is the difference between a nuclear reactor and a nuclear bomb?
- The common-base DC current gain of a transistor is 0.967 :

(i) If the emitter current is . What is the value of base current.

(ii) A certain transistor has an emitter current of and collector current of . Calculate the value of the base current.

**56. Section D**

**57. Attempt any Three of the following questions:**

- Somehow, an ant is stuck to the rim of a bicycle wheel of diameter . While the bicycle is on a central stand, the wheel is set into rotation and it attains the frequency of in 10 seconds, with uniform angular acceleration. Calculate:

(i) Number of revolutions completed by the ant in these 10 seconds. (ii) Time taken by it for first complete revolution and the last complete revolution.

- State the law of equipartition of energy and hence calculate molar specific heat of mono- and diatomic gases at constant volume and constant pressure.
- Monochromatic electromagnetic radiation from a distant source passes through a slit. The diffraction

pattern is observed on a screen from the slit. If the width of the central maximum is , what is the slit width if the wavelength is:

(i) (visible light);

(ii) (infrared radiation);

(iii) -rays)?

- Figure shows a cylindrical wire of diameter a carrying a, current I. The current density which is in the direction of the central axis of the wire varies linearly with radial distance from the axis according to the relation . Obtain the magnetic field inside the wire at a distance from its centre. What will be the magnetic field inside the wire at distance from its centre, if the current density is uniform across the cross-section of the wire?

- A long solenoid consisting of turns has an area of cross-section of . A coil C, consisting of 150 turns is wound tightly around the centre of the solenoid. Calculate for a current of in the solenoid.

(i) The magnetic flux density at the centre of the solenoid,

(ii) The flux linkage in the coil ,

(iii) The average emf induced in coil if the current in the solenoid is reversed in direction in a time of

**58. Answer Key**

**59. Section A**

- (i) (b)

(ii) (c)

(iii) (b) 1.12

(iv) (a) Constant, decreases, decreases

(v) (d) Neither cell nor auxiliary battery

(vi) (a) Remain unchanged

(vii) (d) , upward

(viii) (a) BLv

(ix) (b)

(x) (d) to

- (i) When a two-wheeler takes a turn along an unbanked road, the force of friction provides the centripetal force. The two-wheeler leans inward to counteract a torque that tends to topple it outward. Firstly, friction cannot be relied upon to provide the necessary centripetal force on all road conditions. Secondly, the friction results in the wear and tear of the tyres. On a banked road at a turn, any vehicle can negotiate the turn without depending on friction and without straining the tyres.

(ii) We know that a bubble has two surfaces in contact with air, so the total surface area of the bubble will be

Now,

Work done Surface tension Increase in surface ared

(iii)

(iv) Yes. Work is always done when a gas is compressed by any external agent, no matter what type of system is there.

(v) Suppose the parallel-plate capacitor has capacitor , plates of area and separation d. Assume the metal sheet introduced has the same area .

Case (1): Finite thickness . Free electrons in the sheet will migrate towards the positive plate of the capacitor. Then, the metal sheet is attracted towards whichever capacitor plate is closet and gets stuck to it, so that its potential is the same as that of that plate. The gap between the capacitor plates is reduced to – so that the capacitance increases.

Case (2): Negligible thickness. The thin metal sheet divides the gap into two of thicknesses and of capacitances and in series.

Their effective capacitance is

i.e, the capacitance remains unchanged.

(vi) The capacitance of a spherical capacitor is where and are radii of the concentric inner and outer conducting shells. Hence, the capacitance decreases if the difference is increased.

(vii) Given:

The voltmeter reading.

(viii) The potential gradient depends upon the potential difference the ends of the wire and the length of the wire.

**60. Section B**

- Definition: The radius of gyration of body rotating about an axis is defined as the distance between the axis of rotation and the point at which the entire mass of the body can be supposed to be concentrated so as to give the same moment of inertia as that of the body about the given axis.

The moment of inertia (MI) of a body about a given rotation axis depends upon (i) the mass of the body and (ii) the distribution of mass about the axis of rotation. These two factors can be separated by expressing the as the product of the mass and the square of a particular distance from the axis of rotation. This distance is called the radius of gyration and is defined as given above. Thus,

Physical significance: The radius of gyration is less if is less, i.e., the mass is distributed close to the axis and it is more if is more, i.e, if the mass is distributed away from the axis. Thus, it gives an ideal about the distribution of mass about the axis of rotation.

- Given,

Radius of big drop

Radius of small drop

No. of droplets ?

droplets will be formed.

- Athermanous substances: Substances that don’t allow transmission of infrared through them are called athermanous substances.

For example, wood, metal, , water vapour etc.

Diathermanous substances: Substances that allow transmission of infrared radiation through them are called diathermanous substances.

For example, rock salt, pure air, glass, etc.

- Work done in isobaric process is given by.

- The period of a simple pendulum at a given place is

where is the length of the simple pendulum and is the acceleration due to gravity at that place. From the above expression, the laws of a simple pendulum are as follows:

- Law of length: The period of a simple pendulum at a given place ( constant) is directly proportional to the square root of its length.

- Law of acceleration due to gravity: The period of a simple pendulum of a given length ( constant) is inversely proportional to the square root of the acceleration due to gravity.

- Law of mass: The period of a simple pendulum does not depend on the mass of material of the bob of the pendulum.
- Law of isochronism: The period of a simple pendulum does not depend on the amplitude of oscillations, provided that the amplitude is small.

- Harmonic frequencies are whole number multiples of the fundamental frequency or the lowest frequency of vibration.

Consider a vibrating string. The modes of vibration are all multiples of the fundamental and are related to the string length and wave velocity. High frequencies are found via the relationship , wavelength where is the string length.

An overtone is a name given to any resonant frequency above the fundamental frequency of fundamental tone.

The list of successive overtones for an object is called the overtone series. The first overtone as well as subsequent overtones in the series may or may not be an integer multiple of the fundamental. Sometimes the relationship is that simple and other times it is more complex, depending on the properties and geometry of the vibrating object.

- The circuit used in a dc power supply to remove the ripple is called a filter. A filter circuit can produce a very smooth waveform that approximates the waveform produced by a battery. The most common technique used for filtering is a capacitor connected across the output of a rectifier.
- Given

- 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

- For an LR circuit, the impedance,

where is the reactance of the inductor.

When a capacitor of capacitance is added in series with and , the impedance,

because in the case of an inductor the current lags behind the voltage by a phase angle of while in the case of a capacitor the current leads the voltage by a phase angle of . The decrease in net reactance decreases the total impedance .

- We have , where is the stopping potential, is the magnitude of the charge on the electron, his Planck’s constant, is the speed of light in free space, is the wavelength of the electromagnetic radiation incident on a metal surface and is the work function for the metal, and are constants, is constant for a particular metal.

Hence, it follows that as increases, increases.

The plot of verses is linear. This is because the energy associated with a quantum of radiation (photon) is directly proportional to the frequency of the radiation and hence inversely proportional to the wavelength of radiation.

- Given,

Frequency

**61. Section C**

- A spherical shape has the minimum surface area to volume ratio of all geometric forms. When two drops of a liquid are brought in contact, the cohesive forces between their molecules coalesce the drops into a single larger drop. This is because the volume of the liquid remaining the same, the surface area of the resulting single drop is less than the combined surface area of the smaller drops. The resulting decrease in surface energy is released into the environment as heat.

Proof: Let droplets each of radius coalesce to form a single drop of radius . As the volume of the liquid remains constant, volume of the drop volume of droplets.

Surface area of droplets

Surface area of the drop

The change in the surface area

Surface area of drop – Surface area of droplets

Since the bracketed term is negative, there is a decrease in surface area and a decrease in surface energy.

- (i) Heat is generated into the resistor due to the passage of electric current. In the usual notation heat generated .

(ii) Yes, water receives heat from the resistor.

(iii)

(iv)

Here, current through the resistor, resistance of the resistor, time for which the current is passed through the resistor, mass of the water, specific heat of water. rise in the temperature of water, pressure which the work is done by the water, increase in the volume of the water.

- Given: The time period of the particle

Speed of the particle

To find: The time required by the particle to travel a distance of bm.

The velocity of particle is given by

By substituting the values

The time taken to travel distance from mean position is

Further time taken by the particle to reach mean position is

The time required by the particle is .

- (i) Verification of law of length: According to this law, if and are constant. To verify this law, the sonometer wire of given linear density is kept under constant tension T. The length of the wire is adjusted for the wire to vibrate in unison with tuning forks of different frequencies , be the corresponding resonating lengths of the wire. It is found that within experimental errors, This implies that the product, constant, which verifies the law of length.

(ii) Verification of law of tension: According to this law, , if and are constant. To verify this law, the vibrating length of the sonometer wire is given linear density is kept constant.

A set of tuning forks of differentfrequencies is used. The tension in the wire is adjusted for the wire to vibrate in unison with tuning forks of different frequencies be the corresponding tensions. It is found that, within experimental errors,

This implies constant which verifies the law of tension.

(iii) Verification of linear density: According to this law, , if and are constant. To verify this law, two wires having different linear densities and are kept under constant tension T. A tuning fork of frequency is used. The lengths of the wires are adjusted for the wires to vibrate in unison with the tuning fork. Let and be the corresponding resonating lengths of the wires. It is found that within experimental errors, . This implies constant. According to the law of length of a vibrating string, .

, which verifies the law of linear density.

- Reflection of a plane wavefront of light at a plane surface

Where MN: Plane mirror.

and : Incident rays,

AP: Normal to MN,

AB: Incident wavefront,

i. Angle of incident,

CE: Reflected wavefront.

. Angle of reflection

When wavefront is incident on the mirror, at first, point A becomes a secondary source and emits secondary

waves in the same medium. If is the time taken by the incident wavefront to travel from to , then . During this time, the secondary wave originating at A covers the same distance, so that the secondary spherical wavelet has a radius at time .

To construct the reflected wavefront, a hemisphere of radius is drawn from point . Draw a tangent to the secondary wavelet.

The arrow AE shows the direction of propagation of the reflected wave.

is the normal to at .

angle of incidence and

angle of reflection

In and ,

and are congruent.

Also, as is perpendicular to and is perpendicular to ,

From equations (1) and (2),

Thus, the angle of incidence is equal to the angle of reflection. This is the first law of reflection. Also, it can be seen from the figure that the incident ray and reflected ray lie on the opposite sides of the normal to the reflecting surface at the point of incidence and all of them lie in the same plane. This is the second law of reflection. Thus, the laws of reflection of light can be deduced by Huygen’s construction of 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 .

- A battery of stable emf is used to set up a potential gradient V/L, along the potentiometer wire, where potential difference across length of the wire. The positive terminal of the cell 1 is connected to the higher potential terminal of the potentiometer; the negative terminal is connected to the galvanometer G through the reversing key. The other terminal of the galvanometer is connected to a pencil jockey. The cell 2 is connected across the remaining two opposite terminals of the reversing key. The other terminal of the galvanometer is connected to a pencil jockey. The emf should be greater than the emf ; this can be adjusted by trial and error.

Two plugs are inserted in the reversing key in positions 1-1. Here, the two cells assist each other so that the net emf is . The jockey is tapped along the wire to locate the null point . If the null point is a distance from ,

**62. Comparison to two emf’s using a potentiometer by**

the combination method (the sum and difference method): For the same potential gradient (without changing the rheostat setting), the plugs are now inserted into position 2.2 (instead of 1-1). The emf

then opposes

and the net emf is

. The new null point

is, say, a distance

from

and

.

Here, the emf should be greater than . The experiment is repeated for different potential gradients using the rheostat.

- 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 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

current area of the loop

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 his 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:

The capacitive reactance

If the frequency is doubled, the capacitive reactance will be halved and the current will be doubled.

24. Given:

(i) For (threshold frequency)

Hence, no photoelectrons are emitted.

(ii) Maximum kinetic energy of electrons ejected

(iii) Maximum kinetic energy of electrons ejected

- Nuclear Reactor: A nuclear reactor is a machine where electricity and heat energy is generated by utilizing the power of atoms. In this mechanism, nuclear chain reactions are produced, controlled and contained releasing a tremendous amount of energy. This controlled energy is used to electricity generation and radioactive isotopes production. These isotopes are used in the treatment and research of cancer in the medical field. All operating nuclear reactors are “critical”. When reactors are running at a constant power level, they are said to be in a “critical condition”.

These reactors use heavy atoms as fuel instead of fossils fuels. Fast-moving electrons strike a radioactive nucleus such as Plutonium-239 or Uranium-235 causing the nucleus to split. This splitting process is known as fission. In the process of fission, a tremendous amount of energy. radiation and free electrons are released. These free electrons that are released guided to strike other nuclei and so on causing a chain reaction.

Neutron moderators and neutron poisons control these fast-moving electrons and slow them down while

becoming absorbed in other nuclei, thus managing the output of electricity from a reactor. The moderators are heavy water, water and solid graphite.

Nuclear Bomb: In a nuclear bomb, there is a nuclear device having massive destructive power coming from uncontrolled fusion and fission reactions. The fusion and fission processes generate a tremendous amount of energy with a small amount of matter. This matter is usually the unstable nuclei of Plutonium-239 and Uranium-235. An atom bomb is categorized as a fission bomb and a hydrogen bomb as a fusion bomb are both weapons of mass destruction. In World War II, Hiroshima and Nagasaki are recent examples of such mass destruction. In fusion bombs, nuclear fusion is the result of a huge amount of released energy while in the case of fission bombs the released energy in the result of fission reactions.

- Current gain

Emitter current

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

(i) 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 .

(ii) Given:

Therefore, the base current,

**63. Section D**

- Given:

(i) Angular acceleration being constant, the average angular speed,

The angular displacement of the wheel in time ,

revolutions.

(ii)

For ,

For ,

The time for the last, i.e, the 10th, revolution is

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

For example, for a monoatomic molecule, each molecule has 3 degrees of freedom. According 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: , where is the Avogadro number.

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

For an ideal gas,

where is molar specific heat at constant pressure.

Specific heat capacity of diatomic gas: The molecules of a monatomic gas have 5 degrees of freedom, 3 translational 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 soft diatomic molecule

In this case,

and

[Note: For a monatomic gas, adiabatic constant.

- Given:

(i)

(ii)

(iii)

Let be the slit width.

(1)

(ii)

- Consider an annular differential element of radius and width . The current through the area of this element

To apply the Ampere’s circuital law of the circular path of integration, we note that the wire has perfect cylindrical symmetry with all the charges moving parallel to the wire. So, the magnetic field must be tangent to circles that are concentric with the wire. The enclosed current is the current within radius . Thus,

which is the required expression.

The cross-section of a long straight wire of radius that carries a current of the page. Because the current is uniformly distributed over the cross-section of the wire, the magnetic field due to the current must be cylindrically symmetrical. Thus, along the Amperian loop of the radius , symmetry suggests that is tangent to the loop.

Because the current is uniformly distributed, the current enclosed by the loop is proportional to the area encircled by the loop; that is

By right-hand rule, the sign of is positive. Then by Ampere’s law.

By right-hand rule, the sign of is positive. Then by

- Given:

(i) Magnetic flux density inside the solenoid,

(ii) Flux per unit turn through the coils of the solenoid,

Since the coil is wound tightly over the solenoid, the flux linkage of is

(iii) Initial flux through coil ,

Reversing the current in the solenoid reverses the flux through coil , the magnitude remaining the same. But since the flux enters through the other face of the coil, the final flux through is

Therefore, the average emf induced in coil ,