Code No. 55/1/2 Series : SKS / 1 Candidates must write the Code on the title page of the answer-book. R.No. Please check that this question paper contains 7 printed pages. Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate. Please check that this question paper contains 26 questions. Please write down the Serial Number of the question before attempting it. 15 minutes time has been allotted to read this question paper. The question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. PHYSICS (Theory) Time allowed : 3 hours Maximum Marks : 70 General Instructions 1. 2. All questions are compulsory. There are 26 questions in all. This question paper has five sections: Section A, Section B, Section C, Section D and Section E. 3. Section A contains five questions of one mark each, Section B contains five questions of two marks each, Section C contains twelve questions of three marks each, Section D contains one value based question of four marks and Section E contains three questions of five marks each. 4. There is no overall choice. However, an internal choice has been provided in one question of two marks, one question of three marks and all the three questions of five marks weightage. You have to attempt only one of the choices in such questions. 5. You may use the following values of physical constants wherever necessary. c = 3 × 108 m/s h = 6.63 × 10–34 Js e = 1.6 × 10–19 C µo = 4 ×10–7 T m A–1 –12 2 C N–1 m–2 0 = 8.854 × 10 1 4 0 9 109 N m2 C–2 me = 9.1 10–31 kg mass of neutron = 1.675 × 10–27 kg mass of proton = 1.673 × 10–27 kg Avogadro Numbers = 6.023 ×1023 per gram mole Boltzmann constant = 1.38 × 10–23 JK–1 SIROHI CLASSES PH-9810252244/9555552244/34 Page 1 Section A ( 5×1= 5 Marks) 1) Two charges of magnitudes –2Q and + Q are located at points (a, 0) and (4a, 0) respectively. What is the electric flux due to these changes through a sphere of radius '3a' with its centre at the origin? 2) How does the mutual inductance of a pair of coils change when (i) distance between the coils is increased and (ii) number of turns in the coils is increased ? 3) The graph shown in the figure represents a plot of current versus voltage for a given semiconductor. Identify the region, if any, over which the semiconductor has a negative resistance. 4) Two identical cells, each of emf E, having negligible internal resistance, are connected in parallel with each other across an external resistance R. What is the current through this resistance ? 5) The motion of copper plate is damped when it is allowed to oscillate between the two poles of a magnet. What is the cause of this damping ? Section B (5×2= 10 Marks) 6) Define the activity of a given radioactive substance. Write its SI unit. 7) Welders wear special goggles or face masks with glass windows to protect their eyes from electromagnetic radiations. Name the radiations and write the range of their frequency. 8) Write the expression for the de Broglie wavelength associated with a charged particle having charge 'q' and mass 'm', when it is accelerated by a potential V. 9) Draw typical output characteristics of an n-p-n transistor in CE configuration. Show how these characteristics can be used to determine output resistance. SIROHI CLASSES PH-9810252244/9555552244/34 Page 2 10) A parallel beam of light of 500 nm falls on a narrow slit and the resulting diffraction pattern is observed on a screen 1 m away. It is observed that the first minimum is at a distance of 2.5 mm from the centre of the screen. Calculate the width of the slit. Or Describe briefly, with the help of a circuit diagram, how a potentiometer is used to determine the internal resistance of a cell. Section C (12×3 = 36 Marks) 11) A slab of material of dielectric constant K has the same area as that of the plates of a parallel plate capacitor but has the thickness d/2, where d is the separation between the plates. Find out the expression for its capacitance when the slab is inserted between the plates of the capacitor. 12) Write the order of frequency range and one use of each of the following electromagnetic radiations. i) Microwave ii) Ultra violet rays iii) Infra-red ray iv) Gamma ray 13) A convex lens of focal length f1 is kept in contact with a concave lens of focal length f2. Find the focal length of the combination. 14) In the block diagram of a simple modulator for obtaining an AM signal, shown in the figure, identify the boxes A and B. Write their functions. 15) (a) For a given a.c., i = im sin t, show that the average power dissipated in a resistor R over a complete cycle is 1 2 im R . 2 (b) A light bulb is rated at 100 W for a 220 V a.c. supply. Calculate the resistance of the bulb. 16) A rectangular conductor LATNO is placed in a uniform magnetic field of 0.5 T. The field is directed perpendicular to the plane of the conductor. When the arm MN of length of 20 cm is moved towards left with a velocity of 10 m s–1, calculate the emf induced in the arm. Given the SIROHI CLASSES PH-9810252244/9555552244/34 Page 3 resistance of the arm to be 5 Ω (assuming that other arms are of negligible resistance) find the value of the current in the arm. Or A wheel with 8 metallic spokes each 50 cm long is rotated with a speed of 120 rev/min in a plane normal to the horizontal component of the Earth's magnetic field. The Earth's magnetic field at the place is 0.4 G and the angle of dip is 60°. Calculate the emf induced between the axle and the rim of the wheel. How will the value of emf be affected if the number of spokes were increased ? 17) Define the current sensitivity of a galvanometer. Write its SI unit. Figure shows two circuits each having a galvanometer and a battery of 3 V. When the galvanometers in each arrangement do not show any deflection, obtain the ratio R1/R2. 18) A wire AB is carrying a steady current of 12 A and is lying on the table. Another wire CD carrying 5 A is held directly above AB at a height of 1 mm. Find the mass per unit length of the wire CD so that it remains suspended at its position when left free. Give the direction of the current flowing in CD with respect to that in AB. [Take the value of g = 10 m / s2] 19) Draw V-I characteristics of a p-n junction diode. Answer the following questions, giving reasons: a) Why is the current under reverse bias almost independent of the applied potential upto a critical voltage ? SIROHI CLASSES PH-9810252244/9555552244/34 Page 4 b) Why does the reverse current show a sudden increase at the critical voltage ? Name any semiconductor device which operates under the reverse bias in the breakdown region. 20) Draw a labelled ray diagram of a refracting telescope. Define its magnifying power and write the expression for it. Write two important limitations of a refracting telescope over a reflecting type telescope. 21) Write Einstein's photoelectric equation and point out any two characteristic properties of photons on which this equation is based. Briefly explain the three observed features which can be explained by this equation. 22) Name the type of waves which are used for line of sight (LOS) communication. What is the range of their frequencies ? A transmitting antenna at the top of a tower has a height of 20 m and the height of the receiving antenna is 45 m. Calculate the maximum distance between them for satisfactory communication in LOS mode. (Radius of the Earth = 6.4 × 106 m) Section D ( 1×4 = 4 Marks) 23) One day Chetan's mother developed a severe stomach ache all of a sudden. She was rushed to the doctor who suggested for an immediate endoscopy test and gave an estimate of expenditure for the same. Chetan immediately contacted his class teacher and shared the information with her. The class teacher arranged for the money and rushed to the hospital. On realising that Chetan belonged to a below average income group family, even the doctor offered concession for the test fee. The test was conducted successfully. Answer the following questions based on the above information : a) Which principle in optics is made use of in endoscopy ? b) Briefly explain the values reflected in the action taken by the teacher. c) In what way do you appreciate the response of the doctor on the given situation ? Section E (3×5 = 15 Marks) 24) A long straight wire of circular cross section of radius a carries a steady current I. The current is uniformly distributed across the cross-section. Apply Ampere’s circuital to calculate the magnetic field at a point in the region for (i) r < a (ii) r = a (iii) r > a. Also show the variation of magnetic field B with distance r with the help of graph. Or SIROHI CLASSES PH-9810252244/9555552244/34 Page 5 Explain briefly with the help of a labelled diagram, the principle and working of a moving coil galvanometer. Define the term 'current sensitivity' of a galvanometer. How is it that increasing current sensitivity may not necessarily increase its voltage sensitivity? Explain. What is the nature of the magnetic field in a moving-coil galvanometer and why? 25) (a) Define electric dipole moment. Is it a scalar or a vector ? Derive the expression for the electric field of a dipole at a point on the equatorial plane of the dipole. (b) Draw the equipotential surfaces due to an electric dipole. Locate the points where the potential due to the dipole is zero. Or Using Guass' law deduce the expression for the electric field due to a uniformly charged spheri• cal conducting shell of radius R at a point (i) outside and (ii) inside the shell. Plot a graph showing variation of electric field as a function of r > R and r < R (r being the distance from the centre of the shell) 26) Using Bohr's postulates, derive the expression for the frequency of radiation emitted when electron in hydrogen atom undergoes transition from higher energy state (quantum number n) to the lower state, (n f). When electron in hydrogen atom jumps from energy state ni= 4 to nf= 3, 2, 1, identify the spectral series to which the emission lines belong. Or (a) Draw the plot of binding energy per nucleon (BE/A) as a function of mass number A. Write two important conclusions that can be drawn regarding the nature of nuclear force. (b) Use this graph to explain the release of energy in both the processes of nuclear fusion and fission. (c) Write the basic nuclear process of neutron undergoing -decay. Why is the detection of neutrinos found very difficult ? SIROHI CLASSES PH-9810252244/9555552244/34 Page 6 SOLUTIONS 1) Two charges of magnitudes –2Q and + Q are located at points (a, 0) and (4a, 0) respectively. What is the electric flux due to these changes through a sphere of radius '3a' with its centre at the origin ? Ans: Electric flux through a sphere of radius 3a is 2) How does the mutual inductance of a pair of coils change when (iii) distance between the coils is increased and (iv) number of turns in the coils is increased ? Ans: (i) When the distance between the two coils is increased, mutual inductance of the pair of coils is decreased because now whole magnetic flux per unit turn of primary coil is not linked with the secondary coil. (ii) As mutual inductance M = 0 N1 N 2 A l , hence, it is clear that on increasing number of turns N1 and N2 in two coils the mutual inductance will definitely increase. 3) The graph shown in the figure represents a plot of current versus voltage for a given semiconductor. Identify the region, if any, over which the semiconductor has a negative resistance. Ans: In the region BC of plot the semiconductor has a negative resistance. It is because voltage is increasing here but the current is decreasing, hence R = V is negative. I 4) Two identical cells, each of emf E, having negligible internal resistance, are connected in parallel with each other across an external resistance R. What is the current through this resistance ? Ans: Net emf of parallel combination = Current through the resistance R, I = R 5) The motion of copper plate is damped when it is allowed to oscillate between the two poles of a magnet. What is the cause of this damping ? Ans: The cause of damping is setting up of induced eddy currents in copper plate when magnetic flux linked with it changes on account of its oscillation between the two poles of a magnet. 6) Define the activity of a given radioactive substance. Write its SI unit. Ans: Rate of decay (R) or the activity of a radioactive sample is defined as the number of disintegrations taking place per unit time. SI unit of activity of a given radioactive substance is a becquerel (1 Bq), where 1 Bq =1 disintegration per second. 7) Welders wear special goggles or face masks with glass windows to protect their eyes from electromagnetic radiations. Name the radiations and write the range of their frequency. SIROHI CLASSES PH-9810252244/9555552244/34 Page 7 Ans: Welders wear special glass goggles or face masks with glass windows to protect their eyes from large amount of ultraviolet rays produced by welding arcs. Frequency range of ultraviolet radiation is from 7 x 1014 Hz to 5 x 1017 Hz. 8) Write the expression for the de Broglie wavelength associated with a charged particle having charge 'q' and mass 'm', when it is accelerated by a potential V. Ans: de Broglie wavelength h . 2mqV 9) Draw typical output characteristics of an n-p-n transistor in CE configuration. Show how these characteristics can be used to determine output resistance. Ans: Typical output characteristics of an n-p-n transistor in CE configuration are shown in the following figure : To determine dynamic output resistance of a transistor at a given value of V CE (say 10 V) and IB (say 30μA), we select two points on eitherside of the operating point (say 10 ± 2 V or 12 V and 8 V respectively). Observe the value of collector currents from the characteristics corresponding to these voltages. Then 10) A parallel beam of light of 500 nm falls on a narrow slit and the resulting diffraction pattern is observed on a screen 1 m away. It is observed that the first minimum is at a distance of 2.5 mm from the centre of the screen. Calculate the width of the slit. Ans: It is given here that wavelength of light = 500 nm = 5 ×10–7 m, Distance of screen from the slit D = 1 m and distance of first minimum on the screen from the central maxima x1 = 2.5 mm = 2.5 × 10–3 m SIROHI CLASSES PH-9810252244/9555552244/34 Page 8 Or Describe briefly, with the help of a circuit diagram, how a potentiometer is used to determine the internal resistance of a cell. Ans: Determination of internal resistance of a cell : The labelled circuit diagram is shown in figure. Put the plug in key K1 and adjust rheostat Rh so as to allow flow of a constant current I through the potentiometer wire due to the driver cell. Slide the pencil jockey and adjust the length on potentiometer wire to l1 such that galvanometer shows null deflection. Obviously, fall in potential across 11 length is equal to emf s of given cell i.e., = k l1 where k = potential gradient Now, put plug in key K2 also and insert a suitable resistance R from R.B. Again slide the pencil jockey and find the length l 2 to obtain null deflection. Now V = k l2 ...(ii) Internal resistance of given cell 11) A slab of material of dielectric constant K has the same area as that of the plates of a parallel plate capacitor but has the thickness d/2, where d is the separation between the plates. Find out the expression for its capacitance when the slab is inserted between the plates of the capacitor. Ans: Let a dielectric slab of thickness t = d/2 is introduced between the plates of a parallel plate capacitor without touching the plates, the electric field in air E0 of dielectric the electric field inside the dielectric changes to E = but on account of polarization 0 E0 . K If potential difference between the plates of capacitor be V now, then clearly SIROHI CLASSES PH-9810252244/9555552244/34 Page 9 12) Write the order of frequency range and one use of each of the following electromagnetic radiations. i) Microwave ii) Ultra violet rays iii) Infra-red ray iv) Gamma ray SOL: (i) Microwave : 1×109 to 3×1011 Hz use: (1) radar systems for aircraft navigation. (2) detecting speed of tennis or cricket ball or of moving Vehicle. (3) microwave ovens (4) in satellite communications. (5) in analysis of very fine details of atomic & molecular structure. (ii) Ultra- violet ray : 7.5 x 1014 Hz to 3 x 1017 Hz use: (1) to destroy bacteria & germs in water purifiers. (2) in sterilizing surgical instruments. (3) in detection of forged documents, Finger prints in forensic labs. (4) to preserve food stuffs. (5) in burglar alarm (6) in LASIK eye surgery. (7) checking mineral samples by fluorescence. (8) in study of electrons arrangement in external shells by UV absorption spectra. (iii) Infra red : 3 x 1011 Hz to 4 x 1014 H use (1) treatment of mascular pain. (2) producing dehydrated fruits. (3) taking photographs in fog, haze etc. (4) in green house effect to keep plants warm. (5) in reading secret writings on ancient walls. (6) in solar heaters (7) infrared detectors in earth satellites (8) semiconductor LED emit infrared rays & are Used in remotes of TV, VCR & hifi systems. (9) in checking purity of chemicals & study of molecular structure. (iv) Gamma ray : 3 x 1020 Hz to 1022 Hz use: (1) in radio therapy. (2) to initiate some nuclear reactions. (3) to preserve food stuffs for long time. (4) to study structure of atomic nuclei. (5) to manufacture polyethylene from ethylene. 13) A convex lens of focal length f1 is kept in contact with a concave lens of focal length f2. Find the focal length of the combination. Ans: Consider a thin convex lens L1 of focal length f1 in contact with a thin concave lens L2 of focal length f2 as shown in figure : Let a point object O be placed at the common principal axis of two lenses, at a distance 'u'. The convex lens L1, in the absence of concave lens L2, forms a real image of object O at point .I1 at a distance where 1 1 1 . v1 u f1 However, the refracted ray is further deviated on refraction at concave lens and final image, due to combination of two lenses, is formed at point I at a distance 'v' from the lenses. We can consider that for concave lens, the point I1 behaves as an object and the image is formed at I. Thus, from lens formula, we have 1 1 1 v v1 f 2 ……..(i) Adding (i) and (ii), we have If the lens combination is considered as equivalent to a single lens of equivalent focal length 'feq' then we have 1 1 1 v u f eq ….....(iv) Comparing (iii) and (iv), we get 1 1 1 f eq f1 f 2 SIROHI CLASSES PH-9810252244/9555552244/34 Page 10 As per sign convention followed fi is +ve but f2 is —ve. Hence, we have 14) In the block diagram of a simple modulator for obtaining an AM signal, shown in the figure, identify the boxes A and B. Write their functions. Ans: Box A in the block diagram is a "square law device". It is a non-linear device which transforms the input signal x(t) = Am sin m t + AC C t and produces an output of the form y (t) = B x(t) + C x2(t), where B and C are two constants. Box B in the block diagram is a "band pass filter" centred at frequency C It allows frequencies C and ( C ± m ) only. 15) (a) For a given a.c., i = im sin t, show that the average power dissipated in a resistor R over a complete cycle is 1 2 im R . 2 (b) A light bulb is rated at 100 W for a 220 V a.c. supply. Calculate the resistance of the bulb. Ans: (a) When an alternating voltage V = Vm sin t is applied across an ideal resistor, the current in the circuit is given by I = Im sin t, where Average power dissipated per complete cycle of a.c. is given by (b) V2 R V 2 220 220 R 484 P 100 P 16) A rectangular conductor LATNO is placed in a uniform magnetic field of 0.5 T. The field is directed perpendicular to the plane of the conductor. When the arm MN of length of 20 cm is moved towards left with a velocity of 10 m s–1, calculate the emf induced in the arm. Given the resistance of the arm to be 5 Ω (assuming that other arms are of negligible resistance) find the value of the current in the arm. SIROHI CLASSES PH-9810252244/9555552244/34 Page 11 Ans: Here B =.0.5 T, length of moving conductor 1= 20 cm = 0.2 m and speed v = 10 m s–1 Induced emf | e | = B l v = 0.5 ×0.2 × 10 = 1.0 volt As resistance of rectangular conductor R = 5 Ω Current I = R 1.0 0.2 A 5 Or A wheel with 8 metallic spokes each 50 cm long is rotated with a speed of 120 rev/min in a plane normal to the horizontal component of the Earth's magnetic field. The Earth's magnetic field at the place is 0.4 G and the angle of dip is 60°. Calculate the emf induced between the axle and the rim of the wheel. How will the value of emf be affected if the number of spokes were increased ? Ans: As per question number of spokes in wheel n = 8, length of each spoke l = 50 cm = 0.50 m. angular speed = 120 rpm = 120 rps = 2 rps = 4π rad s–1, earth's magnetic field BE = 0.4 G = 60 4.0×10–5 T and angle of dip δ = 600. As the wheel is rotated in a plane normal to the horizontal component BH of the earth's magnetic field, hence emf induced between the axle and the rim of the wheel The value of emf remains unaffected if the number of spokes were increased because all the spokes are connected in parallel. 17) Define the current sensitivity of a galvanometer. Write its SI unit. Figure shows two circuits each having a galvanometer and a battery of 3 V. When the galvanometers in each arrangement do not show any deflection, obtain the ratio R1/R2. Ans: Current sensitivity of a galvanometer is defined as the deflection given by galvanometer when a unit current is passed through it. SI unit of current sensitivity is "division A–1". SIROHI CLASSES PH-9810252244/9555552244/34 Page 12 In circuit number (I) as galvanometer does not give any deflection, hence the bridge is balanced one and we have 18) A wire AB is carrying a steady current of 12 A and is lying on the table. Another wire CD carrying 5 A is held directly above AB at a height of 1 mm. Find the mass per unit length of the wire CD so that it remains suspended at its position when left free. Give the direction of the current flowing in CD with respect to that in AB. [Take the value of g = 10 m / s2] Ans: The wire CD will remain steady above AB at a height of d =1 mm = 10–3 m, if force on it due to magnetic field produced on account of current flowing in wire AB just balances its weight. As weight acts vertically downward, magnetic force must act in vertically upward direction and it is possible when current in wire CD is flowing along a direction opposite to that in AB as shown in figure. If m be mass of wire CD per unit length, then 19) Draw V-I characteristics of a p-n junction diode. Answer the following questions, giving reasons : a) Why is the current under reverse bias almost independent of the applied potential upto a critical voltage ? b) Why does the reverse current show a sudden increase at the critical voltage ? Name any semiconductor device which operates under the reverse bias in the breakdown region. Ans: The V-I characteristics of a p-n junction diode in forward bias and reverse bias arrangement are shown below : (i) In reverse bias the junction width increases. The higher junction potential restricts the flow of majority charge carriers. However, such a field favours flow of minority carriers. Thus, reverse bias SIROHI CLASSES PH-9810252244/9555552244/34 Page 13 current is due to flow of minority carriers only. Since the number of minority carriers is very small, the current is small and almost independent of the applied potential upto a critical (before breakdown) voltage. (ii) At a critical (breakdown) voltage the reverse bias current shows a sudden increase. Under high reverse bias, the high junction field may strip an electron from the valence band, which can tunnel to the n-side through the thin depletion layer. This mechanism of emission of electrons after a critical applied voltage leads to a high reverse (breakdown) current. A zener diode operates under the reverse bias in the breakdown region. 20) Draw a labelled ray diagram of a refracting telescope. Define its magnifying power and write the expression for it. Write two important limitations of a refracting telescope over a reflecting type telescope. Ans: A ray diagram showing the image formation in a refracting telescope for normal adjustment is being shown in figure. Magnifying power of telescope in normal adjustment is defined as the ratio of angle subtended (β) by the final image at the eye to the angle (α) subtended by the object at the eye. It is found that magnifying power is given by : m fo fe where fo = focal length of objective and fe = focal length of eye piece. A refractive type astronomical telescope suffers with the following drawbacks as compared to a reflecting type telescope : (i) The image formed has both chromatic as well as spherical aberration. (ii) It is extremely difficult to design and maintain the mechanical support of the telescope. 21) Write Einstein's photoelectric equation and point out any two characteristic properties of photons on which this equation is based. Briefly explain the three observed features which can be explained by this equation. Ans: Einstein's photoelectric equation is : According to photon nature of radiation, radiation energy is built up of discrete units, the photons. Each photon has energy E = h , where h is Planck's constant and the frequency of radiation. Moreover, one photon may interact only with one electron present in a metal surface. Einstein's photoelectric equation successfully explains all the laws of photoelectric emission, as given below : (i) Since one photon ejects one photoelectron from a metal surface, number of photoelectrons emitted per second is proportional to the number of incident photons per second i.e., the intensity of incident light. (ii) If < o, then kinetic energy is negative, which is impossible. Hence, photoelectric emission SIROHI CLASSES PH-9810252244/9555552244/34 Page 14 does not take place for the incident radiation below threshold frequency. (iii) For > o, the maximum kinetic energy Kmax is proportional to frequency v of incident light. If on increasing intensity of light more photons fall on metal surface, it may result in ejection of greater number of electrons but their energy remains unchanged. (iv) Photoelectric emission is a consequence of elastic collision between the photon and an electron inside the metal. Thus, absorption of energy by the electron in the metal from the incident photon is a single event involving transfer of energy at once without any time lag. As a result, photo-emission takes place instantaneously. 22) Name the type of waves which are used for line of sight (LOS) communication. What is the range of their frequencies ? A transmitting antenna at the top of a tower has a height of 20 m and the height of the receiving antenna is 45 m. Calculate the maximum distance between them for satisfactory communication in LOS mode. (Radius of the Earth = 6.4 × 106 m) Ans: For line of sight (LOS) communication "space waves" are used which travel in straight line from transmitter antenna to receiver antenna through troposphere. Range of frequencies of space waves used for LOS communication usually varies from about 50 MHz to few giga hertz. As per question hT = 20 m, hR = 45 m and R = 6.4 × 106 m Maximum distance between two antennas for satisfactory communication in LOS mode : =16+24 = 40km 23) One day Chetan's mother developed a severe stomach ache all of a sudden. She was rushed to the doctor who suggested for an immediate endoscopy test and gave an estimate of expenditure for the same. Chetan immediately contacted his class teacher and shared the information with her. The class teacher arranged for the money and rushed to the hospital. On realising that Chetan belonged to a below average income group family, even the doctor offered concession for the test fee. The test was conducted successfully. Answer the following questions based on the above information : a) Which principle in optics is made use of in endoscopy ? b) Briefly explain the values reflected in the action taken by the teacher. c) In what way do you appreciate the response of the doctor on the given situation ? Ans: (a) Principle of total internal reflection of light is used in endoscopy. (b) The teacher has a concern and caring for her student and his family. She gave him moral support as well as arranged money needed for test of Chetan's mother. She also rushed to hospital and supervised herself the progress of tests etc. (c) The doctor played a role of responsible citizen. On realising that Chetan belonged to a low income group family, he gave concession for test fee without compromising the procedure of test. 24) A long straight wire of circular cross section of radius a carries a steady current I. The current is uniformly distributed across the cross-section. Apply Ampere’s circuital to calculate the magnetic field at a point in the region for (i) r < a (ii) r = a (iii) r > a. Also show the variation of magnetic field B with distance r with the help of graph. SOL: MAGNETIC FIELD DUE TO CURRENT THROUGH A LONG CYLINDERICAL WIRE ( THICK WIRE ) : Consider an infinite long cylinder of radius R with axis XY. Let I be the current passing through the cylinder. A magnetic field is set up due to current through the cylinder in the form of circular magnetic lines of force, with their centres lying on the axis of cylinder. These lines of force are perpendicular to the length of cylinder. Case I. Point P is lying outside the cylinder. Let r be the perpendicular distance of point P from the axis of cylinder, where r > R. Let B be the magnetic field induction at P. It is acting tangential to the magnetic line of force at P directed into the paper SIROHI CLASSES PH-9810252244/9555552244/34 Page 15 Case II. Point P is lying inside cylinder. Here r < R. We may have two possibilities. (i) If the current is only along the surface of cylinder which is so if the conductor is a cylindrical sheet of metal, then current through the closed path L is zero. Using Ampere circutal law, we have B = 0 (ii) If the current is uniformly distributed throughout the cross-section of the conductor, then the current through closed path L is given by The field at the centre (r = 0) of the thick wire (or cylinder) will he zero. If we plot a graph between magnetic field induction B and distance from the axis of cylinder for a current flowing through a solid cylinder, we get a curve of the type as shown in Fig. Or Explain briefly with the help of a labelled diagram, the principle and working of a moving coil galvanometer. Define the term 'current sensitivity' of a galvanometer. How is it that increasing current sensitivity may not necessarily increase its voltage sensitivity? Explain. What is the nature of the magnetic field in a movingcoil galvanometer and why? SOL: MOVING COIL GALVANOMETER Moving coil galvanometer: It is a device used for the detection and measurement of small electric currents or small p.d.’s of the order of mA and mV. Principle: A current-carrying coil placed in a magnetic field experiences a torque, the magnitude of which. depends on the strength of current. Construction: It consists of a coil having a large number of turns of insulated copper wire wound on a metallic frame. The coil is suspended by means of a phosphorbronze strip and is surrounded by a horse-shoe magnet SIROHI CLASSES PH-9810252244/9555552244/34 Page 16 NS. A hair spring is attached to lower end of the coil. The other end of the spring is attached to the scale through a pointer. The coil is placed symmetrically between the concave poles of a permanent horse-shoe magnet. There is a cylindrical soft iron core which not only makes the field radial but also increases the strength of the magnetic field. Theory and Working: As the field is radial, the plane of the coil always remains parallel to the field B . When a current flows through the coil, a torque acts on it. It is given by = Force Perpendicular distance = NIbB a sin 90° = NIB (ab) = NIBA Here =90°, because the normal to the plane of coil remains perpendicular to the field B in all positions ( Magnetic field is radial ) or plane of coil is parallel to the direction of magnetic field. Due to deflecting torque, the coil rotates and suspension wire gets twisted. A restoring torque is set up in the suspension fibre. If is angle through which the coil rotates and k is the restoring torque per unit angular twist, then restoring torque, τ = k . In equilibrium, Deflecting torque = Restoring torque BNA BINA = k or I k BNA where = Galvanometer constant (Figure of merit) k I Thus the deflection produced in the galvanometer coil is proportional to the current flowing through it. (i)A uniform magnetic field provides a linear current scale. (ii)A soft iron core makes the field radial. It also increases the strength of the magnetic field and hence increases the sensitivity of the galvanometer. Significance of radial magnetic field : In a radial field, the plane of the coil remains parallel to the lines of force in all orientations of the coil and so the magnitude of the deflecting torque remains constant. Hence the deflection of the galvanometer is directly proportional to the current in its coil. Figure of merit of a galvanometer: It is defined as the current which produces a deflection of one scale division in the galvanometer and is given by I k BNA Example: A galvanometer having 20 divisions on its scale and a resistance of 50 Ω , when joined in series to a 1.5-V cell through a resistance of 100 Ω gives full-scale deflection. Find the figure of merit of the galvanometer. Sol: The current giving full-scale deflection in the galvanometer is given by Sensitivity : A galvanometer is said to be sensitive if a small current passed through it produces a sufficiently large deflection. The sensitivity is of two types : current sensitivity and voltage sensitivity. The current sensitivity of a galvanometer is defined as the deflection produced in the galvanometer when a unit current flows through it. SIROHI CLASSES PH-9810252244/9555552244/34 Page 17 If a current I produces a deflection in the galvanometer, then the current sensitivity is I , we have I BNA k The reverse of current sensitivity is known as 'figure of merit' of the galvanometer. Voltage Sensitivity It is the deflection produced in the galvanometer, when a unit potential difference is applied across its coil V IR BNA kR the sensitivity of a galvanometer can be increased by increasing the values of N, A and B, and decreasing the value-of k. However, N and A cannot be increased beyond a limit, otherwise the resistance and also the mass of the galvanometer coil would increase undesirably. Therefore, B is made as large as possible and k as small as possible. The magnetic field B is increased by taking a very strong horse-shoe magnet and placing a soft-iron core within the coil. The core concentrates the lines of force. The torsional constant k is made small by using a long and fine suspension strip and a thin spiral spring of phospher-bronze or quartz. The quartz fibres are silvered to make them conducting. Factors on which the sensitivity of a moving coil galvanometer depends : 1. Number of turns N in its coil. 2. Magnetic field B. 3. Area A of the coil. 4. Torsion constant k of the spring and suspension wire. Factors by which the sensitivity of a moving coil galvanometer can be increased : 1. By increasing the magnetic field B. 2. By decreasing the value of torsion constant k. 3. By increasing the number of turns N of the coil.(not preferred ) By increasing the area A of the coil.(not preferred ) Radial. (i)Maximum torque is experienced. (ii)Torque is uniform for all positions of the coil. (iii)Plane of the coil, is parallel to the direction of magnetic field (normal to the plane of coil remains perpendicular to the field B ). 25) (a) Define electric dipole moment. Is it a scalar or a vector ? Derive the expression for the electric field of a dipole at a point on the equatorial plane of the dipole. (b) Draw the equipotential surfaces due to an electric dipole. Locate the points where the potential due to the dipole is zero. Ans: (a) Two equal and opposite charges of + q and – q separated by a small distance '2a' constitute an electric dipole of dipole moment p = q.2 a. Generally, the distance '2a' between the charges of dipole is extremely small. Electric dipole moment p is a vector and its direction is from –ve charge to +ve charge. Let us calculate the electrostatic field intensity at a point P on the equatorial line at a distance 'r' from mid-point O of an electric dipole AB. Let us resolve E A and E B along and perpendicular to the dipole axis. We find that components EA sinθ and EB sinθ nullify each other and hence SIROHI CLASSES PH-9810252244/9555552244/34 Page 18 (b) Equipotential surfaces due to an electric dipole are closed loops around the two charges as shown in figure. Potential due to the dipole is zero at all the points (P1, P2, O, P3, P4 etc.) situated on its equatorial plane. Or Using Guass' law deduce the expression for the electric field due to a uniformly charged spheri• cal conducting shell of radius R at a point (i) outside and (ii) inside the shell. Plot a graph showing variation of electric field as a function of r > R and r < R (r being the distance from the centre of the shell) Ans: (i) Electric field outside a conducting spherical shell : Consider a uniformly charged thin spherical shell of radius R and having a charge Q. To find electric field intensity at a point P outside the shell situated at a distance r (r > R) from the centre of shell, consider a sphere of radius r as the Gaussian surface. All points on this surface are equivalent relative to given charged shell and, thus, electric field E at all points of Gaussian surface has same magnitude E and E and n are parallel to each other. Total electric flux over the Gaussian surface From the result it is clear that for any point outside the shell, the effect is, as if whole charge Q is concentrated at the centre of the spherical shell. SIROHI CLASSES PH-9810252244/9555552244/34 Page 19 (ii) Electric field inside a conducting spherical shell : Consider a hollow charged conducting shell of radius R and having charge Q. To find electrical field at a point P inside the shell, consider a sphere through point P and having centre O, i.e., r = OP (where r < R) as the Gaussian surface. The electric flux through the Gaussian surface 26) Using Bohr's postulates, derive the expression for the frequency of radiation emitted when electron in hydrogen atom undergoes transition from higher energy state (quantum number n) to the lower state, (n f). When electron in hydrogen atom jumps from energy state ni= 4 to nf= 3, 2, 1, identify the spectral series to which the emission lines belong. Ans: Bohr gave following three postulates for hydrogen atom : 1. An electron revolves round the nucleus in certain specified circular orbits in which it does not radiate energy. The centripetal force required for uniform circular motion in such a stationary orbit is provided by electrostatic force of attraction. Thus, 2. For an orbit to be stationary (or non-radiating), the angular momentum of the electron must be an integer multiple of h , where h is the Planck's constant. Thus, 2 3. Whenever an electron shifts from one of its specified non-radiating orbit to another such orbit, it emits/absorbs a photon whose energy is equal to the energy difference between the initial and final states. Thus, SIROHI CLASSES PH-9810252244/9555552244/34 Page 20 If an electron undergoes transition from higher energy state (quantum number n) to the lower state (nf), then as per equation (iii), we have When electron in hydrogen atom jumps from energy state ni = 4 to nf = 3, 2 and 1 respectively, the emission line belongs to Paschen series (4 - 3), Balmer series (4 - 2) and Lyman series (4 -1). Or (a) Draw the plot of binding energy per nucleon (BE/A) as a function of mass number A. Write two important conclusions that can be drawn regarding the nature of nuclear force. (b) Use this graph to explain the release of energy in both the processes of nuclear fusion and fission. SIROHI CLASSES PH-9810252244/9555552244/34 Page 21 (c) Write the basic nuclear process of neutron undergoing -decay. Why is the detection of neutrinos found very difficult ? Ans: (a) The plot of binding energy per nucleon (BE /A) as a function of mass number (A) is shown in figure. Two important conclusions which can be drawn regarding the nature of nuclear force are : (i) From the constancy of the binding energy per nucleon in the range 30 < A < 170 we conclude that the nuclear force is short ranged one. A particular nucleon will be under the influence of only some of the neighbouring nucleons which come within the range of the nuclear force: (ii) As binding energy is always positive, it suggests that nuclear force is attractive in nature and much stronger than electrostatic repulsion between the protons. (b) (i) Binding energy per nucleon for heavier nuclei is comparatively small (for 92 U 238 , it is 7.6 MeV/nucleon) as compared to nuclides having mass numbers in the middle range (about 8.5 MeV/ nucleon). It means that heavier nuclides are less stable and prone to disintegrate via nuclear fission. As a results, of fission we obtain product nuclei having higher binding energy per nucleon, thereby releasing a large amount of energy (equal to B.E. of product nuclides minus the B.E. of nucleusundergoing fission). The energy released will be about 235 × (8.5 – 7.6) MeV = 200 MeV. (ii) Consider fusion of 1 H 2 nuclei into a single 2He4 nucleus as an example. B.E. per nucleon of 2 4 1H is 1.11 MeV but B.E. per nucleon of 2He is 7.07 MeV. Thus, when two deuterium nuclei fuse into a helium nucleus, an energy of about 23.8 MeV will be released, being the difference between binding energies of a helium nucleus and that of 2 deuterium nuclei. (c) Beta-particles (or electrons) as such are not present inside a nucleus. However, at the time of decay a neutron decays into a proton, an electron and an antineutrino as per equation : on 0 n 1 p 1 e Q where mass and charge of antineutrino particle is zero. Out of the particles formed, the proton remains within the nucleus itself but electron along with antineutrino come out of nucleus. It is this electron which is being emitted as beta-particle. 1 1 0 It is very difficult to detect neutrino ( ) and antineutrino ( ) particles because these are massless and chargeless. SIROHI CLASSES PH-9810252244/9555552244/34 Page 22
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