P18 – Magnetic Effects of Electric Current 1 Fourth Edition P18. Magnetic Effects of Electric Current TABLE OF CONTENTS P18. Magnetic Effects of Electric Current 1 P18.1 Moving Charge in a Magnetic Field ......................................................................................................................................2 P18.2 Biot-Savart’s Law and Ampere’s Law..................................................................................................................................7 P18.3 Magnetic field and Current Carrying Loop ....................................................................................................................11 Subjective Questions..............................................................................................................................................................................14 Test Practice Problems .........................................................................................................................................................................16 Answer Key ................................................................................................................................................................................................21
P18 – Magnetic Effects of Electric Current 2 PRE-TEST Q1. Electric current is a __________ (scalar/vector) quantity. Q2. If 2 non-collinear vectors ���⃗��� and ���⃗⃗��� are taken, then the direction of ���⃗��� × ���⃗⃗��� is perpendicular to __________. (���⃗���/���⃗⃗���/both) Q3. Acceleration is the rate of change of __________ of a body. (velocity/speed/displacement) Q4. The net electric force on a charge ������ in an electric field ���⃗⃗��� is ___________. Q5. The power delivered by a battery of emf ������ and producing a current ������ is ___________. Q6. If 2 parallel vectors ���⃗��� and ���⃗⃗��� with magnitudes ������ and ������ are taken respectively, the value of ���⃗���. ���⃗⃗��� is ___________. (Answers at the end of this topic) Score 1 point per correct answer. SCORE If you score less than 4, Please take some time to revise the following topics. PRE-REQUISITES P17 Current Electricity P15 Electrostatics P4 Motion in Two Dimensions P3 Vectors and Calculus P18.1 Moving Charge in a Magnetic Field CONCEPTS 1. Magnetism 2. Magnetic force on a moving charge in a magnetic field 3. Radius, pitch and frequency of motion of a charged particle in a magnetic field 4. Magnetic force on various configurations of current carrying conductors 5. Cyclotron 6. Velocity selector and particle separator 7. Motion of a charged particle in an electromagnetic field PRE-READING Category Book Name (Edition) Chapter Section REQUIRED NCERT, Class 12, Part 1 4– Moving Charges and 4.1 to 4.4 Magnetism ADDITIONAL Concepts of Physics – H. C. Verma, Part-2 34-Magnetic Field 34.1 & 34.5 (Optional) P18.1
P18 – Magnetic Effects of Electric Current 3 PRE-READING EXERCISE Q1. It is given that the electric force on a charge ������ moving with velocity ���⃗��� is ���������⃗⃗��� and the magnetic force on it is ���������⃗��� × ���⃗⃗���. The net force on the charge (defined as Lorentz force) is: A) ���������⃗⃗��� − ���������⃗��� × ���⃗⃗��� B) ���������⃗⃗��� + ���������⃗��� × ���⃗⃗��� C) −���������⃗⃗��� − ���������⃗��� × ���⃗⃗��� D) −���������⃗⃗��� + ���������⃗��� × ���⃗⃗��� Q2. A charge with its velocity perpendicular to the uniform magnetic field at all times moves in a __________. (Circle/Straight line/ Helix) Q3. Magnetic force on a straight current carrying conductor of length ������, carrying a current ������, kept in a magnetic field ���⃗⃗��� is ������(���⃗��� × ���⃗⃗���). (True/ False) Q4. Magnetic field exerts force which is parallel to the velocity of motion. (True/ False) Q5. Magnetic field exerts force on moving as well as stationary charge. (True/ False) Q6. The radius of circle described by a charged particle of charge ������ and mass ������ moving in a region of uniform magnetic field of magnitude ������ which is perpendicular to the velocity of the particle at all instances is : A) ������������ B) ������������2 C) ������������ D) 2������ ������������2 ������������ 2������ ������������ IN CLASS EXERCISE LEVEL 1 Q1. A charge of 2.0 ������������ moves with a speed of 2.0 × 106 ������/������−1 along the positive ������ −axis. A magnetic field ���⃗⃗��� of strength (0.20 ������̂ + 0.40 ���̂���)������ exists in space. Find the magnetic force acting on the charge. Q2. A charged particle is projected in a magnetic field ���⃗⃗��� = (3������̂ + 4������̂) × 10−2������. The acceleration of the particle is found to be ���⃗��� = (������������̂ + 2������̂) ������/������2. Find the value of ������(Neglect gravity). Q3. A particle of mass ������ and charge ������ is projected with a speed ������ into a region having a perpendicular uniform magnetic field ������ of width ������. Find the angle of deviation ������ of the particle as it comes out of the magnetic field. LEVEL 2 Q4. A square of side 2.0 ������ is placed in a uniform magnetic field ���⃗⃗��� = 2.0 ������ in a direction perpendicular to the plane of the square inwards. Equal current ������ = 3.0 ������ is flowing in the directions shown in the figure in arms ������������������, ������������ and ������������������. Find the magnetic force on the loop. Here, take the direction out of the page to be the positive ������ − axis, the upwards direction to be the positive ������ − axis and the right direction be the positive ������ − axis. Q5. When a proton has a velocity ���⃗��� = (2������̂ + 3������̂) × 106 ������⁄������, it experiences a force ���⃗��������� = −(1.28 × 10−13���̂���)������ due to the uniform magnetic field it is traveling in. When its velocity is along the ������ −axis, it moves undeflected. What is the magnetic field? Q6. A beam of charged particle, all having kinetic energy 103 ������������, containing two kinds of particles with masses 8 × 10−27kg and 1.6 × 10−26 ������������ and all having the same charge of 1.6 × 10−19������ emerge from the end of an accelerator tube. There is a plate at distance 10−2 ������ from the end of the tube and placed perpendicular to the beam. Calculate the minimum magnitude of magnetic field which can prevent the beam from striking the plate.
P18 – Magnetic Effects of Electric Current 4 Q7. A cyclotron’s oscillator frequency is 10 ������������������ . What should be the operating magnetic field for accelerating protons? If the radius of its Dees is 60 cm, what is the kinetic energy (������������ ������������������) of the proton beam produced by the accelerator? (Take ������ = 22) 7 (������ = 1.60 × 10−19������, ������������ = 1.67 × 10−27������������, 1 ������������������ = 1.6 × 10−13������) Q8. A stream of protons and deuterons in a vacuum chamber enters a uniform magnetic field. Both protons and deuterons have been subjected to same acceleration potential, hence the kinetic energies of the particles are the same. If the ion-stream is perpendicular to the magnetic field and the protons move in a circular path of radius 15 ������������, find the radius of the path traversed by the deuterons. Given that mass of deuterons is twice that of a proton. LEVEL 3 Q9. A proton (charge 1.6 × 10−19������, mass= 1.67 × 10−27������������) is shot with a speed 8 × 106 ������/������ at an angle of 30° with the ������ −axis . A uniform magnetic field ������ = 0.30������ exists along positive ������ −axis. Show that path of the proton is a helix. Find the radius and pitch of the helix. (Take (1.67)(4) = 13.9, 4√3 = 6.93, 2������×1.67 = 21.9, 6.93 × 2.19 = (1.6)×0.3 (1.6)×0.3 15.18) Q10. A particle of specific charge ������ enters a uniform magnetic field ���⃗⃗��� = −������0���̂��� with velocity ���⃗��� = ������0������̂ from the origin. Find the time dependence of velocity and position of the particle. HOMEWORK LEVEL 1 Q1. A wire is bent in the form of an equilateral triangle ������������������ of side 10 ������������ and carries a current of 5.0 ������. It is placed in a magnetic field ������ of magnitude 2.0 ������ directed perpendicularly into the plane of the loop into the plane of paper. Find the forces on the three sides of the triangle. Q2. What is the smallest value of ������ (magnitude of magnetic field) that can be set up at the equator to permit a proton of speed 107 ������/������ to circle around the Earth? Radius of the Earth is ������������ and mass of a proton is ������������. [������������ = 6.4 × 106 ������, ������������ = 1.67 × 10−27 ������������] Assume that there is no magnetic field of the Earth. Q3. A charge ������ = −4 ������������ has an instantaneous velocity ���̂��� = (2������̂ − 3������̂ + ���̂���) × 106 ������������−1 in a uniform magnetic field ���̂��� = (2������̂ + 5������̂ − 3���̂���) × 10−2 ������. What is the force on the charge? P18.1
P18 – Magnetic Effects of Electric Current 5 LEVEL 2 Q4. A potential difference of 600 ������ is applied across the plates of a parallel plate condenser. The separation between the plates is 3 ������������. An electron projected vertically, parallel to the plates, with the velocity of 2 × 106 ������������−1moves undeflected between the plates. Find the magnitude and direction of the minimum magnetic field in the region between the condenser plates (Neglect the edge effects. Charge of the electron = −1.6 × 10−19������ ). Q5. Figure shows two long fixed metal rails placed horizontally and parallel to each other at a separation ������ . A uniform magnetic field ������ exists in the vertically downward direction. A wire of mass ������ can slide on the rails. The rails are connected to a constant current source which drives a current ������ in the circuit. The friction coefficient between the rails and the wire is ������. I. What should be the minimum value of ������ which can prevent the wire from sliding on the rails? II. Describe the motion of the wire if the value of ������ is half the value found in the previous part. Q6. A wire of 60 ������������ length and mass 16 ������ is suspended by a pair of flexible leads in a magnetic field of induction 0.40 ������. What are the magnitude and direction of the current required to remove the tension in the supporting leads? (Take ������ = 10 ������������−2) ( 1.6 = 6.7) 0.24 Q7. A magnetic field of (4.0 × 10−3���̂���)������ exerts a force (4.0������̂ + 3.0������̂) × 10−10������ on a particle having a charge 10−9������ and moving in the ������ − ������ plane. Find the velocity of the particle. Q8. Deuterons in a cyclotron describe a circle of radius 32.0 ������������ just before emerging from the ������′������. The frequency of the applied alternating voltage is 10 ������������������. Find, I. The magnetic field II. The energy (in ������������������) and speed of the deuterons upon emergence. (Note:1������������������ = 1.6 × 10−13������) Q9. A proton beam passes without deviation through a region of space where there are uniform, transverse, mutually perpendicular electric and magnetic fields with ������ = 120 ������������ and ������ = ������ 50 ������������ as shown in the figure. Then the beam strikes a grounded target and stops. Find the magnitude of force with which the beam acts on the target if the beam current is equal to ������ = 0.8 ������������. Mass of protons = 1.67 × 10−27������������ , take 8.35 × 2.4 = 20.04. Q10. An electron accelerated through a potential difference of 2.5������������, moves horizontally into a region of space in which there is a downward directed uniform electric field of magnitude 10 ������������ ������−1. I. In what direction must a magnetic field be applied so that the electron moves undeflected? Ignore the gravitational force. What is the magnitude of the smallest magnetic field possible in this case? II. What happens to the magnetic field’s magnitude if the charge is a proton, in place of the above electron, that passes through the same electric field with the same velocity and orientation?
P18 – Magnetic Effects of Electric Current 6 Q11. A charged particle, with charge ������ = 1������������, moving in a uniform magnetic field with velocity ������1 = 106 ������/������ at angle 45° with ������ −axis in the ������ − ������ plane, experiences a force ������1 = 5√2 ������������ along the negative ������ − ������������������������. When the same particle moves with velocity ������2 = 106 ������/������ along the ������ −axis, it experiences a force ������2 in ������ − direction. Find, I. Magnitude and direction of the minimum magnetic field, II. The magnitude of the force ������2. LEVEL 3 Q12. An electron accelerated by a potential difference ������ = 1.0������������ moves in a uniform magnetic field at an angle ������ = 30° to the vector ������ whose modulus is ������ = 29 ������������. Find the pitch of the helical trajectory of the electron. (Take the mass of electron ������������ = 9.1 × 10−31 ������������, the charge of an electron ������ = −1.6 × 10−19 ������ and √32 × √3 = 1.624) 9.1 2 Q13. A charged particle (charge ������, mass ������) enters a uniform magnetic field ���⃗⃗��� at an angle ������ as shown in figure with speed ������0. Find, I. The angle ������ at which it leaves the magnetic field II. Time spent by the particle in magnetic field & III. The distance ������������. Q14. A loop of flexible conducting wire of length 0.5 ������ lies in a magnetic field of 1.0 ������ perpendicular to the plane of the loop. Show that when a current (as shown in figure) is passed through the loop, it opens into a circle. Also, calculate the tension developed in the wire if the current is 1.57 ������. Q15. A particle of specific charge ������ is projected from origin with velocity ���⃗��� = ������0������̂ − ������0���̂��� in a uniform magnetic field ���⃗⃗��� = −������0���̂��� . Find the time dependence of velocity and position of the particle. Q16. A current ������, indicated by the crosses in figure, is established in a strip of copper of height ℎ and width ������. A uniform magnetic field of magnitude ������ is applied at right angle to the strip. I. Calculate the drift velocity ������������ of the electrons. II. What are the magnitude and direction of the magnetic force ������ acting on the electrons? III. What should the magnitude and direction of a homogeneous electric field ������ be in order to counterbalance the effect of magnetic field? IV. Calculate voltage ������ necessary between two sides of the conductor in order to create this field ������? Between which sides of the conductor would this voltage have to applied? V. If no electric field is applied from the outside, the electrons will be pushed somewhat to one side and therefore will give rise to a uniform electric field ������������ across the conductor until the forces of this electrostatic field ������������ balance the magnetic forces encountered in part (b). What will be the magnitude and direction of field ������������? (This is called the Hall Effect) Assume that ������ , the number of conductor electrons per unit volume is 1.1 × 1029/������3 , ℎ = 0.02������, ������ = 0.1 ������������, ������ = 50 ������ and ������ = 2 ������. P18.1
P18 – Magnetic Effects of Electric Current 7 P18.2 Biot-Savart’s Law and Ampere’s Law CONCEPTS 1. Biot-Savart’s Law 2. Magnetic force on one current carrying conductor due to another current carrying conductor 3. Ampere's Law 4. Magnetic field due to a solenoid and toroid PRE-READING Category Book Name (Edition) Chapter Section REQUIRED NCERT, Class 12, Part 1 4 – Electric Charges and Fields 4.5 to 4.9 ADDITIONAL Concepts of Physics – H. C. Verma, Part-2 35 - Magnetic Field due to a Current 35.1 to 35.8 PRE-READING EXERCISE Q1. SI unit of magnetic field is __________. Q2. If ������ = mass, L= length, ������ = time and ������ = electric current, then the dimensional formula for the magnetic permeability of free space, ������0 is : A) [������0] = [������1������1������−2������−2] B) [������0] = [������1������−4������3������2] C) [������0] = [������1������−4������3������−1] D) [������0] = [������1������1������0������−1] Q3. Magnetic flux through an Amperian loop is directly proportional to the __________ (charge/current) enclosed in the loop. Q4. The direction of the magnetic field is given by the left-hand thumb rule. (True/False) Q5. Turns in a solenoid are insulated from each other. (True/ False) Q6. According to Biot-Savart’s Law, the magnetic field due to a current carrying element is inversely proportional to __________ (current/length of current element/distance/square of the distance). IN CLASS EXERCISE LEVEL 1 Q1. The given figure shows three current carrying conductors and three imaginary loops. Calculate the current enclosed by each of the loops.
P18 – Magnetic Effects of Electric Current 8 Q2. A pair of stationary and infinitely long bent wires are placed in the ������ − ������ plane as shown in the given figure. Each wire carries current of 10 ������. The segments ������������ and ������������ are along the ������ −axis. The segments ������������ and ������������ are parallel to the ������ −axis such that ������������ = ������������ = 0.02 ������. Find the magnitude and direction of the magnetic induction at the origin ������. LEVEL 2 Q3. Shown in the figure below is a conductor carrying current ������. Find the magnetic field intensity at point ������. Q4. Consider a coaxial cable which consists of an inner wire of radius ������ surrounded by an outer shell of inner and outer radii ������ and ������ respectively. The inner wires carries an electric current ������0 and the outer shell carries an equal current in opposite direction. Find the magnetic field at a distance ������ from the axis where (I) ������ < ������, (II) ������ < ������ < ������, (III) ������ < ������ < ������ (IV) ������ > ������. Assume that the current density is uniform in the inner wire and also uniform in the outer shell. Q5. A solenoid of length 0.4 ������ and diameter 0.6 ������ consists of a single layer of 1000 ������������������������������ of fine wire carrying a current of 5.0 × 10−3������. Calculate the magnetic field intensity on the axis at the middle and at the ends of the solenoid. LEVEL 3 Q6. Two long straight parallel wires are 2.0 ������ apart, perpendicular to the plane of the paper. The wire A carries a current of 9.6 ������, directed into the plane of the paper. The wire ������ carries a current such that the magnetic field of induction at the point ������, at a distance of 10/11 ������ (As shown in the figure) from the wire ������ is zero. Find: I. The magnitude and direction of the current in ������. II. The magnitude of the magnetic field at point ������. III. The magnitude of magnetic force per unit length on the wire ������. P18.2
P18 – Magnetic Effects of Electric Current 9 Q7. A sphere of radius ������, uniformly charged with the surface charge density ������, rotates around the axis passing through its centre at an angular velocity ������. Find the magnetic induction at the centres of the rotating sphere. Q8. A square frame carrying a current ������ = 0.90������ is located in the same plane as a long straight wire carrying a current ������0 = 5.0������. The frame side has a length ������ = 8.0������������. The axis of the frame passing through the mid-points of the opposite sides is parallel to the wire and is separated from it by a distance ℎ = 15 times greater than the side of the frame. Find: I. Magnitude of magnetic force acting on the frame. II. The mechanical work to be performed in order to turn the frame through 180° about its axis, with the current maintained constant. Q9. A long cylinder of uniform cross section and radius ������ is carrying a current ������ along its length and current density is uniform. There is a cylindrical cavity of uniform cross section and radius ������ in the cylinder parallel to its length. The axis of the cylindrical cavity is separated by a distance ������ from the axis of the cylinder. Find the magnitude of magnetic field at the axis of cylinder. HOMEWORK LEVEL 1 Q1. If two parallel wires carry current in opposite directions A) The wires attract each other B) The wires repel each other C) The wires experience neither attraction nor repulsion D) The forces of attraction or repulsion do not depend on current direction. Q2. Figure below shows a square loop made from a uniform wire. Find the magnetic field at the center of the square if a battery is connected between the points ������ and ������. Q3. Three identical long solenoids ������, ������ and ������ are connected to each other as shown in figure below. If the magnetic field at the centre of ������ is 2.0 ������, what would be the field at the center of ������? Assume that the field due to any solenoid is confined within the volume of that solenoid only.
P18 – Magnetic Effects of Electric Current 10 Q4. A toroid has ������ = 90 turns with an inside diameter of ������������������ = 4 ������������ and an outside diameter ������������������������ = 5 ������������ . Determine the magnitude of magnetic field intensity ������ along the mean path length within the toroid with a current ������ = 2.5 ������. Q5. Find the magnetic field at the point ������ in figure below. The curved portion is a semicircle with its center at ������ and the straight wires are long. LEVEL 2 Q6. A current of 2.00 ������ exists in a square loop of edge 10.0 ������������. Find the magnetic field ������ at the centre of the square loop. Q7. Figure below shows a cross section of a large metal sheet carrying an electric current along its surface. The current in a strip of width ������������ is ������������������ where ������ is a constant. Find the magnitude of magnetic field at a point ������ at a distance ������ from the metal sheet. Q8. Consider the situation described in the previous example. A particle of mass ������ having a charge ������ is placed at a distance ������ from the metal sheet and is projected towards it. Find the maximum velocity of projection for which the particle does not hit the sheet. Q9. Two parallel horizontal conductors are suspended by two light vertical threads each 75.0 ������������ long. Each conductor has a mass per unit length of 40.0 ������������/������, and when there is no current they are 0.5 ������������ apart. Equal current in the two wires result in a separation of 1.5 ������������. Find the values and directions of currents. Take ������ = 9.8 ������/������2. Q10. Two long parallel wires are carrying currents 12 ������ and 8 ������ in the same direction. The wires are 10 ������������ apart. Where should a third parallel wire be placed so as to experience no force? Q11. Two straight infinitely long and thin parallel wires are spaced 0.1������ apart and current of 10 ������ each. Find the magnetic field at a point at a distance 0.1 ������ from both wires in two cases when the currents are in the (I) same and (II) opposite directions. Q12. A battery is connected between two points ������ and ������ on the circumference of a uniform conducting ring of radius ������ and resistance ������ as shown in figure below. One of the arcs ������������ of the ring subtends angle ������ at the centre. Show that the magnetic field at the centre of the coil is zero and independent of ������. LEVEL 3 Q13. Two long parallel wires carry currents of equal magnitude but in opposite directions. These wires are suspended from rod ������������ by four cords of same length ������ as shown in figure below. Determine the value of ������ assuming it to be small. Assume the length of each conductor be ������0 and the mass per unit length of the conductors be ������ and the current through each conductor be ������. Q14. A square loop of wire, of edge length ������, carries a current ������. Compute the magnitude of the magnetic field produced at a point on the axis of the loop at a distance ������ from the center of the square. P18.2
P18 – Magnetic Effects of Electric Current 11 P18.3 Magnetic field and Current Carrying Loop CONCEPTS 1. Torque and potential energy of a closed current carrying loop in an external magnetic field 2. Magnetic dipole 3. Magnetic moment due to a current carrying wire loop 4. Effect of parameters such as radius, number of turns, distance from the center of coil etc of a coil on the magnetic field produced by it 5. Bohr Magneton, angular quantum number and spin quantum number for an electron moving in a loop 6. Moving coil galvanometer, its conversion to Ammeter and voltmeter, and current and voltage sensitivity PRE-READING Category Book Name (Edition) Chapter Section REQUIRED NCERT, Class 12 4 – Moving charges and 4.10 & 4.11 magnetism ADDITIONAL Concepts of Physics – H. C. Verma, Part-2 36 - Permanent magnets 36.1, 36.2 & 36.9 (Optional) PRE-READING EXERCISE Q1. The net force on a current carrying loop placed in a uniform magnetic field is always zero. (T/F) Q2. The net torque on a current carrying loop placed in a uniform magnetic field is always zero. (T/F) Q3. The net force on a current carrying loop placed in a non-uniform magnetic field is always zero. (T/F) Q4. The net torque on a current carrying loop placed in a non-uniform magnetic field is always zero. (T/F) Q5. Magnetic dipole is a current carrying loop. It behaves the same for magnetic field as an electric dipole does for electric field. (T/F) Q6. Electric dipole is a combination of negative and positive point charges whereas magnetic dipole is a combination of negative and positive separable magnetic charges. (T/F) Q7. Moving coil galvanometer and electric motor are all based on the concept of magnetic dipole. (T/F) IN CLASS EXERCISE LEVEL 1 Q1. A galvanometer having 30 divisions has current sensitivity of 20 ������������/division. Find the maximum current it can measure. Q2. A circular loop and a square loop are formed from the same wire and the same current is passed through them. Find the ratio of their dipole moments.
P18 – Magnetic Effects of Electric Current 12 LEVEL 2 Q3. A circular coil of 200 ������������������������������ and radius 10 ������������ is placed in a uniform magnetic field of 0.5 ������, normal to the plane of the coil. If the current in the coil is 3.0 ������, calculate the: I. total torque on the coil, II. total force on the coil, III. and magnitude of average force on each electron in the coil, due to the magnetic field. Assume the area of cross- section of the wire to be 10−5 ������2 and the free electron density as 1029 ������−3 Q4. A rectangular coil of area 5.0 × 10−4 ������2 and 60 turns is pivoted about one of its vertical sides. The coil is in a radial horizontal field of 9.0 ������������ (radial here means the field lines are in the plane of the coil for any rotation). What is the torsional constant of the hair spring connected to the coil, if a current of 2.0 ������������ produces an angular deflection of 18°. Leave answer in terms of ������. Q5. An electron is moving in a circular orbit of radius ������ with an angular speed of ������ as shown in the figure. Find the magnitude of magnetic moment produced due to the revolving electron. Q6. What is the magnitude of magnetic moment of an electron in ������������ℎ orbit of hydrogen atom? Q7. A moving coil galvanometer gives full scale deflection, when a current of 0.005 ������ is passed through its coil. It is converted into a voltmeter reading upto 5 ������ by using an external resistance of 976 Ω. What is the resistance of the galvanometer coil? Q8. The coil of galvanometer is of the dimensions 0.02 ������ × 0.8 ������. It consists of 200 turns of the wire and is in a magnetic field of 0.20 ������ . The restoring torque constant of suspension fibre is 10−5������������/������������������������������������ . Assuming magnetic field to be radial I. What is the maximum current that can be measured by this galvanometer if scale can accommodate 45° deflection? II. What is the smallest current that can be detected if minimum observed deflection is 0.1°? LEVEL 3 Q9. A galvanometer of resistance 100 Ω gives a full scale deflection of 500. The galvanometer experiences a radial magnetic field of 0.1 ������, the number of turns of the coil is 200 and the area of the coil is 1 ������������2.The restoring torque constant for the coil is 0.4 × 10−9 ������������/������������������������������������. To convert it into an ammeter capable of measuring upto 1 ������, what resistance should we connect with it and in what combination? HOMEWORK LEVEL 1 Q1. Magnetic field at the center of a circular loop of area ������ is ������. Then find the magnitude of magnetic moment of the loop. Q2. A wire of length ������ is bent in the form of a circular coil of some turns. A current ������ flows through the coil. The coil is placed in a uniform magnetic field ������. What is the magnitude of maximum torque on the coil? P18.3
P18 – Magnetic Effects of Electric Current 13 Q3. A moving coil galvanometer has 100 ������������������������������ and each turn has an area of 2 ������������2. The magnitude of magnetic field produced by the magnet is 0.01 ������. The deflection in the galvanometer coil is 0.05 ������������������ when a current of 10 ������������ is passed through it. Find the torsional constant of the spiral spring, given that the magnetic field is always parallel to the area vector. Q4. A current carrying loop is placed in a uniform magnetic field in four different orientations, ������, ������������, ������������������ and ������������ , arrange them in descending order of potential energy: LEVEL 2 Q5. A metallic wire is folded to form a square loop of side ������. It carries a current ������ and is kept perpendicular to a uniform magnetic field. If the shape of the loop is changed from square to a circle without changing the length of the wire and current, what is the amount of work done in doing so? Q6. A galvanometer can be converted into voltmeter to measure upto I. ������ volts by connecting a resistance ������1 in series with coil. II. ������/2 volts by connecting a resistance ������2 in series with its coil. Find the resistance ������ in terms of ������1 and ������2 required to convert it into a voltmeter that can read up to 2������ volts. Q7. A conducting ring of mass 2 ������������ and radius 0.5 ������ is placed on a smooth horizontal plane. The ring carries a current ������ = 4 ������. A horizontal magnetic field with a magnitude, ������ = 10 ������ is switched on at time ������ = 0 as shown in figure. Find the initial angular acceleration of the ring. Leave the answer in terms of ������. Q8. A current of 5.0 ������ is passed through the coil of a galvanometer having 500 ������������������������������ and each ������������������������ has an average area of 3 × 10−4 ������2. If a torque of 1.5 ������������ is required for this coil carrying same current to set it parallel to a uniform magnetic field, calculate the strength of the magnetic field. Q9. A voltmeter has resistance of 2000 ������ and it can measure upto 2 ������. If we want to increase its range by 8 ������, then what is the required resistance in series? Q10. A rigid circular loop of radius ������ and mass ������ lies in the x-y plane on a flat table and has a current ������ flowing in it, at this particular place, the earth’s magnetic field is ���⃗⃗��� = ������������������̂ + ���������������̂���. What is the value of ������ so that one edge of the loop lifts from the table? LEVEL 3 Q11. In the figure, a coil of single turn is wound on a sphere of radius ������ and mass ������. The plane of the coil is parallel to the plane and lies in the equatorial plane of the sphere. Current in the coil is ������. Assume no friction acts between the sphere and the incline. Find the value of ������ if the angular velocity of the sphere remains constant.
P18 – Magnetic Effects of Electric Current 14 Subjective Questions Q1. What is a shunt? Mention its important uses. Q2. I. What is the importance of a radial magnetic field and how is it produced? II. Why is it that while using a moving coil galvanometer as a voltmeter a high resistance in series is required whereas in an ammeter a shunt is used? Q3. State the underlying principal of working of a moving coil galvanometer. Write two reasons why a galvanometer cannot be used as such to measure current in a given circuit. Name any two factors on which the current sensitivity of a galvanometer depends. Q4. How can a galvanometer of resistance ������ be converted into a voltmeter to read maximum potential difference of ������ volts? Support your answer with related mathematical expression. Q5. An electron does not suffer any deflection while passing through a region of uniform magnetic field. What is the direction of the magnetic field? Q6. An electron and a proton, moving parallel to each other in the same direction with equal momenta, enter into a uniform magnetic field which is at right angles to their velocities. Trace their trajectories in the magnetic field. Q7. A charged particle enters an environment of a strong and non-uniform magnetic field varying from point to point both in magnitude and direction, and comes out of it following a complicated trajectory. Would its final speed equal the initial speed if it suffered no collisions with the environment? Q8. Draw a schematic sketch of the cyclotron. State its working principle. Show that the cyclotron frequency is independent of the velocity of the charged particles. Q9. Using Biot-Savart law, derive an expression for the intensity of magnetic field at a point near a long current carrying straight conductor. Q10. State Biot-Savart law. Apply it to calculate the magnetic field at a point near a current carrying long straight wire. Q11. A straight wire of length ������ is bent into a semicircular loop. Use Biot-Savart law to deduce an expression for the magnetic field at its centre due to the current ������ passing through it. Q12. State Ampere’s circuital law. Write the expression for the magnetic field at the centre of a circular coil of radius ������ carrying a current ������. Draw the magnetic field lines due to this coil. Q13. A straight wire carrying a current of 12 ������ is bent into a semi-circular arc of radius 2.0 ������������ as shown in Fig. (������) Consider the magnetic field ������ at the centre of the arc. (I) What is the magnetic field due to the straight segments? (II) In what way the contribution to ������ from the semicircle differs from that of a circular loop and in what way does it resemble? (III) Would your answer be different if the wire were bent into a semi-circular are of the same radius but in the opposite way as shown in Fig. (������) Q14. Depict the field-line pattern due to a current carrying solenoid of finite length. I. In what way do these lines differ from those due to an electric dipole? II. Why can’t two magnetic field lines intersect each other? Q15. Magnetic field lines can be entirely confined within the core of a toroid, but not within a straight solenoid, Why? Subj.
P18 – Magnetic Effects of Electric Current 15 Q16. Depict the magnetic field lines due to two straight, long, parallel conductors carrying currents ������1 and ������2 in the same direction. Hence, deduce an expression for the force acting per unit length on one conductor due to the other. Is this force attractive or repulsive? Q17. A current-carrying circular loop is located in a uniform external magnetic field. If the loop is free to turn, what is its orientation of stable equilibrium? Show that in this orientation, the flux of the total field (external field + field produced by the loop) is maximum. Q18. Two wires of equal lengths are bent in the form of two loops. One of the loops in square shaped whereas the other loop is circular. These are suspended in a uniform magnetic field and the same current is passed through them. Which loop will experience greater torque? Give reasons. Q19. Write two factors by which voltage sensitivity of a galvanometer can be increased. Q20. Write two properties of a material used as a suspension wire in a moving coil galvanometer. Q21. Define the current sensitivity of a moving coil galvanometer. “Increasing the current sensitivity may not necessarily increase the voltage sensitivity.” Justify this statement.
P18 – Magnetic Effects of Electric Current 16 Test Practice Problems No. of questions: 30 Total time: 90 mins Time per question: 3 mins Purpose: To practice a mixed bag of questions in a speed based format similar to what you will face in entrance examinations. In most entrance examinations, you will get not more than 3 minutes to attempt a question. Hence, you need to be able to attempt a question in less than 3 minutes, and at the end of 3 minutes skip the question and move to the next one. Approach: Attempt the Test Practice Problems only when you have the stipulated time available at a stretch. Start a timer and attempt the section as a test. DO NOT look at the answer key / solutions after each question. DO NOT guess a question if you do not know it. Competitive examinations have negative marking. Fill the table at the end of the TPP and evaluate the number of attempts, and accuracy of attempts, which will help you evaluate your preparedness level for the chapter. Q1. Current ������ flows in an infinitely long wire with cross-section in the form of a semi-circular ring of radius ������. The magnitude of the magnetic induction along its axis is: A) ������0������ B) ������0������ C) ������0������ D) ������0������ 2������2������ ������2������ 2������������ 4������������ Q2. A steady electric current is flowing through a cylindrical conductor A) The magnetic field in the vicinity of the conductors is zero B) The electric field in the vicinity of the conductors is non-zero C) The magnetic field at the axis of the conductor is zero D) None of these Q3. A horizontal overhead power line is at a height of 4 ������ from the ground and carries a current of 100 ������ from east to west. The magnetic field directly below it on the ground is (������0 = 4������ × 10−7������ ������������−1) A) 2.5 × 10−7 ������, Southward B) 5.0 × 10−6 ������, Northward C) 5.0 × 10−6 ������, Southward D) 2.5 × 10−7 ������, Northward Q4. Two concentric coils of 10 ������������������������������ each are placed in the same plane. Their radii are 20 ������������ and 40 ������������ and carry 0.2 ������ and 0.3 ������ current respectively in opposite directions. The magnetic field (in tesla) at the centre is: A) 3 ������0 B) 5 ������0 C) 7 ������0 D) 9 ������0 4 4 4 4 Q5. A long straight wire of radius ������ carries a steady current ������. The current is uniformly distributed across its cross- section. The ratio of the magnetic field at ������ and 2������ is 2 A) 1 B) 4 C) 1 D) 1 4 2 Q6. A current ������ flows along the length of an infinitely long, straight, thin walled pipe. Then A) The magnetic field is zero only on the axis of the pipe B) The magnetic field is different at different points inside the pipe C) The magnetic field at any point inside the pipe is zero D) The magnetic field at all point inside the pipe is the same, but not zero T.P.P.
P18 – Magnetic Effects of Electric Current 17 Q7. A long solenoid has 200 ������������������������������/������������ and carries a current ������. The magnetic field at its centre is 6.28 × 10−2 ������������ ������−2. Another long solenoid has 100 ������������������������������/������������ and it carries a current ������/3. The value of the magnetic field at its centre is A) 1.05 × 10−2 ������������������−2 B) 1.05 × 10−5 ������������������−2 C) 1.05 × 10−3 ������������������−2 D) 1.05 × 10−4 ������������������−2 Q8. A current flows in a conductor from east to west. The direction of the magnetic field at a point above the conductor is A) Towards east B) Towards west C) Towards north D) Towards south Q9. Two concentric coils each of radius equal to 2������ are placed at right angles to each other. 3������ and 4������ are the currents flowing in each coil respectively. The magnitude of magnetic induction, in ������������������−2, at the centre of the coils will be (������0 = 4������ × 10−7������������������������−1) A) 12 × 10−5 B) 10−5 C) 5 × 10−5 D) 7 × 10−5 Q10. At the centre of a circular coil of radius 5 ������������ carrying current, magnetic field due to earth is 0.5 × 10−5 ������������������−2. What should be the current flowing through the coil so that it equals the earth’s magnetic field? A) 40 ������ B) 4 ������ C) 0.4 ������ D) 0.2 ������ Q11. An electron and a proton are moving on straight parallel paths with same velocity in positive ������ −direction. They enter a region of uniform magnetic field (direction of magnetic field perpendicular to the velocity) starting at ������ = 0 and extending to ∞ along the positive ������ −axis. Which of the following statement(s) is/are true? A) They will never come out of the magnetic field region B) They will never come out travelling along parallel paths C) They will never come out at the same time D) They will never come out at different times. Q12. An electron and a proton of equal linear momentum enter in a region of uniform magnetic field perpendicular to their velocities. If the radii of their circular paths be ������������ and ������������ respectively, then ������������ , is equal to (������������ = mass of ������������ electron, ������������=mass of proton) A) (������������)1/2 B) ������������ C) (������������ 1/2 D) 1 ������������ ������������ ������������ ) Q13. A charged particle (charge ������) is moving in a circle of radius ������ with uniform speed ������. The associated magnetic moment ������ is given by A) ������������������ B) ������������������2 C) ������������������2 D) ������������������ 2 2 Q14. The proton with energy 1 ������������������ describes a circular path in a plane at right angles to a uniform magnetic field of 6.28 × 10−4 ������. The mass of the proton is 1.7 × 10−27 kg. The cyclotron frequency of the proton is very nearly equal to A) 107 ������������ B) 105 ������������ C) 106 ������������ D) 104 ������������ Q15. Under the influence of a uniform magnetic field a charged particle is moving in a circle of radius ������ with constant speed ������. The time period of the motion A) Depends on ������ and not on ������ B) Depends on the both ������ and ������ C) Is independent of both ������ and ������ D) Depends on ������ and not on ������
P18 – Magnetic Effects of Electric Current 18 Q16. An electron having mass (9.1 × 10−31������������) and charge (1.6 × 10−19 ������) moves in a circular path of radius 0.5 ������ with a velocity 106 ������������−1 in a magnetic field. Strength of magnetic field is A) 1.13 × 10−5������ B) 5.6 × 10−6������ C) 2.8 × 10−6������ D) None of these Q17. A charged particle of mass ������ and charge ������ enters a region of uniform magnetic field ������ perpendicular of its velocity ������. The particle initially at rest was accelerated by a potential difference ������ (volts) before it entered the region of magnetic field. What is the diameter of the circular path followed by the charged particle in the region of magnetic field? A) 2 √������������������ B) 2 √2������������ ������ C) ������√2������������ ������ D) ������ √2������������ ������ ������ ������ ������ Q18. An electron (������ = 1.6 × 10−19������) is moving at right angle to the uniform magnetic field 3.534 × 10−5 ������. The time taken by the electron to compete a circular orbit is A) 2 ������������ B) 4 ������������ C) 3 ������������ D) 1 ������������ Q19. Two ions having masses in the ratio 1:1 and charges 1:2 are projected into uniform magnetic field perpendicular to the field with speeds in the ratio 2:3. The ratio of the radii of circular paths along which the two particles move is A) 4:3 B) 2:3 C) 3:1 D) 1:4 Q20. A charge is moving through a magnetic field. The force acting on the charge is maximum when the angle between the direction of motion of charge and the magnetic field is A) Zero B) ������ C) ������ D) ������ 4 2 Q21. A thin flexible wire of length ������ is connected to two adjacent fixed points and carries a current ������ in the clockwise direction, as shown in the figure. When the system is put in a uniform magnetic field of strength ������ going into the plane of the paper, the wire takes the shape of a circle. The tension in the wire is A) ������������������ B) ������������������ C) ������������������ ������ 2������ D) ������������������ 4������ Q22. Two long parallel wires are at a distance 2������ apart. They carry steady equal current flowing out of the plane of the paper as shown. The variation of the magnetic field along the line ������������′ is given by A) B) C) D) T.P.P.
P18 – Magnetic Effects of Electric Current 19 Q23. Two circular concentric loops of radii ������1 = 20 ������������ and ������2 = 30 ������������ are placed in the ������ − ������ plane as shown in the figure. A current ������ = 7 ������ is flowing through them as shown in the figure. The magnetic moment of this loop system is (Assume upward direction to be positive y direction and right direction to be positive x direction) A) +0.4���̂���(������������2) B) −1.5���̂���(������������2) C) +1.1���̂���(������������2) D) +1.3������̂(������������2) Q24. A current carrying straight wire is kept along axis of a circular loop carrying a current. The straight wire A) Will exert an inward force on the circular loop B) Will exert an outward force on the circular loop C) Will exert a force on the circular loop parallel to itself D) Will not exert any force on the circular loop Q25. A coil of 10 ������������������������������ and area 2 × 10−2 ������2, pivoted about a vertical diameter in a uniform magnetic field carries a current of 5 ������. When the coil is held with its plane in north-south direction, it experiences a torque of 0.3 ������������. When the plane is in East-West direction, the torque is 0.4 ������������. The value of magnetic induction is (Neglect earth’s magnetic field) A) 0.2 ������ B) 0.3 ������ C) 0.4 ������ D) 0.5 ������ Q26. An electron is moving in an orbit of radius ������ with a time period ������ as shown in the figure. The magnetic moment produced may be given by (where |������| represents the charge on an electron) A) |������| = 2������|������|������ B) |������| = |������|������ ������ ������ C) |������| = 2|������|������ D) |������| = ������ |������|������ ������ ������ Q27. Two long conductors, separated by a distance ������ carry currents ������1 and ������2 in the same direction. They exert a force ������ on each other. Now the current in one of them is increased to two times and its direction is reversed. The distance is also increased to 3������. The new value of the force between them is A) −2������ B) ������ C) − 2������ D) − ������ 3 3 3 Q28. Three long straight wires ������, ������ and ������ are carrying currents as shown in the figure. Then the resultant force on ������ is directed A) Perpendicular to the plane of paper and outward B) Perpendicular to the plane of paper and inward C) Toward A D) Towards C Q29. The figure shows three situations when an electron with velocity ������ travels through a uniform magnetic field ������ . In each case, what is the direction of magnetic force on the electron? A) +������������ ������ −axis, −������������ ������ −axis, +������������ ������ −axis B) – ������������ ������ −axis, −������������ ������ −axis and zero C) +������������ ������ −axis, +������������ ������ −axis and zero D) – ������������ ������ −axis, +������������ ������ −axis and zero
P18 – Magnetic Effects of Electric Current 20 Q30. A conducting loop carrying a current ������ is placed in a uniform magnetic field pointing into the plane of the paper as shown. The loop will have a tendency to A) Contract B) Expand C) Move towards +ve ������ −axis D) Move towards -ve ������ −axis DATA ANALYSIS Guide A # of questions Total problems in TPP B # Attempts Total attempts in OMR C # Correct Total questions correct D # Incorrect Out of the ones marked in OMR E # Unattempted ������ − ������ F Percentage attempts ������ G Percentage Accuracy ������ × 100 ������ ������ × 100 Question type # Correct (C) # Incorrect (I) # Unattempted (U) Level 1 Level 2 Level 3 Tip: To begin with, your accuracy must be high, typically > 60%. Percentage attempts should be > 50% As time progresses, your percentage attempts should increase without a reduction in accuracy. Additionally, you should be able to get > 80% Easy questions correct, as they involve basic recall of the concepts and formulae of the chapter. T.P.P.
P18 – Magnetic Effects of Electric Current 21 Answer Key PRE-TEST Q4. ���������⃗⃗��� Q5. ������������ Q1. Scalar Q6. ������������ Q2. Both Q3. Velocity P18.1 MOVING CHARGE IN A MAGNETIC FIELD PRE-READING EXERCISE Q3. −16(������̂ + 2������̂ + 4���̂���) × 10−2������ Q1. B LEVEL 2 Q2. Circle Q3. True Q4. 0.1 ������, into the plane of the paper. Q4. False Q5. False Q5. I. ������������������ Q6. A ������������ IN CLASS EXERCISE II. Moves towards right with an acceleration of LEVEL 1 Q1. (0.8���̂��� − 1.6������̂)������ ������������������ Q2. − 8 2������ 3 Q6. 0.67 ������ Q3. sin−1 (������������������) Q7. −75������̂ + 100������̂ ������/������ ������������ Q8. I. 1.30 T II. 2.01 × 107 ������ , 4.22 ������������������ LEVEL 2 Q4. −36(������̂) + 36(������̂) ������ ������ Q5. (−0.4 ������̂)������ Q6. 1.414 ������ Q9. 20.04 ������������ Q7. 0.67 ������, 7.5 ������������������ Q8. 0.212 ������ Q10. I. 3.38 × 10−4������ II. 3.38 × 10−4������ Q11. I. (10−2������̂)������ II. 10−2������ LEVEL 3 Q12. 2 ������������ Q13. I. ������ = ������ II. 2������������ LEVEL 3 ������������ Q9. Radius = 0.139 ������, Pitch = 1.518 ������ III. 2������������0 sin ������ ������������ Q10. ���⃗���(������) = ������������������̂ + ������������������̂ = ������0 cos ������ ������̂ + ������0 sin ������ ������̂; Q14. 0.12 ������ ���⃗���(������) = ������0 [sin(������0������������) ������̂ + {1 − cos(������0������������)}������̂] Q15. ���⃗���(������) = ������0 cos(������0������������) ������̂ + ������0 sin(������0������������) ������̂ − ������0���̂���; ������0������ Q16. ������0 ���⃗���(������) = ������0������ [sin(������0������������)������̂ + {1 − cos(������0������������)}������̂] HOMEWORK −������0���������̂��� LEVEL 1 I. 1.4 × 10−4������/������ Q1. 1.0 ������ each, perpendicular to the respective side II. 4.5 × 10−23������ (in downward direction) towards the inside of the triangle. III. 2.8 × 10−4������/������ (in downward direction) Q2. 1.6 × 10−8 ������ IV. 5.6 × 10−6������ V. 2.8 × 10−4������/������ (in downward direction)
P18 – Magnetic Effects of Electric Current 22 P18.2 BIOT-SAVART’S LAW AND AMPERE’S LAW PRE-READING EXERCISE Q9. ������0������ × ������2 2������������ (������2−������2) Q1. Tesla (T) Q2. A HOMEWORK Q3. Current Q4. True LEVEL 1 Q5. True Q6. square of distance Q1. B Q2. 0 IN CLASS EXERCISE Q3. 1.0 ������ LEVEL 1 Q4. 2 ������������ Q1. Loop 1: −������1; Loop 2: ������3 − ������2; Loop 3: ������3 − ������1 − ������2 Q5. ������0������ (1 + 2), out of the plane of paper Q2. 10−4 ������, into the plane of the paper 2������ ������ LEVEL 2 LEVEL 2 Q6. 22.6 × 10−6 ������, perpendicular to the plane of the paper Q3. 5������0������������ ���̂��� Q7. 1 ������0������ 2 24������������ Q4. I. ������0������0������ Q8. ������0������������������, in the direction of the normal to the sheet 2������������2 2������ II. ������0������0 towards it. 2������������ Q9. 14 ������, in opposite directions III. ������0������0 (������������22−−������������22) 2������������ Q10. 6 ������������ from ������ IV. 0 Q5. 2.775������ × 10−6 ������, 2������ × 10−6������ Q11. I. 2√3 × 10−5 ������ along negative ������ −axis II. 2 × 10−5 ������ along positive ������ −axis LEVEL 3 Q12. Proof Q6. I. 3 ������ LEVEL 3 II. 13 × 10−7������ III. 2.88 × 10−6 ������/������ Q13. ������√4������������������0������������ Q7. 2 ������0������������������(���̂���) 3 4������0������������2 Q8. I. 40 ������������ Q14. II. 9.6 ������������ ������(4������2+������2)√4������2+2������2 P18.3 MAGNETIC FIELD AND CURRENT CARRYING LOOP PRE-READING EXERCISE Q7. True Q1. True IN CLASS EXERCISE Q2. False LEVEL 1 Q3. False Q1. 0.6 ������������ Q4. False Q2. 4 Q5. True Q6. False ������ Ans.
P18 – Magnetic Effects of Electric Current 23 LEVEL 2 Q2. ������������������2 Q3. I. 0⃗⃗ 4������ II. 0⃗⃗ III. 1.5 × 10−24 ������ Q3. 4.0 × 10−5������������ ������������������−1 Q4. (I)>(IV)>(II)>(III) Q4. 54 × 10−7 ������������/������������������ LEVEL 2 ������ Q5. ������������������2 (1 − 4) Q5. |������|������������2 ������ 2 Q6. 3������1 − 2������2 Q7. 40 ������ ������̂ ������������������/������2 Q6. ������������ℎ Q8. 2 ������ Q9. 8000 Ω 4������������������ Q10. ������������ Q7. 24 Ω ������������������������ Q8. I. 7.03 × 10−4������ LEVEL 3 II. 1.56 × 10−6������ Q11. ������������ LEVEL 3 ������������ ������ Q9. 1 ������Ω, in parallel connection HOMEWORK LEVEL 1 Q1. 3 2������(������)2 ������0√������ TEST PRACTICE PROBLEMS Q. No. Ans. Level Mark (C) / (I) / (U) Q. No. Ans. Level Mark (C) / (I) / (U) as appropriate as appropriate Hard Easy Q1. D Easy Q16. A Medium Q2. C Easy Q17. B Medium Q3. C Medium Q18. D Easy Q4. B Medium Q19. A Easy Q5. C Easy Q20. D Hard Q6. C Medium Q21. C Hard Q7. A Easy Q22. A Medium Q8. C Medium Q23. C Easy Q9. C Medium Q24. D Medium Q10. C Medium Q25. D Easy Q11. C Easy Q26. B Easy Q12. D Medium Q27. C Medium Q13. A Easy Q28. D Easy Q14. D Easy Q29. B Easy Q15. C Q30. B
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