JEE ADVANCED LEVEL ‘SECTION A: MAGNETIC FIELD DUE TOA ‘STRAIGHT WIRE 1. Figure shows a straight wire of length /carrying @ current /, Find the magnitude of magnetic fled produced by the current at point P. 2. Six wires of current I, = 1A, 1, = 2A,1,= 34,1, = IA, 1, = 5A and I, = 44 cut the page perpendicularly at the points 1,2,3,4,5 and 6 respectively as shown in the figure. Find the value. ofthe integral Eid around the closed path '3. A system of long four parallel conductors whose ‘sections with the plane of the drawing lie at the vertices of a square there flow four equal currents. hob ‘The directions of these currents are as follows : those marked & point away from the reader, while those marked with a dot point towards the reader. How ts the vector of magnetic induction directed at the centre of the square? ‘SECTION B : MAGNETIC FIELD DUE TO A CIRCULAR LOOP: 5 4. Two circular coils A.and 8 of radius\7> cm and 5 em respectively carry current|5 Amp and 5 B Amp respectively. The’ plane of B is perpendicular to plane of A and their centres ‘coincide, Find the magnetic field at the centre, ‘Sol, '5. Two circular coils of wire each having a radius of 4 cm and 10 tums have a common axis and ‘are 6 cm apart. Ifa current of 1 A passes through each coil in the opposite direction find the ‘magnetic Induction. (1) At the centre of either coil ; (i) At a point on the axis, midway between them.6. Find the ratio of magnetic field magnitudes at a distance 10 m along the axis and at 60° from the axis, from the centre of a coil of radius 1 cm, carrying a current 1 amp. Sol. SECTION C: MAGNETIC FIELD DUETO A ‘STRAIGHT WIRE AND CIRCULAR ARC 7, Find the magnetic induction at the origin in the figure shown, 1B. Find the magnetic induction at point O, if the ‘current carrying wire is in the shape shown in the figure. 19. Find the magnit ide of the miagnetic induction B of a magnetic fleld generated by a system of thin conductors along which a current /is lowing atea point A(O, R, 0), that is the centre of a circular conductor of radius R. The ring is in yz plane.‘SECTION D : MAGNETIC FIELD DUETO A CYLINDER, LARGE SHEET, SOLENOID, ‘TOROID AND AMPERE’S LAW 10. A cylindrical conductor of radius R carries @ ‘current its length. The current density 3, however, It Is not uniform over the cross section of the conductor but is a function of the radius according to = br, where bis a constant. Find an expression for the magnetic field B. 4 Sets . measured from the axis Sal, SECTION E : MAGNETIC FORCE ON A CHARGE 11. Electric charge ais uniformly distributed over a rod of length J. The rod is placed parallel to a long wire carrying a current /. The separation between the rod and the wire is a. Find the force needed to move the rod along Its length with a uniform velocity v. 12. A charged particle (charge q, mass m) has velocity v, at origin in +x direction, In space there sa uniform magnetic field B in -z direction. Find the y coordinate of particle’ When is crosses ¥ axis. 13. A charged particle having mass m and charge q is accelerated by a potential difference ¥, it flies through a uniform transverse magnetic field B. The field occupies a region of space d. Find the time interval for which it remains inside the magnetic field. Sol.‘SECTION F : ELECTRIC AND MAGNETIC FORCE ON A CHARGE 14, An electron moving with a velocity 5 x 10* ms-'i., In the uniform electric field of 5 x 10” Vv" j. Find the magnitude and direction of a minimum uniform magnetic field in tesla that will cause the electron to move undeviated along Its original path. Sol. 15. A particle of charge +a and mass m moving Under the influence of a uniform electric field E i and a magnetic feld 8 & enters in quadrant of a coordinate system at a point (0, a) with initial velocity v j and leaves the quadrant at 2 point (22, 0) with velocity - 2v j. Find ) Magnitude of electric field Sol. (b) Rate of work done by the electric field at point (0, a) Sol. (c) Rate of work done by both the fields at (2a, 0) Sol. 16. A proton beam passes without deviation through a region of space where there are uniform transverse mutually perpendicular electric and ‘magnetic field with E and 8. Then the beam strikes 3 grounded target. Find the force imparted by the beam on the target if the beam current is equal tol. Sol. SECTION G: MAGNETIC FORCE ON A CURRENT (CARRYING WIRE 17. An infinitely long straight wire carries a ‘conventional current I as shown in the figure. Lc ‘The rectangular loop carries a conventional current T’in the clockwise direction. Find the net force ‘on the rectangular loop.18. An arc of a circular loop of radius R is kept in the horizontal plane and a constant magnetic field Bis applied in the vertical direction as shown int the figure. If the arc carries current:I then find the force on the arc. 19. Two long straight parallel conductors are ‘separated by a distance of r, = 5 cm and carry currents, = 10.81, = 20 A. What work per unit length of a conductor must be done to increase the separation between the conductors to r, = 10 cm if, currents low in the same direction 3 Soh MAGNETIC DIPOLE MOMENT 20. A conducting circular loop of radius r carries @ constant current I. It is placed in a uniform ‘magnetic field 6, such that 6, Is perpendicular to the plane of the loop. Find the magnetic force acting on the loop is ‘Sol.21. A rectangular loop of wire is oriented with the left corner at the origin, one edge along X- axis and the other edge along Y-axis as shown in the figure. A magnetic field is into the the page and has a magnitude that is given by fi = ay va ok where a Is constant. Find the total magnetic force on the loop ifit carries carrent i. Sol. 22. A square current carrying loop made of thin wire and having a mass m = 10g can rotate without friction with respect to the vertical axis 00,, passing through the centre of the loop at right angles to two opposite sides of the loop. The Joop Is placed In a homogeneous magnetic field with an induction B = 10-'T directed at right angles to the plane of the drawing, A current I = 2s flowing in the loop. Find the period of small oscillations that the loop performs about its position of stable equilibrium. 8LeveL, 1. Three infinitely long conductors R, S and T are lying in a horizontal plane as shown in the figure. ‘The currents in the respective conductors are rat L atsin(at + 2) 1, = 1, sin (ot) T= Lsin (ot - 22) Find the amplitude of the vertical component of the magnetic field at a point P, distance ‘a’ away from the central conductor S. 2. Four long wires each carrying current 1 as shown n the gure are placed atthe points A,B, C and D. Find the mageitude and direction of ¥ (i) magnetic field at the centre of the square. (ii) force per metre acting on wire at point D. 3. An infinite wie, placed along z-axis, has current 1, in positive z-direction. A conducting rod placed in xy plane parallel to y-axis has current I, in positive y-direction. The ends of the rod subtend +30° and -60° at the origin with positives x- direction. The rod is at a distance a from the origin. Find net force on the rod. A square cardboard of side / and mass m is suspended from a horizontal axis XY as shown in figure. A single wire is wound along the penphery ‘of board and carrying a clockwise current I. At t = 0, a vertical downward magnetic field of induction Bis switched on. Find the minimum value of B sa that the board will be able to rotate up to horizontal level, ht segment OC (of length L meter) of a circuit carrying a current I amp is placed along the x-axis. Two infinitely line straight wires A and B, each extending form z= ~ xto + x, are fixed at y = ~ ametre and y = +a metre respectively, 2s shown in the figure. If the wires A and Beach carry a current I amp into plane of the paper. Obtain the expression for the force acting on the segment OC. What will be the force OC if current in the wire Bis reversed? 6. A very long straight conductor has a circular cross-section of radius R and carries’s\current density J. Inside the\conductor there is a cylindrical hole of radius a whose axis is parallel to the axis of the\conductor and a distance b fromit. Let the 2-axis be the axis of the conductor, andilet the axis of the hole be at x = b. Find the ‘magnetic field (2) on the x=axis at x = 2R. (b) on the y = axis at y = 2R. 7..Q charge is uniformly distributed over the same surface of a right circular cone of semi-vertical ‘angle @ and height h. The cone is uniformly rotated about its axis at angular velocity m. Calculated ‘associated magnetic dipole moment.8 Awire loop carrying current Is placed in the XY plane as shown in the figure. LL, (2) If a particle with charge +Q and mass m is placed at the centre P and given a velocity along 'NP (fig). Find its instantaneous acceleration (b) If an external uniform magnetic induction field 8 =Bi is applied, find the torque acting on the oop due to the field, 9. Along straight wire carries a current of 10 A directed along the negative y-axis as shown in figure. Auniform magnetic field B, of magnitude 10* T is directed parallel to the x-axis. What is the resultant magnetic field at the following points? (2)x=0, z= 2m; (b) x= 2m, 2=0; ()x=0, 2=-0.5m 10. A stationary, circular Wall clock has 2 foce with 2 radius of 15cm. Six tus of wire are wound around its perimeter, the wire carries a current 2.0 A in the dockwise direction. The dock is located, where there is a constant, uniform ‘external magnetic field of 70 mT (but the clock sull keeps perfect time) at exactly 1: 00 pm, the hhour hand of the clock points in the direction of the external magnetic field (2) After how many minutes will the minute hand point inthe direction ofthe torque on the winding due to the magnetic field? (b) What is the magnitude of this torque. 11, A U-shaped wire of mass m and length 1 is immersed with its two ends in mercury (see figure). The wire Is in a homogeneous field of magnetic induction 8. If a charge, that is, 8 current pulse q = fist, is sent through the wire, the wire will jump up. Calculate, from the height h that the wire reaches, the size of the charge or current pulse, assuming that the time of the current pulse Is vey small in Comparision with the time of flight. Make use of the fact that impulse of force equals [F dt, which equals my. Evaluate q for B = 0.1 Wb/mm, m = 10 gm + = 20cm &h = 3meters [9 = 10 m/s!) ee * xm 7 Hg 12. A current j, indicated by the crosses infig. is established In a strip of copper of height h and width w. A uniform field of ‘magnetic induction Bis applied at right angles to the strip. swe (a) Calculate the drift speed vjfor the electrons. (b) What are the magnitude and direction of the magnetic force'F acting of the electrons? (©) What would’the magnitude & direction of homogeneous electric field E have to be in order to:counter balance the effect of the magnetic field? () What is the voltage V necessary between two sides of the conductor in order to create this eld E? (e) If no electric field is applied form the outside the electrons will be pushed somewhat to one side & thereforce will give rise to a uniform electric field E, across the conductor untill the force of this electrostatic field E, balance the magnetic forces encountered in part (b). What will be the ‘magnitude and direction of the field B,? Assume that n, the number of conduction electrons per unit volume, is 1.1 x 10%/m? & that h = 0,02 meter, w = 0.1cm, i = 50 amp, & B = 2 webers/ meter. 13. (a) A rigid circular loop of radius r & mass m lies in the xy plane on a flat table and has 2 current I flowing in it. At this particular place, the earth’s magnetic fleld is 6 -6,i+6,j. How large must I be before one edge of the loop will lit from table ? (b) Repeat lf, 6-8.) 8,%14, Zeeman effect. In Bohr's theory of the hydrogen atom the electron can be thought of 8 moving ina circular orbit of radius r about the proton, Suppose that such an atom is placed in a magnetic field, with the plane of the orbit at right angle to B. (2) If the electron Is circulating clockwise, as viewed by an observer sighting along 8, will the angular freugency increase or decrease ? (b) What if the electron is circulating counterclockwise? Assume that the orbit radius. does not change. 15. In above problem show that the change in frqvency of rotation caused by the magnete field is gven approximately by av = 2. such frequency shifts were actually observed by Sooner i 16. A square loop of wire of edge a carries @ se (2) Show that B for 2 point on the axis of the loop and a distance x from its centre is given by, s- Augie? © fax? + a? Mai? + 2a?) * () can tne reut te above oxglem be feruced togive hea aoe 07 (c) Does the square loop behave like a dipole for. somes suartnty 50 7g tie abel roree? 17. A conductor carrying a current | is placed pore toate’ Unt sith ane thos 2 shown te gure. Pathe cep ok Teo an the ences, 1 ; 18. tte nh ad power reeestomene thon rena str neon tar tine ase direction at 2 rotation! frequency are revauor pe set iste 8 veer purely eve ron Sine tare howe ftv ined bye we A, 19. The figure shows a conductor of weight 1.0 N and length L = 0.5 m placed on a rough indined plane making an angle 30° with the horizontal so that conductor is perpendicular to @ uniform horizontal magnetic field of induction B = 0.10. The coefficient of static friciton between the ‘conductor and the plane is 0.1. A current of I = 110A flows through the conductor inside the plane of this paper as shown. What is the force needed tobe the appited parallel to the inclined plane to sustaining the conductor at rest? 8 fa = 30° 20. An electron gun G emits electron of energy 2kev traveling In the (+) ve x-direction. The electron are required ta hit the'spot S where GS = 0.1 m8 the line GS makes anangle of 60° with the x-axis, as shown in thefig: A uniform magnetic field & pardilel_.to GS exists in the region out ‘sides to electron gun. Find the minimum value of B needed tomake the electron hit S. Bs oor, lan x 21. The magnetic fleld due to a current carrying square loop of side a at a point located ‘symmetrically at 8 distance of 2/2 from its centre (as shown is) 22. A charged particle of specific charge a is released from origin at time t = 0 with velocity V-Wpis Vg in magnetic field 6,1. The coordinates of the particle at time t = (specific charge a = a/m) Boa Fe