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MINISTRY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF TECHNICAL AND VOCATIONAL EDUCATION Questions and Solutions For EcE-03011 ENGINEERING ELECTROMAGNETIC Semester-II B.Tech (First Year) Electronic Engineering EcE-03011 ENGINEERING ELECTROMAGNETIC Questions and Solutions 1.(a) Find H at the center of a square current loop of side L. (page 143) (b) A long straight conductor cross section with radius a has a magnetic field strength H=( I r /2 a2 ) awithin the conductor (r<a) and H=( I / 2 r) a for r > a. Find J in both regions. (page 139) 2.(a) A current filament of 5.0 A in the ay direction is parallel to the y axis at x=2 m, z= - 2 m. z Find H at the origin. y 2 r x 5.0 A (page 143) -2 (b) In cylindrical coordinates, B=(2.0 / r) a (T). Determine the magnetic flux crossing the plane surface defined by 0.5 r 2.5 m and 0 z 2.0 m. z 2.0 dS B (page 149) 0 0.5 2.5 3.(a) A thin cylindrical conductor of radius a , infinite in length, carries a current I. Find H at all points using Ampere's law. (page 144) (b) A radial field H = (2.39 x 106 / r ) cos ar A/m exists in free space. Find the magnetic flux crossing the surface defined by -/4 /4, (page 148) 0 z m. z I dS P/ P/ 4.(a) Find H on the axis of a circular current loop of radius a. Specialize the result to the center of the loop. (page 145) (b) Given A = ( y cos ax) ax + (y + ex ) az , find x A at the origin. (page 146) 5.(a) Calculate the curl of H in cartesian coordinates due to a current filament along the z- axis with current I in the az direction. filament I in free space. (page 147) (b) Investigate the vector magnetic potential for the infinite, straight current (page 142) 6.(a) Find the flux crossing the portion of the plane = /4 defined by 0.01 < r < 0.05 m and 0 < z < 2 m. A current filament of 2.50 A along the z axis is in the az direction. 2 (page 140) B dS x /4 0.05 0.01 2.50 A 0 (b) A current sheet, K=6.0 ax A/m, lies in the z=0 plane and a current filament is located at y=0, z=4 m. Determine I and its direction if H=0 at (0, 0, 1.5)m. z (0,0,4) I (0 ,0,1.5) y z K =6.0 ax (page 146) 7. The parallel magnetic circuit shown in figure is silicon steel with the same cross- sectional area throughout, S=1.30 cm2. The mean lengths are l 1= l3=25 cm, l2=5 cm. The coils have 50 turns each. Given that 1=90 Wb and 3=120 Wb, find the coil currents. Use B-H curve. (page 187) L F F L L 8.(a) Find the inductance per unit length of a coaxial conductor such as that shown in figure. I (page 170) = const a b L (b) A solenoid with N1=1000, r1=1.0 cm, and L1=50 cm is concentric within a second coil of N2=2000, r2=2.0 cm, and L2=50cm. Find the mutual inductance assuming free space conditions. 9. (page 173) The parallel magnetic circuit shown in figure is silicon steel with the same cross- sectional area throughout, S=1.30 cm2. The mean lengths are l1=l3=25 cm, l2=5 cm. The coils have 50 turns each. Given that 1=90 Wb and 3=120 Wb, draw the equivalent magnetic circuit using reluctances for three legs and calculate the flux in the core using F 1=19.3 A and F3=37.5 A. Use B-H curve. (page 187-188) L F F L L 10.(a) Find the inductance of an ideal solenoid with 300 turns, L=0.50 m, and a circular cross-section of radius 0.02 m. (page 170) (b) Use the energy integral to find the internal inductance per unit length of a cylindrical conductor of radius a. 11. (page 179) The magnetic circuit shown in figure has a C-shape cast steel part,1, and a cast iron part ,2. Find the current required in the 150-turn coil if the flux density in the cast iron is B2=0.45 T. Use B-H curve. (page 180) cast steel 2 cm NI 2 cm c a s t i r o n 2 cm 14 cm 2 cm 12 cm 1.8 cm 12.(a) The cast iron core has an inner radius of 7 cm and outer radius of 9 cm. Find the flux if the coil mmf is 500 A. Use B-H curve. (page 180) NI 2 cm (b) The cast iron magnetic core has an area Si=4 cm2 and a mean length 0.438 m. The 2 mm air gap has an apparent aea Sa=4.84 cm2. Determine the air gap flux . Use B-H curve. F =1000 A 2 mm (page 181) 13. A coaxial capacitor with inner radius 5 mm, outer radius 6 mm and length 500 mm has a dielectric for which r = 6.7 and an applied voltage 250 sin 377 t (V). Determine the displacement current iD and compare with the conduction current ic. where E =6.0 x 10-6 sin 9.0 x 109 t (V/m). U= 2.5 sin 103 t az (m/s). z U y B x 0.20 (page 197) 14.(a) Moist soil has a conductivity of 10-3 S/m and r =2.5. Find JC and JD, (page 198) (b) Find the induced voltage in the conductor where B= 0.04 ay T and (page 198) 15.(a) An area of 0.65 m2 in the z=0 plane is enclosed by a filamentary conductor. Find the induced voltage ,given that B= 0.05 COS103 t ( a a y z 2 z ) (T). (page 198) dS y x i (b) The circlar loop conductor lies in the z=0 plane ,has a radius of 0.10 m and a resistance of 5.0 . Given B= 0.2 sin 103 t az (T), determine the current. (page 199) 16.(a) In a material for which = 5.0 S/m and r =1 the electric field intensity is E = 250 sin 1010 t(V/m). Find the conduction and displacement current densities, and the frequency at which they have equal magnitudes. (page 197) (b) A conductor 1 cm in length is parallel to the z axis and rotates at a radius of 25 cm at 1200 rev/min. Find the induced voltage if the radial field is given by 17. B= 0.5 ar T. (page 200) A rectangular conducting loop with resistance R = 0.20 turns at 500 rev/min. The vertical conductor at r1=0.03 m is in a field B1=0.25 ar T, and the conductor at r2=0.05 m is in a field B2= 0.80 arT. Find the current in the loop. (page 201) R i 0.50 m 18. A conducting cylinder of radius 7 cm and height 15 cm rotates at 600 rev/min in a radial field B = 0.20 arT. Sliding contacts at the top to a negative terminal of voltmeter and bottom connect to a voltmeter positive terminal . Find the induced voltage. (page 200) 19. In region 1,B1=1.2 ax+0.8ay+0.4az (T). Find H2(i.e.,H at z=+0) and the angles between the field vectors and a tangent to the interface. z 2 r2 =5 1 r1 =3 (page 209) 20. Region 1,where r1=4 is the side of the plane y+z=1 containing the origin . In region 2, r2=6. B1 = 2.0 ax+1.0 ay (T), find B2 and H2. (page 210) z an r2 =6 2 y 1 r1 =4 0 21. A current sheet, K = 9.0 ay A/m, is located at z=0, the interface between region 1, z<0, with r1 = 4 and region 2, z>0, r2 = 3. Given that H2=14.5 ax+ 8.0 az (A/m), find H1. z 2 r2 = 3 vv (page 211) H2 x r1 = 4 1 K=9.0 ay 22. In region 1,for whichr1 = 3, is defined by x<0 and region 2, x>0, has . r2= 5. Given H1= 4.0 ax+3.0 ay- 6.0 az (A/m), show that 2 = 19.7 and H2= 7.12 23. A/m. (page 209) In region 1, defined by z<0, r1=3 and H1= 1/ ( 0.2 ax +0.5 ay+ 1.0 az) (A/m). Find H2 if it is known that 2 = 45 . . (page 210) 24.(a) A current sheet, K = 6.5 az A/m , at x=0 separates region 1,x<0, where H1= 10ay A/m and region 2, x>0. Find H2 at x=+0. (page 211) (b) Region 1, z < 0, has r1 = 1.5 while region 2, z > 0, has r2 = 5. Near (0,0,0), B1=2.40 ax+ 10.0 az(T), B2=.25.75 ax - 17.7 ay+ 10.0 az(T), If the interface carries a sheet current, what is its density at the origin? (page 211) 25. A travelling wave is described by y=10sin(z-t). Sketch the wave at t=0 and at t=t1, when it has advanced /if the velocity is 3x108 m/s and the angular frequency =106 rad/s. Repeat for =2x106 rad/s and the same t1. (page 226-227) At what freuencies may earth be considered a perfect dielectric, if = 5x10-3 S/m, r =1 and r=8? Can be assumed zero at these frequencies? (page229) 26. 27.(a) In free space, E(z,t)=103sin(t-z) ay (V/m). Obtain H(z,t). (page 227) (b) For the wave of above problem(a) determine the propagation constant , given that the frequency is f= 95.5 MHz. 28. z = 0 plane,for t= 0, /4, /2, 3/4 and . (page 227) Examine the field E (z,t) = 10 sin (t + z) ax+ 10 cos (t + z) ay in the (page 227) 29.(a) Find the skin depth at a frequency of 1.6 MHz in aluminum, where 38.2 MS/m and r =1. Also find and the wave velocity u. (page 229) (b) An H field travels in the - az direction in free space with a phaseshift constant of 30.0 rad/m and an amplitude of (1/3)(A/m). If the field has the direction - ay when t=0 and z=0, write suitable expressions foe E (page 228) and H. Determine the frequency and wavelength. and 0.25 pS/m, if the wave frequency is 1.6 MHz. 30.(a) Determine the propagation constant for a material having r =1, r = 8, (page 228) (b) Determine the conversion factor between the neper and the decibel. (page 228-229) **************** H (A/m) B.H curves, H <400 A/m H (A/m) B.H curves, H >400 A/m