+.x2yaz V/m and B = y 2 ax + z2a 1. + x2az Wb/m 2. Find the force on the charge at P.

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1 396 CHAPTER 8 MAGNETC FORCES, MATERALS, AND DEVCES Section 8.2-Forces Due to Magnetic Fields 8.1 A 4 mc charge has velocity u = l.4ax - 3.2ay - az mis at point P(2, 5, - 3) in the presence of E = 2xyzax + x2za 1 +.x2yaz V/m and B = y 2 ax + z2a 1 + x2az Wb/m 2. Find the force on the charge at P. 8.2 An electron (m = 9.11 X kg) moves in a circular orbit of radius 0.4 X m with an angular velocity of 2 X rad/ s. Find the centripetal force required to hold the electron. 8.3 A 1 mc charge with velocity loax - 2a 1 + 6az mis enters a region where the magnetic flux density is 25az Wb/m 2 (a) Calculate the force on the charge. (b) Determine the electric field intensity necessary to make the velocity of the charge constant. 8.4 A particle of mass 2 kg and charge 10 mc starts at the origin with velocity u = (2ax + Say) X 10 3 m/s. f the electric field in the region is 30 ax kv m, calculate its velocity and position at t = 4 s. 8.5 A - 2 m C charge starts at point ( 0, 1, 2) with a velocity of Sax ml s in a magnetic field B = 63.y Wb/m 2 Determine the position and velocity of the particle after 10 s, assuming that the mass of the charge is 1 gram. Describe the motion of the charge. *8.6 By injecting an electron beam normally to the plane edge of a uniform field B 0 az, electrons can be dispersed according to their velocity as in Figure (a) Show that the electrons would be ejected out of the field in paths parallel to the input beam as shown. (b) Derive an expression for the exit distance d above entry point. 8.7 Two large conducting plates are 8 cm apart and have a potential difference 12 kv. A drop of oil with mass 0.4 g is suspended in space between the plates. Find the charge on the drop. 8.8 Given that B = 4ax - 8a 2 Wb/m, find the force it exerts on a 0.2 -m conductor on the y-axis with a current 2 A in the - ar direction. *8.9 Three infinite lines L 1, L 2, and L 3 defined by x = 0, y = O; x = 0, y = 4; x = 3, y = 4, respectively, carry filamentary currents A, 200 A, and 300 A along az Find the force per unit length on (a) 1 2 due to 1 1 (b) 1 1 due to 1 2 (c) 1 3 due to 1 1 (d) 1 3 due to 1 1 and1 2 State whether each force is repulsive or attractive Two infinitely long parallel wires are separated by a distance of 20 cm. f the wires carry current of 10 A in opposite directions, calculate the force on the wires A conductor 2 m long carrying a current of 3 A is placed parallel to the z-axis at distance p 0 = 10 cm as shown in Figure 8.3S. f the field in the region is cos(</j/3) ap Wb/m 2, how much work is required to rotate the conductor one revolution about the z-axis? *8.12 A conducting triangular loop carrying a current of 2 A is located close to an infinitely

2 Problems 397 z 1; ' d3 d 2 1 di l 0 0 Electron...,..m:: x beam 0 0 B x 3A // J / / / -- / '-J -/ / FGURE 8.34 For Problem 8.6. FGURE 8.35 For Problem long, straight conductor with a current of 5 A, as shown in Figure Calculate (a) the force on side 1 of the triangular loop and (b) the total force on the loop. *8.13 A three-phase transmission line consists of three conductors that are supported at points A, B, and C to form an equilateral triangle as shown in Figure At one instant, conductors A and B both carry a current of 75 A while conductor C carries a return current of 150 A. Find the force per meter on conductor Cat that instant A current sheet with K = 1 Oax Alm lies in free space in the z = 2 m plane. A filamentary conductor on the x-axis carries a current of 2.5 A in the ax -direction. Determine the force per unit length on the conductor The magnetic field in a certain region is B = 40 ax m Wb/m 2. A conductor that is 2 m in length lies in the z-axis and carries a current of 5 A in the az-direction. Calculate the force on the conductor. z SA y 2m 2m 4m FGURE 8.36 For Problem FGURE 8.37 For Problem 8.13.

3 398 CHAPTER 8 MAGNETC FORCES, MATERALS, AND DEVCES Sections 8.3 and 8.4-Magnetic Torque, Moments, and Dipole *8.16 The rectangular loop in Figure 8.38 carries current = 3 A and is inside a magnetic field B = 0.5~ y Wb/m 2 Calculate the torque on the coil A 60-turn coil carries a current of 2 A and lies in the plane x + 2y - 5z = 12 such that the magnetic moment m of the coil is directed away from the origin. Calculate m, assum - ing that the area of the coil is 8 cm The earth has a magnetic moment of about 8 X A m 2 and its radius is 6370 km. magine that there is a loop around the equator and determine how much current in the loop would result in the same magnetic moment A triangular loop is placed in the x-z plane, as shown in Figure Assume that a de current = 2 A flows in the loop and that B = 30az m Wb/m exists in the region. Find the forces and torque on the loop A loop with 50 turns and surface area of 12 cm 2 carries a current of 4 A. f the loop rotates in a uniform magnetic field of 100 mwb/m 2, find the torque exerted on the loop High-current circuit breakers typically consist of coils that generate a magnetic field to blow out the arc formed when the contacts open. An arc 30 mm long carries a current of 520 A in a direction perpendicular to a magnetic flux density of 0.4 m Wb/m 2 Determine the magnetic force on the arc. Section 8.5-Magnetization in Materials 8.22 For a linear, isotropic, and homogeneous magnetic medium, show that M = ( Xm ) B. µ,o 1 + Xm 8.23 A block of iron (µ = 5000µ, 0 ) is placed in a uniform magnetic field with 1.5 Wb/m 2 f iron consists of 8.5 X atoms/m 3, calculate (a) the magnetization M, (b) the average magnetic moment. z 0.6 y B y 5 x 5 FGURE 8.38 For Problem FGURE 8.39 For Problem 8.19.

4 Problems n a magnetic material, with Xm = 6.5, the magnetization is M = 24y23z Alm. Find µ,r, H, and J at y = 2 cm n a ferromagnetic material(µ, = 4.5µ, 0 ), calculate (a) Xm' (b) H, (c) M, (d) Jb An electromagnet is made of a ferromagnetic material whose magnetization curve can be approximated by B(H) = B 0 Hl(H 0 + H) mwblm 2 where B 0 = 2 Wb/m 2 and H 0 Find µ,r when H = 250 Alm. = 100 Alm 8.27 An infinitely long cylindrical conductor of radius a and permeability µ, 0 µ,r is placed along the z-axis. f the conductor carries a uniformly distributed current along av find M and 1b for 0 < p < a. Section 8.7-Magnetic Boundary Conditions *8.28 (a) For the boundary between two magnetic media such as is shown in Figure 8.16, show that the boundary conditions on the magnetization vector are Mit M2t µ µ = K and -M = -M X ml X m2 X ml rn2 n X 2n (b) f the boundary is not current free, show that instead of eq. (8.49), we obtain _ta_n_0_1 = _JL1 [ 1 + _K_µ,_2_ ] tan 0 2 JL2 B 2 sin Region 1, for which µ 1 = 2.5µ, 0 is defined by z < 0, while region 2, for which µ 2 = 4µ, 0 is defined by z > 0. f B 1 = 6ax - 4.2ay + l.8az mwblm 2, find H 2 and the angle H 2 makes with the interface Region 1 is y > 0 with µ 1 = µ 0, while region 2 is y < 0 with µ 2 = 12µ 0 f B 2 = l.4ax + 0.6ay - 2a 2 Wblm 2, find H 1 and B Region 1 (x < 0) is free space, while region 2 (x > O) is a magnetic material withµ = 50µ 0 f B 1 = 40ax - 30ay + loaz mwblm 2, find H Suppose space is divided into region 1 (y < 0, µ, 1 = µ 0 µ,r 1 ) and region 2 (y < 0, µ, 2 = µ, 0 µ,r 2 ). f H 1 = aax + ~ay +8azAlm, find H f µ, 1 = 2µ, 0 for region 1 ( 0 < <b < 7T) and µ, 2 = Sµ., 0 for region 2 ( 7r < <b < 27r) and B 2 = loap + 15a<f> - media. 20a 2 m Wblm 2 Calculate (a) B 1, (b) the energy densities in the two *8.34 Region 1 is defined by x - y + 2z > 5 with µ 1 = 2µ 0, while region 2 is defined by x - y + 2z < 5 with µ 2 = 5µ 0 fh 1 = 40~ + 20~ - 30a 2 Alm, find (a) H 1 ni (b) H 2 t, (c) B 2

5 400 CHAPTER 8 MAGNETC FORCES, MATERALS, AND DEVCES 8.35 nside a right circular cylinder, µ 1 = 800 µ 0, while the exterior is free space. Given that B 1 = µ, 0 (22aP + 45a4>) Wb/m 2, determine B 2 just outside the cylinder The plane z = 0 separates air (z > 0, µ, = µ,j from iron (z < 0, µ = 200µ 0 ). Given that in air, find B in iron and the angle it makes with the interface Region 0 < z < 2 mis filled with an infinite slab of magnetic material (µ, = 2.5µ, 0 ). f the surfaces of the slab at z = 0 and z = 2, respectively, carry surface currents 30ax Alm and -40ax Alm as in Figure 8.40, calculate H and B for (a) z < 0 (b) 0 < z < 2 (c) z > Medium 1 is free space and is defined by r < a, while medium 2 is a magnetic material with permeability µ 2 and defined by r > a. The magnetic flux densities in the media are: [( l.6a 3 ) ( 0.8a 3 ) l B 1 = B r3 cos Oar r 3 sin (Ja 0 Section 8.8-lnductors and nductance *8.39 (a) f the cross section of the toroid of Figure 7.15 is a square of side a, show that the self-inductance of the toroid is L = µ,0 N 2 a [2p0 + a] ln 21T 2po - a (b) f the toroid has a circular cross section as in Figure 7.15, show that where p 0 >> a. z 11. N 2 a 2 L = _r_o 2po FGURE 8.40 For Problem z = 2-40 ax A/m z = 0 30 ax A/m

6 Problems An air-filled toroid of square cross section has inner radius 3 cm, outer radius 5 cm, and height 2 cm. How many turns are required to produce an inductance of 45 µ,h? 8.41 Determine the inductance per unit length of a cylindrical conductor of radius 3 mm. Assume µ = µ A coaxial cable has an internal inductance that is twice the external inductance. f the inner radius is 6.5 mm, calculate the outer radius A coaxial cable has inner conductor of radius a = 2.5 mm and outer conductor of radius b = 6 mm. Assuming that the space between the conductors is filled with a nonmagnetic material, calculate the inductance per unit length Show that the mutual inductance between the rectangular loop and the infinite line current of Figure 8.4 is Calculate M 12 when a = b = p 0 - µ,b [a +Pol M ln 27r = m. *8.45 Prove that the mutual inductance between the close-wound coaxial solenoids of length f 1 and f 2 ( f 1 >> f 2 ), turns N 1 and N 2, and radii r 1 and r 2 with r 1 = r 2 is Po Section 8.9-Magnetic Energy 8.46 A coaxial cable consists of an inner conductor of radius 1.2 cm and an outer conductor of radius 1.8 cm. The two conductors are separated by an insulating medium(µ, = 4µ, 0 ). f the cable is 3 m long and carries 25 ma current, calculate the energy stored in the medium n a certain region for which Xm = 19, How much energy is stored in 0 < x < 1, 0 < y < 2, -1 < z < 2? 8.48 n a certain medium with µ, = 4.5 µ, 0, H = 200ax + 500a; ma/m. Calculate the total energy stored in a 2 X 2 X 2 cm cubical region centered at the origin. Section 8.10-Magnetic Circuits 8.49 A cobalt ring (µ,r = 600) has a mean radius of 30 cm. f a coil wound on the ring carries 12 A, calculate the number of turns required to establish an average magnetic flux density of 1.5 Wb/m 2 in the ring.

7 402 CHAPTER 8 MAGNETC FORCES, MATERALS, AND DEVCES 0.2 A 500 turns j_ -----, 12cm T 0.6 cm L= 42cm ~ 2cm ~ ,.. * lg = 0.1 cm FGURE 8.41 For Problem FGURE 8.42 For Problem Refer to Figure f the current in the coil is 0.5 A, find the mmf and the magnetic field intensity in the air gap. Assume that µ, = 500µ, 0 and that all branches have the same cross-sectional area of 10 cm The magnetic circuit of Figure 8.41 has a current of 10 A in the coil of 2000 turns. Assume that all branches have the same cross section of 2 cm 2 and that the material of the core is iron with µ, = Calculate R, <JF, and 1/1' for (a) The core (b) The air gap 8.52 Consider the magnetic circuit in Figure Assuming that the core(µ = 1000µ, 0 ) has a uniform cross section of 4 cm 2, determine the flux density in the air gap A ring of silicon steel is 1.5 cm wide and has a rectangular cross section with inner radius of 5 cm and outer radius of 6 cm. f a 500-turn coil produces a flux of 12 m Wb in the ring when the coil current is 2 ma, find B and µ,. Section Force on Magnetic Materials 8.54 An electromagnetic relay is modeled as shown in Figure What force is on the armature (moving part) of the relay ifthe flux in the air gap is 2 mwb? The area of the gap is 0.3 cm 2, and its length 1.5 mm A toroid with air gap, shown in Figure 8.44, has a square cross section. A long conductor carrying current 2 is inserted in the air gap. f J 1 = 200 ma, N = 750, p 0 = 10 cm, a = 5 mm, and.ea = 1 mm, calculate (a) The force across the gap when 1 2 = 0 and the relative permeability of the toroid is 300. (b) The force on the conductor when 1 2 = 2 ma and the permeability of the toroid is infinite. Neglect fringing in the gap in both cases.

8 Problems 403 N FGURE 8.43 For Problem FGURE 8.44 For Problem N FGURE 8.45 For Problem FGURE 8.46 For Problem Nonmagnetic sleeve 8.56 A section of an electromagnet with a plate below it carrying a load is shown in Figure The electromagnet has a contact area of200 cm 2 per pole, and the middle pole has a winding of 1000 turns with = 3 A. Calculate the maximum mass that can be lifted. Assume that the reluctance of the electromagnet and the plate is negligible Figure 8.46 shows the cross section of an electromechanical system in which the plunger moves freely between two nonmagnetic sleeves. Assuming that all legs have the same cross-sectional area S, show that

ELECTRO MAGNETIC FIELDS

ELECTRO MAGNETIC FIELDS SET - 1 1. a) State and explain Gauss law in differential form and also list the limitations of Guess law. b) A square sheet defined by -2 x 2m, -2 y 2m lies in the = -2m plane. The charge density on the