Laws and Continuity Conditions with Magnetization

9.2.1

Return to Prob. 6.1.1 and replace P M. Find _m and _sm.

9.2.2^*

A circular cylindrical rod of material is uniformly magnetized in the y^' direction transverse to its axis, as shown in Fig. P9.2.2. Thus, for r < R, M = M_o [i_x sin + i_y cos ]. In the surrounding region, the material forces H to be zero. (In Sec. 9.6, it will be seen that such a material is one of infinite permeability.)

Figure P9.2.2

(a)

Show that if H = 0 everywhere, both Ampère's law and (9.2.2) are satisfied.

(b)

Suppose that the cylinder rotates with the angular velocity so that = t. Then, B is time varying even though there is no H. A one-turn rectangular coil having depth d in the z direction has legs running parallel to the z axis in the +z direction at x = -R, y = 0 and in the -z direction at x = R, y = 0. The other legs of the coil are perpendicular to the z axis. Show that the voltage induced at the terminals of this coil by the time-varying magnetization density is v = -_o 2 Rd M_o sin t.

Figure P9.2.3

9.2.3

In a region between the planes y = a and y = 0, a material that moves in the x direction with velocity U has the magnetization density M = M_o i_y cos (x - Ut), as shown in Fig. P9.2.2. The regions above and below are constrained so that H = 0 there and so that the integral of H ds between y = 0 and y = a is zero. (In Sec. 9.7, it will be clear that these materials could be the pole faces of a highly permeable magnetic circuit.)

(a)	Show that Ampère's law and (9.2.2) are satisfied if H = 0 throughout the magnetizable layer of material.
(b)	A one-turn rectangular coil is located in the y = 0 plane, one leg running in the +z direction at x = -d (from z = 0 to z = l) and another running in the -z direction at x = d (from z = l to z = 0). What is the voltage induced at the terminals of this coil by the motion of the layer?

Permanent Magnetization

9.3.1^*

The magnet shown in Fig. P9.3.1 is much longer in the z directions than either of its cross-sectional dimensions 2a and 2b. Show that the scalar magnetic potential is

(Note Example 4.5.3.) floating figure GIF #32

Figure P9.3.1

9.3.2^*

In the half-space y > 0, M = M_o cos ( x) exp (- y)i_y, where and are given positive constants. The half-space y < 0 is free space. Show that equation GIF #9.111

9.3.3

In the half-space y < 0, M = M_o sin ( x) exp ( y)i_x, where and are positive constants. The half-space y > 0 is free space. Find the scalar magnetic potential. figure GIF #3

Figure P9.3.4

9.3.4

For storage of information, the cylinder shown in Fig. P9.3.4 has the magnetization density equation GIF #9.112

where p is a given integer. The surrounding region is free space.

(a)	Determine the magnetic potential .
(b)	A magnetic pickup is comprised of an N-turn coil located at = /2. This coil has a dimension a in the direction that is small compared to the periodicity length 2 R/p in that direction. Every turn is essentially at the radius d + R. Determine the output voltage v_out when the cylinder rotates, = t.
(c)	Show that if the density of information on the cylinder is to be high (p is to be high), then the spacing between the coil and the cylinder, d, must be small.

Magnetization Constitutive Laws

9.4.1^*

The toroidal core of Example 9.4.1 and Demonstration 9.4.1 is filled by a material having the single-valued magnetization characteristic M = M_o tanh ( H), where M and H are collinear.

(a)	Show that the B-H characteristic is of the type illustrated in Fig. 9.4.4.
(b)	Show that if i = i_o cos t, the output voltage is
(c)	Show that the characteristic is essentially linear, provided that N₁ i_o/2 R 1.

9.4.2

The toroidal core of Demonstration 9.4.1 is driven by a sinusoidal current i(t) and responds with the hysteresis characteristic of Fig. 9.4.6. Make qualitative sketches of the time dependence of

(a)	B(t)
(b)	the output voltage v(t).

Fields in the Presence of magnetically Linear Insultating Materials

9.5.1^*

A perfectly conducting sheet is bent into a shape to make a one-turn inductor, as shown in Fig. P9.5.1. The width w is much larger than the dimensions in the x - y plane. The region inside the inductor is filled with two linearly magnetizable materials having permeabilities _a and _b, respectively. The cross-section of the system in any x - y plane is the same. The cross-sectional areas of the magnetizable materials are A_a and A_b, respectively. Given that the current i(t) is uniformly distributed over the width w of the inductor, show that H = (i/w)i_z in both of the magnetizable materials. Show that the inductance L = (_a A_a + _b A_b)/w. floating figure GIF #33

Figure P9.5.1

9.5.2

Perfectly conducting coaxial cylinders, shorted at one end, form the one-turn inductor shown in Fig. P9.5.2. The total current i flowing on the surface at r = b of the inner cylinder is returned through the short and the outer conductor at r = a. The annulus is filled by materials of uniform permeability with an interface at r = R, as shown. floating figure GIF #34

Figure P9.5.2

(a)	Determine H in the annulus. (A simple solution can be shown to satisfy all the laws and continuity conditions.)
(b)	Find the inductance.

9.5.3^*

The piece-wise uniform material in the one-turn inductor of Fig. P9.5.1 is replaced by a smoothly inhomogeneous material having the permeability = - _m x/l, where _m is a given constant. Show that the inductance is L = d_m l/2w.

9.5.4

The piece-wise uniform material in the one-turn inductor of Fig. P9.5.2 is replaced by one having the permeability = _m (r/b), where _m is a given constant. Determine the inductance.

9.5.5^*

Perfectly conducting coaxial cylinders, shorted at one end, form a one-turn inductor as shown in Fig. P9.5.5. Current flowing on the surface at r = b of the inner cylinder is returned on the inner surface of the outer cylinder at r = a. The annulus is filled by sectors of linearly magnetizable material, as shown.

(a)	Assume that in the regions (a) and (b), respectively, H = i A/r and H = i C/r, and show that with A and C functions of time, these fields satisfy Ampère's law and the flux continuity law in the respective regions.
(b)	Use the flux continuity condition at the interfaces between regions to show that C = (_a /_b)A.
(c)	Use Ampère's integral law to relate C and A to the total current i in the inner conductor.
(d)	Show that the inductance is L = l _a ln(a/b)/[ + (2 - ) _a /_b ].
(e)	Show that the surface current densities at r = b adjacent to regions (a) and (b), respectively, are K_z = A/b and K_z = C/b.

Figure P9.5.5

9.5.6

In the one-turn inductor of Fig. P9.5.1, the material of piece-wise uniform permeability is replaced by another such material. Now the region between the plates in the range 0 < z < a is filled by material having uniform permeability _a, while = _b in the range a < z < w. Determine the inductance.

Fields in Piece-Wise Uniform Magnetically Linear Materials

9.6.1^*

A winding in the y = 0 plane is used to produce the surface current density K = K_o cos z i_x. Region (a), where y > 0, is free space, while region (b), where y < 0, has permeability .

(a)	Show that
(b)	Now consider the same problem, but assume at the outset that the material in region (b) has infinite permeability. Show that it agrees with the limit of the first expression of part (a).
(c)	In turn, use the result of part (b) as a starting point in finding an approximation to in the highly permeable material. Show that this result agrees with the limit of the second result of part (a) where _o.

9.6.2

The planar region -d < y < d is bounded from above and below by infinitely permeable materials, as shown in Fig. P9.6.2. Region (a) to the right and region (b) to the left are separated by a current sheet in the plane x = 0 with the distribution K = i_z K_o sin ( y/2d). The system extends to infinity in the x directions and is two dimensional.

Figure P9.6.2

(a)	In terms of , what are the boundary conditions at y = d.
(b)	What continuity conditions relate in regions (a) and (b) where they meet at x = 0?
(c)	Determine .

9.6.3^*

The cross-section of a two-dimensional cylindrical system is shown in Fig. P9.6.3. A region of free space having radius R is surrounded by material having permeability which can be considered as extending to infinity. A winding at r = R is driven by the current i and has turns density (N/2R) sin (turns per unit length in the direction). Thus, at r = R, there is a current density K = (N/2R) i sin i_z.

(a)	Show that
(b)	An n-turn coil having a spacing between conductors of 2a is now placed at the center. The magnetic axis of this coil is inclined at the angle relative to the x axis. This coil has length l in the z direction. Show that the mutual inductance between this coil and the one at r = R is L_m = _o a ln N cos /R[1 + (_o / )].

Figure P9.6.3

9.6.4

The cross-section of a motor or generator is shown in Fig. 11.7.7. The two coils comprising the stator and rotor windings and giving rise to the surface current densities of (11.7.24) and (11.7.25) have flux linkages having the forms given by (11.7.26).

infinite, and determine the vector potential in the air gap.

(a) Assume that the permeabilities of the rotor and stator are

(b) Determine the self-inductances L_s and L_r and magnitude of the peak mutual inductance, M, in (11.7.26). Assume that the current in the +z direction at is returned at + .

9.6.5

A wire carrying a current i in the z direction is suspended a height h above the surface of a magnetizable material, as shown in Fig. P9.6.5. The wire extends to "infinity" in the z directions. Region (a), where y > 0, is free space. In region (b), where y < 0, the material has uniform permeability .

images to determine the fields in the two regions.

(a) Use the method of

(b) Now assume that _o and find H in the upper region, assuming at the outset that .

(c) In turn, use this approximate result to find the field in the permeable material.

(d) Show that the results of (b) and (c) are consistent with those from the exact analysis in the limit where _o.

Figure P9.6.5

9.6.6^*

A conductor carries the current i(t) at a height h above the upper surface of a material, as shown in Fig. P9.6.5. The force per unit length on the conductor is f = i x _o H, where i is a vector having the direction and magnitude of the current i(t), and H does not include the self-field of the line current.

f = _o i_y i²/4 h.

(a) Show that if the material is a perfect conductor,

(b) Show that if the material is infinitely permeable, f = - _o i_y i²/4 h.

9.6.7^*

Material having uniform permeability is bounded from above and below by regions of infinite permeability, as shown in Fig. P9.6.7. With its center at the origin and on the surface of the lower infinitely permeable material is a hemispherical cavity of free space having radius a that is much less than d. A field that has the uniform intensity H_o far from the hemispherical surface is imposed in the z direction.

approximate magnetic potential in the magnetizable material is = - H_o a[(r/a) + (a/r)²/2] cos .

(a) Assume _o and show that the

(b) In turn, show that the approximate magnetic potential inside the hemisphere is = - 3H_o z/2.

Figure P9.6.7

9.6.8

In the magnetic tape configuration of Example 9.3.2, the system is as shown in Fig. 9.3.2 except that just below the tape, in the plane y = -d/2, there is an infinitely permeable material, and in the plane y = a > d/2 above the tape, there is a second infinitely permeable material. Find the voltage v_o.

9.6.9^*

A cylindrical region of free space of rectangular cross-section is surrounded by infinitely permeable material, as shown in Fig. P9.6.9. Surface currents are imposed by means of windings in the planes x = 0 and x = b. Show that

Figure P9.6.9

9.6.10^*

A circular cylindrical hole having radius R is cut through a material having permeability _a. A conductor passing through this hole has permeability _b and carries the uniform current density J = J_o i_z, as shown in Fig. P9.6.10. A field that is uniform far from the hole, where it is given by H = H_o i_x, is applied by external means. Show that for r < R, and R < r, respectively, equation GIF #9.117

Figure P9.6.10

9.6.11^*

Although the introduction of a magnetizable sphere into a uniform magnetic field results in a distortion of that field, nevertheless, the field within the sphere is uniform. This fact makes it possible to determine the field distribution in and around a spherical particle even when its magnetization characteristic is nonlinear. For example, consider the fields in and around the sphere of material shown together with its B-H curve in Fig. P9.6.11. where M is a constant to be determined, and show that the magnetic field intensity inside the sphere is uniform, z directed, and of magnitude H = H_o - M/3, and hence that the magnetic flux density, B, in the sphere is related to the magnitude of the magnetic field intensity H by equation GIF #9.118

(a)	Assume that the magnetization density is M = M i_z,
(b)	Draw this load line in the B-H plane, showing that it is a straight line with intercepts 3H_o/2 and 3_o H_o with the H and B axes, respectively.
(c)	Show how (B, H) in the sphere are determined, given the applied field intensity H_o, by graphically finding the point of intersection between the B - H curve of Fig. P9.6.11 and (a).
(d)	Show that if H_o = 4 x 10⁵ A/m, B = 0.75 tesla and H = 3.1 x 10⁵ A/m.

Figure P9.6.11 floating figure GIF #43

Figure P9.6.12

9.6.12

The circular cylinder of magnetizable material shown in Fig. P9.6.12 has the B - H curve shown in Fig. P9.6.11. Determine B and H inside the cylinder resulting from the application of a field intensity H = H_o i_x where H_o = 4 x 10⁵ A/m.

9.6.13

The spherical coil of Example 9.6.1 is wound around a sphere of material having the B - H curve shown in Fig. P9.6.11. Assume that i = 800 A, N = 100 turns, and R = 10 cm, and determine B and H in the material.

Magnetic Circuits

9.7.1^*

The magnetizable core shown in Fig. P9.7.1 extends a distance d into the paper that is large compared to the radius a. The driving coil, having N turns, has an extent in the direction that is small compared to dimensions of interest. Assume that the core has a permeability that is very large compared to _o.

(with defined to be zero at = ) are

(a)	Show that the approximate H and inside the core
(b)	Show that the approximate magnetic potential in the central region is

Figure P9.7.1

9.7.2

For the configuration of Prob. 9.7.1, determine in the region outside the core, r > a.

9.7.3^*

In the magnetic circuit shown in Fig. P9.7.3, an N-turn coil is wrapped around the center leg of an infinitely permeable core. The sections to right and left have uniform permeabilities _a and _b, respectively, and the gap lengths a and b are small compared to the other dimensions of these sections. Show that the inductance L = N² w[(_b d/b) + (_a c/a)].

Figure P9.7.3 floating figure GIF #46

Figure P9.7.4

9.7.4

The magnetic circuit shown in Fig. P9.7.4 is constructed from infinitely permeable material, as is the hemispherical bump of radius R located on the surface of the lower pole face. A coil, having N turns, is wound around the left leg of the magnetic circuit. A second coil is wound around the hemisphere in a distributed fashion. The turns per unit length, measured along the periphery of the hemisphere, is (n/R) sin , where n is the total number of turns. Given that R h w, find the mutual inductance of the two coils.

9.7.5^*

The materials comprising the magnetic circuit of Fig. P9.7.5 can be regarded as having infinite permeability. The air gaps have a length x that is much less than a or b, and these dimensions, in turn, are much less than w. The coils to left and right, respectively, have total turns N₁ and N₂. Show that the self- and mutual inductances of the coils are equation GIF #9.121

Figure P9.7.5 floating figure GIF #48

Figure P9.7.6

9.7.6

The magnetic circuit shown in Fig. P9.7.6 has rotational symmetry about the z axis. Both the circular cylindrical plunger and the remainder of the magnetic circuit can be regarded as infinitely permeable. The air gaps have widths x and g that are small compared to a and d. Determine the inductance of the coil.

9.7.7

Two cross-sectional views of an axisymmetric magnetic circuit that could be used as an electromechanical transducer are shown in Fig. P9.7.7. Surrounding an infinitely permeable circular cylindrical rod having a radius slightly less than a is an infinitely permeable stator having a hole down its center with a radius slightly greater than a. A pair of coils, having turns N₁ and N₂ and driven by currents i₁ and i₂, respectively are wound around the center rod and positioned in slots in the surrounding stator. The longitudinal position of the rod, denoted by , is limited in range so that the ends of the rod are always well inside the ends of the stator. Thus, H in each of the air gaps is essentially uniform. Determine the inductance matrix, (9.7.12). floating figure GIF #49

Figure P9.7.7

9.7.8

Fields in and around the magnetic circuit shown in Fig. P9.7.8 are to be considered as independent of z. The outside walls are infinitely permeable, while the horizontal central leg has uniform permeability that is much less than that of the sides but nevertheless much greater than _o. Coils having total turns N₁ and N₂, respectively, are wound around the center leg. These have evenly distributed turns in the planes x = l/2 and x = -l/2, respectively. The regions above and below the center leg are free space. coordinates. As far as is concerned inside the center leg, what boundary conditions must satisfy if the central leg is treated as the "inside" of an "inside-outside" problem?

(a)	Define = 0 at the origin of the given
(b)	What is in the center leg?
(c)	What boundary conditions must satisfy in region (a)?
(d)	What is , and hence H, in region (a)? (A simple exact solution is suggested by Prob. 7.5.3.) For the case where N₁ i₁ = N₂ i₂, sketch and H in regions (a) and (b).

Figure P9.7.8

9.7.9

The magnetic circuit shown in Fig. P9.7.9 is excited by an N-turn coil and consists of infinitely permeable legs in series with ones of permeability , one to the right of length l₂ and the other to the left of length l₁. This second leg has wrapped on its periphery a metal strap having thickness w, conductivity , and height l₁. With a terminal current i = i_o cos t, determine H within the left leg.

Figure P9.7.9

9.7.10^*

The graphical approach to determining fields in magnetic circuits to be used in this and the next example is similar to that illustrated by Probs. 9.6.11-9.6.13. The magnetic circuit of a high-field magnet is shown in Fig. P9.7.10. The two coils each have N turns and carry a current i. equation GIF #9.123

(a)	Show that the load line for the circuit is
(b)	For N = 500, d = 1 cm, l₁ = 0.8 m, l₂ = 0.2 m, and i = 10 amps, find the flux density B in its air gap.

Figure P9.7.10

9.7.11

In the magnetic circuit of Fig. P9.7.11, the infinitely permeable core has a gap with cross-sectional area A and height a + b, where the latter is much less than the dimensions of the former. In this gap is a material having height b and the M - H relation also shown in the figure. Within the material and in the air gap, H is approximated as being uniform. the field intensity in the material, M, and the driving current i.

(a) Determine the load line relation between H_b,

(b) If Ni/a = 0.5 x 10⁶ amps/m and b/a = 1, what is M, and hence B?

Figure P9.7.11