2.9 DEMAGNETIZING FACTORS

Transcription

1 52 EXPERIMENTAL METHODS Fig Magnetic shielding. 2.9 DEMAGNETIZING FACTORS Returning to the bar magnets of Fig. 2.24, we might ascribe the nonuniformity of the induction inside the magnet to the fact that lines of B leak out of the sides. If we taper the magnet toward each end to make up for this leakage, the induction can be made uniform throughout. It may be shown, although not easily, that the correct taper to achieve this result is that of an ellipsoid (Fig. 2.26). If an unmagnetized ellipsoid is placed in a uniform magnetic field, it becomes magnetized uniformly throughout; the uniformity of M and B are due to the uniformity of H d throughout the volume. This uniformity can be achieved only in an ellipsoid. (These statements require qualification for ferro- and Fig The H field of an ellipsoidal magnet in zero applied field.

2 2.9 DEMAGNETIZING FACTORS 53 ferrimagnetic materials, because they are made up of domains, or small regions magnetized to saturation in different directions. Even an ellipsoidal specimen of such a material cannot be uniformly magnetized, although a condition of uniform M is approached as the domain size becomes small relative to the specimen size. (See Sections 4.1 and 7.2.) The demagnetizing field H d of a body is proportional to the magnetization which creates it: H d ¼ N d M, (2:17) where N d is the demagnetizing factor or demagnetizing coefficient. The value of N d depends mainly on the shape of the body, and has a single calculable value only for an ellipsoid. The sum of the demagnetizing factors along the three orthogonal axes of an ellipsoid is a constant: N a þ N b þ N c ¼ 4p (cgs) N a þ N b þ N c ¼ 1 (SI): (2:18) For a sphere, the three demagnetizing factors must be equal, so N sphere ¼ 4p 3 (cgs) or N sphere ¼ 1 3 (SI): The general ellipsoid has three unequal axes 2a, 2b, 2c, and a section perpendicular to any axis is an ellipse (Fig. 2.27). Of greater practical interest is the ellipsoid of revolution, or spheroid. A prolate spheroid is formed by rotating an ellipse about its major axis 2c; then a ¼ b, c, and the resulting solid is cigar-shaped. Rotation about the minor axis 2a results in the disk-shaped oblate spheroid, with a, b ¼ c. Maxwell calls this the planetary spheroid, which may be easier to remember. Fig Ellipsoids.

3 54 EXPERIMENTAL METHODS Equations, tabular data, and graphs for the demagnetizing factors of general ellipsoids are given by E. C. Stoner [Phil. Mag., 36 (1945) p. 803] and J. A. Osborn [Phys. Rev., 67 (1945) p. 351]. The most important results are as follows [here C 3 ¼ 4p (cgs); C 3 ¼ 1 (SI)]: 1. Prolate spheroid, or rod (cigar). a ¼ b, c. Put c=a ¼ m. Then, N c ¼ C 3 m p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi (m 2 ln(m þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 1) 2 1) 1 m 2 1 (2:19) N a ¼ N b ¼ C 3 N c : (2:20) 2 When m is large (long, thin rod), then N c C 3 ( ln(2m) 1) (2:21) m2 N a ¼ N b C 3 2 : (2:22) The approximation is in error by less than 0.5% for m. 20. N c approaches zero as m becomes large. Example: Form ¼ 10, N c ¼ and N a ¼ N b ¼ (cgs); N c ¼ and N a ¼ N b ¼ (SI). 2. Oblate (planetary) spheroid, or disk. a, b ¼ c, and c/a ¼ m. N c ¼ N b ¼ C (m 2 1) m 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2 1 arcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffi!! m 2 1 m 1 (2:23) N a ¼ C 3 2 N c (2:24) N b and N c approach zero as m becomes large. Example: Form ¼ 10, N b ¼ N c ¼ and N a ¼ (cgs); N b ¼ N c ¼ and N a ¼ (SI). When m is large (thin disk), then p N c ¼ N b C 3 4 m 1 2 m 2 N a C 3 1 p 2 m þ 1 m 2 (2:25) (2:26) This approximation is in error by less than 0.5% for m. 20. For larger values of m, the 1=m 2 terms can be dropped, giving p N c ¼ N b C 3 4 m N a C 3 1 p 2 m (2:27) (2:28) Specimens often encountered in practice are a cylindrical rod magnetized along its axis and a disk magnetized in its plane. Since these are not ellipsoids, the demagnetizing factors calculated according to the previous formulas will be in error to some degree.

4 2.9 DEMAGNETIZING FACTORS 55 Rods and disks are never uniformly magnetized except when completely saturated. The demagnetizing field varies from one point to another in the specimen and so has no single value. Two specific effective demagnetizing factors may be defined and used, depending on the way the magnetization is measured. The ballistic or fluxmetric demagnetizing factor N f is the ratio of the average demagnetizing field to the average magnetization, both taken at the midplane of the sample. It is the appropriate factor to use when the magnetization is measured with a small coil wound around the sample at its midpoint, using a ballistic galvanometer (now obsolete) or a fluxmeter. The fluxmetric demagnetizing factor is useful primarily for rod-shaped samples. The magnetometric demagnetizing factor N m is the ratio of the average demagnetizing field to the average magnetization of the entire sample. It is the appropriate factor to use when the total magnetic moment of the sample is determined using a vibrating-sample, an alternating-gradient, or a SQUID magnetometer (these instruments are described later). Note, however, that strictly speaking these devices measure the total sample moment only when the sample is small enough (relative to the pickup coil dimensions) to act as a point dipole. The samples used in these instruments are commonly disks magnetized along a diameter, although they may also be rods or rectangular prisms. Values of the demagnetizing factor depend primarily on the geometry of the sample, but also on the permeability or susceptibility of the material. Bozorth [R. M. Bozorth, Ferromagnetism, Van Nostrand (1951); reprinted IEEE Press (1993)] gives a table and graphs of demagnetizing factors for prolate and oblate ( planetary) spheroids, and also of fluxmetric demagnetizing factors for cylindrical samples with various values of permeability. Bozorth s curves have been widely reprinted and used. They are shown here as Fig The values for cylinders are based on a selection of early theoretical and experimental results, and should not be regarded with reverence. Note particularly that the demagnetizing factors for cylindrical (nonellipsoidal) samples given by Bozorth are fluxmetric values (although Bozorth does not use this terminology) and are only appropriate for measurements made with a short, centrally-positioned pickup coil around a cylindrical sample. The values in Bozorth s graph for disk samples magnetized along a diameter are calculated for planetary (oblate) ellipsoids, and so do not distinguish between fluxmetric and magnetometric values. It should also be noted that Bozorth plots and tabulates values of N=4p (cgs), not N (cgs), presumably so that the values can be multiplied by B to give demagnetizing fields H d. This is strictly incorrect, but useful for soft magnetic materials where H B and so B 4pM. Since N (SI) ¼ N (cgs) 4p, (2:29) Bozorth s values are numerically correct in SI. Better values for the demagnetizing factors of rods and disks (and other shapes, such as rectangular prisms) can be determined by experiment, or by calculation. The calculations generally assume a material of constant susceptibility x, which is in fact the differential susceptibility dm/dh measured at a point on the magnetization curve. Three specific values of x are of special significance: x ¼ 21, corresponding to a superconductor in the fullyshielded state; x 0, corresponding to a weakly magnetic material such as a para- or diamagnet, or to a fully-saturated ferro- or ferrimagnet; and x ¼ 1, corresponding to very soft magnetic material. The condition x ¼ 21 requires that B ¼ 0 everywhere in the samples. The condition x ¼ 0 requires that the magnetization M be constant throughout the sample, with H d variable. Note that x ¼ dm/dh ¼ 0 does not require M ¼ 0. The condition x ¼ 1 requires that the demagnetizing field be constant throughout the samples, exactly

5 56 EXPERIMENTAL METHODS Fig Demagnetizing factors for various samples. [R. M. Bozorth, Ferromagnmetism, Van Nostrand (1952); reprinted IEEE (1993) pp ]. Values plotted are N cgs /4p, which are numerically equal to N SI. equal and opposite to the applied field, with magnetization M varying from point to point. Demagnetizing factors can be calculated for other values of x, both positive and negative, but the assumption of constant and uniform x makes them of limited usefulness. The values for x ¼ 1 should apply for soft magnetic materials far from saturation, and values for x ¼ 0 to materials at or approaching magnetic saturation. In practice, demagnetizing field corrections are most important at low fields, where values of permeability and remanence are determined. Demagnetizing corrections are relatively unimportant (although not small) as the sample approaches saturation. Values of the coercive field are generally not much affected by demagnetizing effects, since they are determined when the magnetization

6 2.9 DEMAGNETIZING FACTORS 57 Fig Continued. is at or near zero. Permanent magnet materials, in which the values of susceptibility are low and uncertain, are normally measured in closed magnetic circuits where the demagnetizing fields are kept small. A paper by D.-X. Chen, J. A. Brug, and R. B. Goldfarb [IEEE Trans. Mag., 37 (1991) p. 3601] reviews the history of demagnetizing factor calculations and derives new values of N f and N m for rod samples. A later paper [D.-X. Chen, E. Pardo, and A. Sanchez, J. Magn. Mag. Matls., 306 (2006) p. 135] gives improved values for rod samples, and adds some calculated values of N m for disk samples. Similar results for rectangular prisms are given by the same authors [IEEE Trans. Mag., 41 (2005) p. 2077]. All three of these papers include results for a range of values of susceptibility as well as for sample shape.

7 58 EXPERIMENTAL METHODS Fig Calculated SI magnetometric demagnetizing factors for rod samples magnetized parallel to the rod axis. Central dashed line is for a prolate ellipsoid. Dotted curves are for x ¼ 1; solid curves for x ¼ 0. Upper dotted and solid curves are magnetometric factors N m ; lower curves are fluxmetric factors N f. Data in Figs and 2.30 from D.-X. Chen, E. Pardo, and A. Sanchez, J. Magn. Mag. Mater., 306 (2006) p The results are extensive and detailed, and not easy to summarize. Figure 2.29 shows calculated values of N f and N m for rod samples. The central dashed line is for prolate ellipsoids, where N f and N m are the same. The dotted lines are calculated values of N f and N m for x ¼ 1, i.e., for very soft magnetic materials. At large values of m (long, thin rods) N m is slightly above the ellipsoid line, and N f is slightly below. Note that values of m less than about 10 are largely of mathematical interest, since the measurement requires a central coil whose length is small compared to the sample length. The upper solid line is N m for x ¼ 0, and the lower solid line is N f for x ¼ 0. For samples of low susceptibility, or for samples approaching magnetic saturation, the demagnetizing factors can differ from those of the ellipsoid (of the same m value) by a factor approaching 10 when m ¼ 100. For samples of high susceptibility, in low fields, the demagnetizing factor for an ellipsoid of the same m value is generally a reasonable approximation, considering the various uncertanties involved. Figure 2.30 gives some results for the magnetometric demagnetizing factor N m for disk samples magnetized along a diameter. Fluxmetric demagnetizing factors N f are of little interest for disk samples. The dashed curve is for oblate ( planetary) ellipsoids; this is the same curve given by Bozorth. The dotted curve is for x ¼ 1 (high permeability) and the solid curve is for x ¼ 0 (uniform magnetization). In the m range of practical interest, values of N m are always higher than for the ellipsoid of the same m value, and the difference between the x ¼ 1 and the x ¼ 0 values is much less than for rod samples. There are some relevant experimental measurements. Figure 2.31 shows data points from vibrating-sample (VSM) measurements on a series of permalloy disks, together with the calculated curves for x ¼ 1 and x ¼ 0 from Fig The experimental points generally

8 2.9 DEMAGNETIZING FACTORS 59 Fig Calculated SI magnetometric demagnetizing factors for disk samples magnetized along a diameter. Dashed line is for an oblate (planetary) ellipsoid. Dotted line is N m for x ¼ 1; solid line is for x ¼ 0. fall between the two calculated curves. Note that it is not difficult to prepare samples with values of m greater than the highest value for which calculations (other than for ellipsoids) have been made; this is especially true if thin-film samples are measured. The theoretical papers give demagnetizing factors for values of x between 0 and 1, and also for x between 0 and 21. The negative values apply to superconductors, and will be treated in Chapter 16. In the case of nonellipsoidal samples, it is no longer necessarily true that the sum of the three orthogonal demagnetizing factors is a constant, so Equations 2.18 are not exactly correct. Fig Magnetometric demagnetizing factors for disk samples. Data points measured on 80 permalloy disk samples using a vibrating-sample magnetometer [C. D. Graham and B. E. Lorenz, IEEE Trans. Mag., 43 (2007) p. 2743]. Dotted and solid lines are copied from Fig. 2.30, for x ¼ 1 and x ¼ 0.

9 60 EXPERIMENTAL METHODS Clearly in experimental work it is advantageous to make the value of m large, to minimize the demagnetizing correction. Ideally, the worst-case value of H d should be comparable to the uncertainty in the measurement of the applied field; then uncertainty in the value of N becomes unimportant. Permanent magnet samples are usually made in the form of short cylinders or rectangular blocks, and they need to be measured in high fields, so the usual practice is to make the sample part of a closed magnetic circuit. This largely eliminates the demagnetizing effect. See the next section. A common mathematical procedure to calculate the demagnetizing field is to make use of the magnetic pole density on the sample surface, given by r s ¼ M cos u, where M is the magnetization of the sample and u is the angle between M and the normal to the surface. Note that M cos u is the component of the magnetization normal to the surface inside the body, and that M is zero outside. Therefore, the pole density produced at a surface equals the discontinuity in the normal component of M at that surface. If^n is a unit vector normal to the surface, then M cos u ¼ ~M ˆn ¼ r s : (2:30) Note that this agrees with one of the definitions of M as the pole strength per unit area of cross section. The polarity of the surface is positive, or north, if the normal component of M decreases as a surface is crossed in the direction of M. Free poles can also be produced at the interface between two bodies magnetized by different amounts and/or in different directions. If M 1 and M 2 are the magnetizations of the two bodies, then the discontinuity in the normal component is ~M 1 ˆn ~M 2 ˆn ¼ r s : (2:31) This is an important principle, which we shall need later. We also note that, at the interface between two bodies or between a body and the surrounding air, certain rules govern the directions of H and B at the interface: 1. The tangential components of H on each side of the interface must be equal. 2. The normal components of B on each side of the interface must be equal. These conditions govern the angles at which the B and H lines meet the air body interfaces depicted in Fig. 2.24, for example. Free poles may exist not only at the surface of a body, but also in the interior. For example, on a gross scale, if a bar has a winding like that shown in Fig. 2.32a, south poles will be produced at each end and a north pole in the center, for a current i in the direction indicated. On a somewhat finer scale, free poles exist inside a cylindrical bar magnet, as very approximately indicated in Fig. 2.32b. The condition for the existence of interior poles is nonuniform magnetization. An ellipsoidal body can be uniformly magnetized, and it has free poles only on the surface, unless it contains domains. A body of any other shape, such as a cylindrical bar, cannot be uniformly magnetized except at saturation, because the demagnetizing field is not uniform, and so the body always has interior as well as surface poles. Nonuniformity of magnetization means that there is a net outward flux of M from a small volume element, i.e., the divergence of M is greater than zero. But if there is a net outward flux of M, there must be free poles in the volume element to supply this flux. Such a volume element is delineated by dashed lines in Fig. 2.32b, in

10 2.9 DEMAGNETIZING FACTORS 61 Fig Internal poles in a magnetized body. which lines of M have also been drawn, going from south to north poles. (For clarity, the lines of M connecting surface poles on the ends have been omitted.) If r v is the volume pole density ( pole strength per unit volume), then div M ¼rM ¼ r v: (2:32) On the axis of a bar magnet, M decreases in magnitude from the center toward each end, as indicated qualitatively by the density of flux lines in Fig. 2.32b. Suppose the axis of the magnet is the x-axis, and we assume for simplicity that M is uniform over any cross section. Then only the x =@x need be considered. Between the center of the magnet and the north x =@x becomes increasingly negative, which means that r v is positive and that it increases in magnitude toward the end, as depicted in Fig. 2.32b. Although the interior pole distributions in Fig. 2.32a and b differ in scale, both are rather macroscopic; we shall see in Chapter 9 that interior poles can also be distributed on a microscopic scale. The general derivations of Equations and 2.32 may be found in any intermediatelevel text on electricity and magnetism. In summary: 1. Lines of B are always continuous, never terminating. 2. a. If due to currents, lines of H are continuous. b. If due to poles, lines of H begin on north poles and end on south poles. 3. At an interface, a. the normal component of B is continuous, b. the tangential component of H is continuous, and c. the discontinuity in the normal component of M equals the surface pole density r s at that interface. 4. The negative divergence of M at a point inside a body equals the volume pole density at that point. 5. The magnetization of an ellipsoidal body is uniform, and free poles reside only on the surface, unless the body contains domains. See Section 9.5.

11 62 EXPERIMENTAL METHODS 6. The magnetization of a nonellipsoidal body is nonuniform, and free poles exist on the surface and in the interior. (The saturated state constitutes the only exception to this statement. A saturated body of any shape is uniformly magnetized and has poles only on its surface.) 2.10 MAGNETIC MEASUREMENTS IN OPEN CIRCUITS Measurements of this type are usually made with a VSM or alternating gradient magnetometer (AGM), a fluxmeter, or a SQUID magnetometer. In the case of the VSM, AGM, or SQUID, the direct experimental result is a plot of the sample magnetic moment m vs the applied field H a. In the case of the fluxmeter, the usual result is a plot of flux density B vs applied field H a. The problem is to correct values of the applied field H a to values of the true field H tr, by subtracting the values of the demagnetizing field H d. The relationship is H tr ¼ H a H d, (2:33) where H d ¼ N d M and N d is the demagnetizing factor. As discussed above, unless the sample is in the shape of an ellipsoid, there is no single demagnetizing factor N d that applies for all parts of the sample at all levels of magnetization. A workable procedure is to select a value of N f or N m from Figs that is appropriate for the dimensions of the sample, the measurement technique, and the low-field permeability or susceptibility of the sample. It is important to remember that the demagnetizing field is always directed opposite to the direction of magnetization in the sample. If the experiment produces values of M, the correction is straightforward: at each value of M, the demagnetizing field is calculated from Equation 2.17, and the demagnetizing field is subtracted from the applied field (Equation 2.33) to obtain the true field acting on the sample. The corrections are made at fixed values of M, and move the measured M values parallel to the H axis. Since the demagnetizing field is proportional to the magnetization, H d can be represented by the line OD in Fig. 2.33, and the demagnetizing correction can be visualized by rotating the line OD counterclockwise about the origin O until it coincides with the y-axis, and Fig Graphical treatment of demagnetizing fields. (a) Plot of M versus H. (b) Plot of B versus H.

12 2.10 MAGNETIC MEASUREMENTS IN OPEN CIRCUITS 63 simultaneously moving each of the experimental values of M parallel to the H-axis, keeping the distance between the line OD and the value of M fixed. This is sometimes called the shearing correction. When the measurement gives values of flux density B rather then M, the correction becomes more complicated. M and H d are evaluated as follows: B ¼ H tr þ 4pM ¼ H a N d M þ 4pM (cgs) B m 0 ¼ H tr þ M ¼ H a N d M þ M (SI): (2:34) Then M ¼ B H a (cgs) 4p N d B H a H d ¼ N d 4p N d M ¼ (B=m 0) H a (SI) (2:35) 1 N d (SI) (2:36) (cgs) H d ¼ N d (B=m 0 ) H a 1 N d Two simplifications are often possible. First, if N d is reasonably small compared to 4p (cgs) or 1 (SI), the denominator in Eqs and 2.36 may be replaced by 4p or 1. Remembering that demagnetizing factors are not exact or well-defined except for ellipsoids, we may say that if N d is less than about 2% of its maximum value (4p or 1), it may be neglected here. This would mean m greater than 10 for a prolate ellipsoid (cigar) or greater than 30 for an oblate ( planetary) ellipsoid (disk). This does not mean that the demagnetizing field is negligible, just that N d may be neglected in this denominator. Second, in many cases of measurement on soft magnetic materials, H a is small compared to B (cgs), or compared to B/m 0 (SI). If both these conditions hold, Equation 2.33 reduces to H d ¼ N d B 4p (cgs) or H d ¼ N d B m 0 (SI) (2:37) How this works is illustrated by the experimental data in Table 2.2, obtained from a rod of commercially pure iron in the cold-worked condition. The rod was 240 mm long and 6.9 mm in diameter and hence had a length/diameter ratio of 35. The measurements were made with a search coil at the center of the rod, so the fluxmetric demagnetizing factor TABLE 2.2 Magnetization of Iron Rod H a,oe B, G (B 2 H a ), G M, emu/cm 3 H d,oe H, Oe 8.1 1,080 1, ,850 3, ,910 7, ,080 10, ,420 12, ,860 14,810 1, ,220 18,140 1,

13 64 EXPERIMENTAL METHODS Fig Table 2.2. Measured and corrected magnetization curves of cold-worked iron, from the data of applies. The low-field permeability is unknown, but we may make the reasonable assumption that it lies somewhere between 10 2 and 1. From Fig we read N d ¼ 0:002 (SI) or N d ¼ (0:002)(4p) ¼ 0:025 (cgs). Using Equation 2.37, this leads to the demagnetizing fields H d listed in the table (which is in cgs units), and we see that they form a very substantial fraction of the applied fields. The flux density is plotted in Fig as a function of both applied and true fields. It is clear that the apparent permeability, given by B/H a,is much less than the true permeability,orb/h tr. If the two conditions noted above are met, it may be shown that 1 m true ¼ 1 m apparent N d 4p (cgs) 1 m r(true) ¼ 1 m r(apparent) N d (SI). (2:38) This suggests an experimental method to determine a value for the demagnetizing factor. If a material has a high permeability m true, say 5000 or higher, then 1=m true is negligible and Equation 2.38 gives: m apparent ¼ 4p N d (cgs) m r(apparent) ¼ 1 N d (SI): (2:39) A measurement of B vs H a on a sample of this material will give an initial straight line with a slope B ¼ m H apparent ¼ 4p (cgs) a N d or B m 0 H a ¼ m r(apparent) ¼ 1 N d (SI) The reciprocal of the numerical value of this slope gives the experimental value of N d.soa sample of high-permeability material, such as Ni Fe permalloy or high-purity annealed iron or nickel, made in the same size and shape as the sample(s) to be measured, can be

14 2.10 MAGNETIC MEASUREMENTS IN OPEN CIRCUITS 65 used to establish an experimental demagnetizing factor for all high-permeability samples of the same size and shape, measured in the same apparatus. If M is measured directly instead of B, then we can write H M ¼ H a N (2:40) M or, since susceptibility x ¼ M=H, H a M 1 ¼ N: (2:41) x The value of x at low field can be measured on a ring sample where N ¼ 0. For soft magnetic materials, 1=x may be negligible. Then the reciprocal of the slope of the measured M vs H a curve gives an experimental value for N. This is the method used to determine the experimental points in Fig When the field applied to a specimen on open circuit is reduced to zero, the induction remaining is always less than in a ring specimen, because of the demagnetizing field. In Fig. 2.33b the induction in a ring specimen would be B r, because H d ¼ 0 and H a ¼ H. But in an open-circuit specimen, the remanent induction is given by the intersection of the demagnetizing line OC or OC 0 with the second quadrant of the hysteresis loop. If the specimen is long and thin, this line (OC) will be steep and the residual induction B 1 will not differ much from that of the ring. If the specimen is short and thick, the line (OC 0 ) will be so nearly flat that the residual induction B 2 will be very small. The demagnetizing effect can assume huge proportions in short specimens of magnetically soft materials. For example, suppose a sample of iron can be brought to its saturation value of M s ¼ 1700 emu/cm 3 ( A/m) by a field of 10 Oe (800 A/m) when it is in the form of a ring. If it is in the form of a sphere, H d at saturation will be (4p=3)(1700) ¼ 7120 Oe ([1/3][ ] ¼ 570 ka/m), and the applied field necessary to saturate it will be 7120 þ 10 ¼ 7130 Oe or 570 þ 0.8 ¼ 571 ka/m. The M, H a (or B, H a ) curve will be a straight line almost to saturation, with a slope determined by the value of N d, and the details of the true curve, such as the initial permeability, will be unobservable. In open-circuit measurements on very soft magnetic materials, the Earth s magnetic field of 0.5 Oe or 40 A/m may be significant. This influence can be minimized by orienting the long axis of the sample perpendicular to the Earth s field, as determined by a magnetic compass. It may be important to consider that the Earth s field has a component normal to the Earth s surface, except near the equator. We have seen three possible methods of calculating or eliminating the demagnetizing field correction: 1. Make the specimen in the form of an ellipsoid. Then H d can be exactly calculated, but at the cost of laborious specimen preparation. 2. Use a rod or strip or thin film specimen of very large length area ratio. The demagnetizing factor is then so small that any error in it has little effect on the computed value of the true field. 3. Apply a correction using a calculated or tabulated or measured demagnetizing factor. The uncertainties in this approach have been noted above.

15 66 EXPERIMENTAL METHODS A fourth approach, which is often used, is to make the specimen into part of a closed magnetic circuit, so that the free poles causing the demagnetizing field are largely eliminated. This is done by clamping the sample into some form of magnetic yoke; the resulting device is known as a permeameter. It is described in more detail below. If a large number of identical rod samples are to be tested using a fluxmeter, time will be saved by slipping each rod into a single search coil, previously wound on a nonmagnetic form. Since the cross-sectional area A c of the search coil will be larger than the area A s of the specimen, an air-flux correction must be made for the flux in air outside the specimen but inside the search coil: f observed ¼ f specimen þ f air, B apparent A s ¼ B true A s þ H(A c A s ) (cgs) ¼ B true A s m 0 H(A c A s ) (SI) B true ¼ B apparent H A c A s (cgs) ¼ B apparent m A 0 H A c A s (SI): s A s (2:42) 2.11 INSTRUMENTS FOR MEASURING MAGNETIZATION Extraction Method This method is based on the flux change in a search coil when the specimen is removed (extracted) from the coil, or when the specimen and search coil together are extracted from the field. When the solenoid in Fig is producing a magnetic field, the total flux through the search coil is or F 1 ¼ BA ¼ (H þ 4pM)A ¼ (H a H d þ 4pM)A ¼ (H a N d M þ 4pM)A (cgs) F 1 ¼ BA ¼ m 0 (H þ M)A ¼ m 0 (H a H d þ M)A ¼ m 0 (H a N d M þ M)A (SI) (2:43) where A is the specimen or search-coil area. (The two are assumed equal here to simplify the equations, i.e., the air-flux correction is omitted.) If the specimen is suddenly removed Fig Arrangement for measuring a rod sample in a magnetizing solenoid. Two procedures are possible: (1) solenoid current is varied continuously and fluxmeter output is recorded continuously; (2) at a series of constant solenoid current values the sample (not the search coil) is moved to a position of effectively zero field, and the fluxmeter reading is recorded. This is the extraction method. Procedure 1 measures B vs H; procedure 2 measures M(4p 2 N ) (cgs) or m 0 M(1 2 N ) (SI).