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Laboratory Report Materials Physics Laboratory The B-H loop of a Ferrimagnet Yufei Chang • Group X5


Abstract The aim of this experiment was to study the response of the ferrimagnetic material to the applied magnetic field. Readings in volts (Vx, Vy, see Introduction) were taken from the oscilloscope, which were proportional to the magnetic field strength and magnetic induction. By analysing data, the initial magnetisation curve and B-H loop could be plotted. From which the values of saturation magnetisation, coercivity and remanence could be derived. And the permeability could be obtained from either the gradient of the initial magnetisation curve or by plotting permeability as a function of magnetic field. The ferrimagnet showed a hysteresis behavior. Introduction Certain metallic materials had permanent magnetic moment in the absence of an external field, and manifest very large and permanent magnetizations. These were the characteristics of ferromagnetism.1 Some ceramics also exhibited a permanent magnetization, termed ferrimagnetism.2 The initial magnetisation curve (Fig.1 plotted) was the initial response of the material when it was magnetised from the unmagnetised state at H=0, B=0.3 The B-H loop (Fig.2 plotted) was obtained when a magnetic material was subjected to a magnetic field and measuring the magnetic induction produced.4 Magnetic induction represented the magnitude of the internal field strength within a substance that was subjected to a magnetic field.5 Coercivity was the applied magnetic field required to reduce the magnetisation to zero.6 It was strongly depend on the condition of the sample.7 The remaining magnetic induction after the field was removed was called the remanence.8 It was the upper limit for all remanent inductions.9 The permeability or relative permeability of a material was a measure of the degree to which the material could be magnetized, or the ease with which a magnetic induction could be induced in the presence of an external magnetic field.10 Theory H = nmI / L Amps metre-1 where nm was the number of turns in the coil, I was the current flow and L was the length of the toroid. Other formulae used: Experimental procedure

circumference = 2Ď€r gradient = (y2-y1) / (x2-x1)


In this experiment, a toroid magnetic material was provided, with two coils wound around it. Apparatus was set up according to the lab script. An a.c current past through the first coil, generating voltage (Vx) and inducing flux density to the toroid. The second coil responded to the rate of change of magnetic flux, generating a current. The induced voltage was Vy. Vx and Vy were shown on an oscilloscope, where Vx drove the X-deflection and Vy drove the Ydeflection. The magnetising field through the toroid was gradually increased, the trace on the oscilloscope was observed and initial magnetisation curve and B-H loop could be drawn. The size of the loop increased with the current until it reached a saturation point. • Initial magnetisation curve Error The readings taken from the oscilloscope were accurate to half of the small division, for example if the scale was set to be 1.00V per big division, namely, 0.20V per small division, the error was ¹ 0.10V. Scale of vertical axis and error

VY / V

Scale of horizontal axis and error

VX / mV

VX / V


0.200V / big division 0.040V / small division Error was ± 0.020 V 0.500V / big division 0.100V / small division Error was ± 0.050V

1.00V / big division 0.200V / small division Error was ± 0.200V

0.00±0.02 0.04±0.02 0.08±0.02 0.16±0.02 0.30±0.02 0.80±0.05 1.20±0.05 1.60±0.05 1.80±0.05 2.00±0.20 2.20±0.20 2.40±0.20 2.50±0.20 2.60±0.20 2.70±0.20 2.80±0.20 2.80±0.20 2.85±0.20 2.90±0.20 2.90±0.20 2.95±0.20 2.95±0.20 3.00±0.20 3.00±0.20

0.00±1.00 5.00±1.00 10.0±1.00 10.0mV / big divi- 15.0±1.00 sion 21.0±1.00 2.00mV / small 30.0±1.00 division 35.0±1.00 Error was 40.0±1.00 ±1.00mV 45.0±1.00 50.0±1.00 55.0±1.00 20.0mV / big divi- 60.0±2.00 sion 70.0±2.00 4.00mV / small 80.0±2.00 division 90.0±2.00 Error was 100.0±2.00 ±2.00mV 110.0±5.00 120.0±5.00 50.0mV / big divi130.0±5.00 sion 140.0±5.00 10.0mV / small division 150.0±5.00 Error was ± 160.0±5.00 5.00mV 170.0±5.00 180.0±5.00

0.000±0.001 0.005±0.001 0.010±0.001 0.015±0.001 0.021±0.001 0.030±0.001 0.035±0.001 0.040±0.001 0.045±0.001 0.050±0.001 0.055±0.001 0.060±0.002 0.070±0.002 0.080±0.002 0.090±0.002 0.100±0.002 0.110±0.005 0.120±0.005 0.130±0.005 0.140±0.005 0.150±0.005 0.160±0.005 0.170±0.005 0.180±0.005

Table 1 The magnetic field, H, was given by H = K1Vx, and the magnetic flux density, B, was given by B = K2Vy. The current I = V / R, where R = 1000Ω. And also, H = (nm I) / L. Combined the equations, H = (nm I) / L H = [nm x (Vx / R)] / L [nm x (Vx / R)] / L = K1Vx K1 = (nm / RL)

K1 Magnetic length of the toroid L = 2πr = 2π x 0.009 = 5.65 x 10-2 Number of turns on the magnetizing coil nm = 67 Thus K1 = nm / RL K1 = 67 / (0.0565x1000) = 1.18 (Am-1) / Volt


Table 2 K2 Number of turns on the secondary coil Cross-sectional area of the toroid Resistance value Capacitance value Thus K2 = RC / nsA

ns = 44 A = 0.0065x(0.023–0.013)x0.5 = 3.25 x 10-5 R = 1000 Ω C = 100 x 10-9 F K2=1000x100x10-9/(44x3.25x10-5) = 6.99x10-2 Tesla/Volt

Table 3 Worked out the values of H and B (see below) using equations H = K1Vx, B = K2Vy, and Table 1. Error The errors on H and B were calculated by simply multiplying errors on Vx or Vy by K1 or K2 respectively. For instance took the value when Vx was 0.005 ± 0.001 mV, the value of H was H = (0.005 x 1.18482) ± (0.001 x 1.18482) = 5.92x10-3 ± 1.18x10-3 Am-1 (to 2 d.p) The errors on B were worked out using the same method. H / Am-1 0 0.00592 0.0119 0.0178 0.0249 0.0356 0.0415 0.0474 0.0533 0.0592 0.0652 0.0711 0.0829 0.0948 0.107 0.118

Error on H (ΔH)/Am-1 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±2.37x10-3 ±2.37x10-3 ±2.37x10-3 ±2.37x10-3 ±2.37x10-3

B / Tesla 0.00 2.80x10-3 5.59x10-3 1.12x10-2 2.10x10-2 5.59x10-2 8.39x10-2 1.12x10-1 1.26x10-1 1.40x10-1 1.54x10-1 1.68x10-1 1.75x10-1 1.82x10-1 1.89x10-1 1.96x10-1

Error on B (ΔB)/Tesla ±1.40x10-3 ±1.40x10-3 ±1.40x10-3 ±1.40x10-3 ±1.40x10-3 ±3.50x10-3 ±3.50x10-3 ±3.50x10-3 ±3.50x10-3 ±1.40x10-2 ±1.40x10-2 ±1.40x10-2 ±1.40x10-2 ±1.40x10-2 ±1.40x10-2 ±1.40x10-2


0.130 0.142 0.154 0.166 0.178 0.190 0.201 0.213

±5.92x10-3 ±5.92x10-3 ±5.92x10-3 ±5.92x10-3 ±5.92x10-3 ±5.92x10-3 ±5.92x10-3 ±5.92x10-3

1.96x10-1 2.00x10-1 2.03x10-1 2.03x10-1 2.06x10-1 2.06x10-1 2.10x10-1 2.10x10-1

±1.40x10-2 ±1.40x10-2 ±1.40x10-2 ±1.40x10-2 ±1.40x10-2 ±1.40x10-2 ±1.40x10-2 ±1.40x10-2

Table 4 Plotted initial magnetization curve (Fig. 1) using Table 4 (B against H). The errors were represented by the error bars. •

Hysteresis loop

Error The sensitivity of the x-axis was set to be 50.00mV per big division, so every small division was 10.00mV. It was 1.00V per big division, hence 0.20V per small division on the y-axis. Therefore, the errors on Vx were ± 5.00mV, and ± 0.10V on Vy. Top part of the loop Vx / V Vy / V Vx / mV -3.00±0.10 -150±5.00 -0.150±0.005 -2.95±0.10 -140±5.00 -0.140±0.005 -2.95±0.10 -130±5.00 -0.130±0.005 -2.90±0.10 -120±5.00 -0.120±0.005 -2.85±0.10 -110±5.00 -0.110±0.005 -2.80±0.10 -100±5.00 -0.100±0.005 -2.70±0.10 -90±5.00 -0.090±0.005 -2.65±0.10 -80±5.00 -0.080±0.005 -2.50±0.10 -70±5.00 -0.070±0.005 -2.40±0.10 -60±5.00 -0.060±0.005 -2.20±0.10 -50±5.00 -0.050±0.005 -2.00±0.10 -45±5.00 -0.045±0.005 -1.00±0.10 -40±5.00 -0.040±0.005 1.00±0.10 -30±5.00 -0.030±0.005 2.00±0.10 -20±5.00 -0.020±0.005 2.30±0.10 -10±5.00 -0.010±0.005 0.000±0.005 2.40±0.10 0±5.00 0.010±0.005 2.45±0.10 10±5.00

Bottom part of the loop Vx / V Vy / V Vx / mV -0.160±0.005 -3.00±0.10 -160±5.00 -0.140±0.005 -2.95±0.10 -140±5.00 -0.120±0.005 -2.90±0.10 -120±5.00 -0.100±0.005 -2.90±0.10 -100±5.00 -0.080±0.005 -2.85±0.10 -80±5.00 -0.070±0.005 -2.80±0.10 -70±5.00 -0.060±0.005 -2.80±0.10 -60±5.00 -0.050±0.005 -2.75±0.10 -50±5.00 -0.040±0.005 -2.70±0.10 -40±5.00 -0.030±0.005 -2.65±0.10 -30±5.00 -0.020±0.005 -2.60±0.10 -20±5.00 -0.010±0.005 -2.50±0.10 -10±5.00 0.00±0.005 -2.40±0.10 0±5.00 0.010±0.005 -2.30±0.10 10±5.00 0.020±0.005 -1.90±0.10 20±5.00 0.025±0.005 -1.00±0.10 25±5.00 0.030±0.005 0.00±0.10 30±5.00 0.035±0.005 1.00±0.10 35±5.00


2.50±0.10 2.55±0.10 2.60±0.10 2.65±0.10 2.70±0.10 2.70±0.10 2.75±0.10 2.80±0.10 2.90±0.10 2.95±0.10 3.00±0.10

20±5.00 30±5.00 40±5.00 50±5.00 60±5.00 80±5.00 100±5.00 120±5.00 140±5.00 160±5.00 180±5.00

0.020±0.005 0.030±0.005 0.040±0.005 0.050±0.005 0.060±0.005 0.080±0.005 0.10±0.005 0.120±0.005 0.140±0.005 0.160±0.005 0.180±0.005

1.40±0.10 2.00±0.10 2.40±0.10 2.50±0.10 2.60±0.10 2.65±0.10 2.70±0.10 2.70±0.10 2.75±0.10 2.80±0.10 2.90±0.10 2.90±0.10 2.95±0.10 3.00±0.10 3.00±0.10

40±5.00 50±5.00 60±5.00 70±5.00 80±5.00 90±5.00 100±5.00 110±5.00 120±5.00 130±5.00 140±5.00 150±5.00 160±5.00 170±5.00 180±5.00

0.040±0.005 0.050±0.005 0.060±0.005 0.070±0.005 0.080±0.005 0.090±0.005 0.100±0.005 0.110±0.005 0.120±0.005 0.130±0.005 0.140±0.005 0.150±0.005 0.160±0.005 0.170±0.005 0.180±0.005

Table 5 Calculated the values of H and B (see below) using H = K1Vx, B = K2Vy, and Table 5. Error The errors on H and B were derived by the same method as did for Table 4. e.g When Vx was -0.150±0.005V and Vy was -3.00±0.10V. H = [1.18482 x (-0.150)] ± (1.18482 x 0.005) = -0.178 ± 0.00592 Am-1 B = [6.99 x 10-2 x (-3.00)] ± (6.99 x 10-2 x 0.10) = -0.21 ± (6.99 x 10-3) Tesla Since the scale did not change when the values were taken, the errors for all the H values and B values were the same as the errors calculated above. Top part of the loop H / Am-1 B / Tesla -0.178±0.00592 -0.210±6.99x10-3 -0.166±0.00592 -0.206±6.99x10-3 -0.154±0.00592 -0.206±6.99x10-3 -0.142±0.00592 -0.203±6.99x10-3 -0.130±0.00592 -0.199±6.99x10-3 -0.118±0.00592 -0.196±6.99x10-3 -0.107±0.00592 -0.189±6.99x10-3 -0.095±0.00592 -0.185±6.99x10-3 -0.083±0.00592 -0.175±6.99x10-3 -0.071±0.00592 -0.168±6.99x10-3

Bottom part of the loop H / Am-1 B / Tesla -0.190±0.00592 -0.210±6.99x10-3 -0.166±0.00592 -0.206±6.99x10-3 -0.142±0.00592 -0.203±6.99x10-3 -0.118±0.00592 -0.203±6.99x10-3 -0.095±0.00592 -0.199±6.99x10-3 -0.083±0.00592 -0.196±6.99x10-3 -0.071±0.00592 -0.196±6.99x10-3 -0.059±0.00592 -0.192±6.99x10-3 -0.047±0.00592 -0.189±6.99x10-3 -0.036±0.00592 -0.185±6.99x10-3


-0.059±0.00592 -0.053±0.00592 -0.047±0.00592 -0.036±0.00592 -0.024±0.00592 -0.012±0.00592 0.00±0.00592 0.012±0.00592 0.024±0.00592 0.036±0.00592 0.047±0.00592 0.059±0.00592 0.071±0.00592 0.095±0.00592 0.118±0.00592 0.142±0.00592 0.166±0.00592 0.190±0.00592 0.213±0.00592

-0.154±6.99x10-3 -0.140±6.99x10-3 -0.070±6.99x10-3 0.070±6.99x10-3 0.140±6.99x10-3 0.161±6.99x10-3 0.168±6.99x10-3 0.171±6.99x10-3 0.175±6.99x10-3 0.178±6.99x10-3 0.182±6.99x10-3 0.185±6.99x10-3 0.189±6.99x10-3 0.189±6.99x10-3 0.192±6.9 x10-3 0.196±6.99x10-3 0.203±6.99x10-3 0.206±6.99x10-3 0.210±6.99x10-3

-0.024±0.00592 -0.012±0.00592 0.00±0.00592 0.012±0.00592 0.024±0.00592 0.030±0.00592 0.036±0.00592 0.041±0.00592 0.047±0.00592 0.059±0.00592 0.071±0.00592 0.083±0.00592 0.095±0.00592 0.107±0.00592 0.118±0.00592 0.130±0.00592 0.142±0.00592 0.154±0.00592 0.166±0.00592 0.178±0.00592 0.190±0.00592 0.201±0.00592 0.213±0.00592

-0.182±6.99x10-3 -0.175±6.99x10-3 -0.168±6.99x10-3 -0.161±6.99x10-3 -0.133±6.99x10-3 -0.070±6.99x10-3 0.00±6.99x10-3 0.070±6.99x10-3 0.098±6.99x10-3 0.140±6.99x10-3 0.168±6.99x10-3 0.175±6.99x10-3 0.182±6.99x10-3 0.185±6.99x10-3 0.189±6.99x10-3 0.189±6.99x10-3 0.192±6.99x10-3 0.196±6.99x10-3 0.203±6.99x10-3 0.203±6.99x10-3 0.206±6.99x10-3 0.210±6.99x10-3 0.210±6.99x10-3

Table 6 The hysteresis loop (Fig. 2) was plotted using Table 6. •

Permeability as a function of magnetising field

The permeability was given by μ = B/H. Calculated the values of μ using Table 4 and Fig,1. Error According to the formula of combination of errors. If A (±a) x B (±b) = C (±c) or A (±a) / B (±b) = C (±c) then (c/C)2 = (a/A)2 + (b/B)2 where a, b and c were errors on A, B and C respectively.


For instance, took the second set of values from Table 4, where H equaled 0.00592±0.00118 Am-1 and B equaled 2.80 x 10-3±1.40 x 10-3 Tesla. The error on μ, Δμ, was Δμ = μ √ (ΔH/H)2 + (ΔB/B)2 Δμ = [(2.80 x 10-3) / 0.00592] √ {(0.00118 / 0.00592)2 + [(1.40 x 10-3) / 2.80 x 10-3]2} Δμ = 0.255 H/m-1 μ = [(2.80 x 10-3) / 0.00592] ± 0.255 = 0.473 ± 0.255 H/m-1 The errors for the rest values of permeability were calculated using the same formula. µ (= B/H) / H/m-1 -0.473 0.472 0.630 0.844 1.57 2.02 2.36 2.36 2.36 2.36 2.36 2.11 1.92 1.77 1.65 1.50 1.41 1.32 1.22 1.16 1.09 1.04 0.985

Error on µ (Δµ) / H/ m-1 -± 2.55 x 10-1 ± 1.27 x 10-1 ± 8.92 x 10-2 ±6.91 x 10-2 ± 1.11 x 10-1 ± 1.02 x 10-1 ± 9.44 x 10-2 ± 8.39 x 10-2 ± 2.41 x 10-1 ± 2.19 x 10-1 ± 2.12 x 10-1 ± 1.79 x 10-1 ± 1.55 x 10-1 ± 1.37 x 10-1 ± 1.23 x 10-1 ± 1.27 x 10-1 ± 1.15 x 10-1 ± 1.04 x 10-1 ± 9.50 x 10-2 ± 8.77 x 10-2 ± 8.13 x 10-2 ± 7.60 x 10-2 ± 7.11 x 10-2

H / Am-1

Error on H (ΔH)/Am-1

0 0.00592 0.0119 0.0178 0.0249 0.0356 0.0415 0.0474 0.0533 0.0592 0.0652 0.0711 0.0829 0.0948 0.107 0.118 0.130 0.142 0.154 0.166 0.178 0.190 0.201 0.213

±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±1.18x10-3 ±2.37x10-3 ±2.37x10-3 ±2.37x10-3 ±2.37x10-3 ±2.37x10-3 ±5.92x10-3 ±5.92x10-3 ±5.92x10-3 ±5.92x10-3 ±5.92x10-3 ±5.92x10-3 ±5.92x10-3 ±5.92x10-3

Table 7 Plotted permeability against magnetising field (Fig. 3) using Table 7.


Determination of initial and maximum permeability

The initial permeability was the gradient of the beginning part of initial magnetization curve, shown on Fig.1. Taking the point B = 0.05±1.40x10-3 Tesla, H = 0.09±0.00118 Am-1, and the point B=0, H=0.

gradient = (y2 – y1) / (x2 – x1) = (0.05 - 0) / (0.09 - 0) = 0.556 Hm-1

The error on μi was, Δμi = 0.556 √ {[(1.40 x 10-3) / 0.05]2 + (0.00118 / 0.09)2} = 0.0172 Hm-1 Hence,

μi = 0.556±0.0172 Hm-1

The maximum permeability, μmax, was found from Fig. 3, which was the highest point on the graph. μmax = ± Hm-1 Saturation magnetisation of the ferrimagnet Bs = 0.212±6.99x10-3 Tesla Coercivity of the ferrimagnet Hc = - ±0.00592 Am-1 Remanence of the ferrimagnet Br = 0.168±6.99x10-3 Tesla Initial value of the relative permeability µi = 0.556±0.0172 Hm-1 Maximum value of the relative permeabilµmax = Hm-1 ity

Discussion The graphs plotted gave the suggested shape, and the respective error bars showed the experimental data were reasonably reliable. However, in Fig.1, the errors increased with the increase of magnetic field strength. This was due to the change in sensitivity. As the strength of magnetic field increased, the volts per division increased, so did the uncertainty when readings were taken. This was not the case for Fig.2, as the sensitivity did not change throughout the process. In Fig.3, the first point and points on the top of graph showed relatively large errors. Permeability was the gradient of the initial magnetisation curve; there were few readings available in the range where maximum permeability obtained. The errors could be reduced by recording more data in that range. Moreover, there were always subjective errors when taking the readings, which could also be improved by taking more data. In addition, the number of significant figures used in calculation introduced uncertainty to the results. Three significant figures were used for all the results in this experiment. The more significant figures used, the more accurate the results.


The area inside the B-H loop was the energy lost per unit volume per magnetisation-demagnetisation cycle in magnetising material, termed hysteresis lost. The soft magnetic materials had low energy lost. They were used in the device that were subjected to alternating magnetic fields, for instance, transformer cores. Soft magnetic materials were easily magnetised and demagnetised. Hard materials had large energy lost and were good permanent magnets. Hysteresis was caused by two mechanisms. One was the imperfections in the materials, in the form of dislocation or impurity, which caused an increase in the energy lost during the magnetisation process. Another mechanism was caused by magneto-crystalline anisotropy, which was associated with the domain process. Questions Why do ferro- and ferri-magnetic materials display a B-H loop? Both of the ferro- and ferri-magnetic materials had magnetic dipole moments, which were randomly aligned when there was no magnetic field applied. The vector sum of the magnetisation was zero, as the moments with different direction cancelled out each other. However, they had a preferential aligned direction in the presence of a magnetic field and showed hysteresis property. Therefore they displayed a B-H loop. Which value you have measured tells you the magnetic hardness of the material? The coercivity, Hc, was the value told the hardness of the material. If the value of Hc > 10000Am-1, the material was difficult to demagnetise, which were “hard” material.19 Whereas if it was below 1000Am-1, the material was said to be “soft”.20 What type of the magnetic material have you measured: is it magnetically hard or soft? The ferrimagnet used in this experiment was magnetically soft, because the magnetic field applid to demagnetise the material was fairly small. This material is ferrimagnetic. What does mean? Ferrimagnetism was a particular case of antiferromagnetism.(195 mmm) Thus, magnetic moment coupling was parallel for ferromagnets while antiparallel for ferrimagnets.21 Ferrimagnets behaved very much like ferromagnets on a macroscopic scale, the distinction lied in the source of the net magnetic moments. The net ferrimagnetic moment arose from the incomplete cancellation of spin moments, because the ferrimagnet did not have the same spin moments. The saturation magnetisations for ferrimagnetic materials were not as high as for ferromagnets.So ferrimagnets were good electrical insulators. Conclusion


The ferrimagnet showed hysteresis behavior as ferromagnets. The B-H loop gave values of saturation magnetisation, coercivity, remanence and permeability, which were important parameters for magnetic materials. References William D. Callister Jr. “Material science and engineering An introduction”, 7th Edition, web chapter 20, W26. 2. William D. Callister Jr. “Material science and engineering An introduction”, 7th Edition, web chapter 20, W28. 3. Lab script. 4. Lab script. 5. William D. Callister Jr. “Material science and engineering An introduction”, 7th Edition, web chapter 20, W21. 6. Lab script. David Jiles, “Introduction to Magnetism and Magnetic Materials 1.


Lab report the B-H loop of a ferrimagnet  

Imperial college London, material science and engineering,first year lab report copies

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