"PHY-320 Lab Report Measurement of Magnetic Susceptibility by Gouy’s Method Gaurav Bhole, Anirban Sharma Indian Institute of Science Education and Research, Pune April 20, 2015 Abstract Magnetism is a physical phenomenon by which materials assert an attractive or repulsive force or in?uence on other materials. A magnetic ?eld can arise due to several processes within an atom, such as the nuclear spin, electron spin and electron orbital motion. These magnetic ?elds interact with each other as well as an external magnetic ?eld. The property which makes a material magnetic is the measure of the extent to which it responds to a magnetic ?eld, known as magnetic susceptibility. In this report, we demonstrate the measurement of magnetic susceptibility of CuSO\u0000 5H O and MnSO\u0000 H O 4 2 4 2 using Gouy’s Method. We also discuss various experimental errors which might have occured and ways to improve the method further. netic, (ii) Paramagnetic and (iii) Ferromag- I. Introduction netic. We give a brief discussion of each of Magnetism is a property of substances by the above class of magnetic materials in the which they can attract or repel other sub- next section. stances. There are innumerable materials in nature which exhibit magnetism. Materi- Figure 1: Louis Georges Gouy (1854-1926) als like Iron, Cobalt and Nickel are naturally magnetic, whereas others like Manganese and Tin acquire magnetic properties when placed in an external magnetic ?eld. For a systematic study of the magnetic materials, we catego- rize them based upon the extent to which they respond to an external magnetic ?eld. This response is characterised by the magnetic sus- ceptibility of the substance, which is the ratio of induced magnetic ?eld to the applied mag- netic ?eld. When a substance is placed in an external magnetic ?eld H, a magnetic moment M per unit volume is generated within the material. Thus the magnetic susceptibility c of a mate- rial is de?ned as : M c = (1) H where M is the magnetization of the material. In this report, we demonstrate the mea- For isotropic materials, c is a scalar as the surement of magnetic susceptibility of Cop- M and H are parallel to each other. For per Sulfate Pentahydrate(CuSO\u0000 5H O) and 4 2 anisotropic materials, c is a tensor. Based Manganese Sulfate Monohydrate(MnSO\u0000 H O) 4 2 upon the susceptibility of various solids, we using Gouy’s Method. In section II, we give classify the magnetic materials as: (i) Diamag- a brief background of the theory involved in 1PHY-320 Lab Report this experiment. Further, in section III, we rise to a magnetic moment . But due to ther- describe the experimental setup. We describe mal ?uctuations, at room temperature, the the method and the measurements involved magnetic dipoles point in random directions. in section IV. The calculations are discussed If we take the average over a unit volume in section V. We describe our discussions in of the sample, the net magnetic moment is section VI. We have shown our experimen- zero. But as soon we place the substance in tal results in section VII. We have discussed a magnetic ?eld , all the dipoles align them- about the sources of error and the precautions selves parallel to the applied magnetic ?eld to be taken while performing the experiment giving non zero magnetisation. There is no in section VIII. Finally, we conclude in section interaction between the magnetic moments of IX. neighbouring atoms, ions or lattice points in a crystal. II. Theory Ferromagnetic : These substances retain their magnetization even after the external I. Background ? ?eld is removed . There are dipole âAS ¸ dipole All magnetic phenomena are mainly due to interactions between the neighbouring atoms , ions or lattice point in crystals. Quantum electric charges in motion. If we go to atomic mechanical effects also come into play. Due to scale , we can ?nd tiny currents because of these, the substances have domains in which electrons orbiting around nuclei . Electrons all the atoms or entities have the same direc- also spin about their axis . The current loop tion of moment giving rise a net moment of can be treated as magnetic dipoles . A sub- the domain. There is diamagnetism in para- stance placed in a magnetic ?eld results in the magnetic substances too . But diamagnetism allignment of the magnetic dipoles within the being a weak effect, it is very small compared substance. This allignment of dipoles occurs in a particular fashion according to the class to paramagnetism. of magnetic material involved. The vector Table 1: Volume susceptibility c for various sub- v ?eld that expresses the density of permanent stances or induced magnetic dipole moments in a magnetic material is known as its magneti- \u0000 5 Substance Category c \u0000 10 zation or magnetic polarisation. The ratio of v the magnetization to the magnetising ?eld Mercury Diamagnetic \u0000 2.9 is known as the magnetic susceptibility (c) Aluminium Paramagnetic 2.2 of the substance. Depending on whether we Iron Oxide Paramagnetic 720 take 1 gram or 1 c.c. of the substance, the dif- ferent susceptibilities are respectively known as speci?c susceptibility and volume suscepti- II. Formulae bility. Based on the susceptibility of materials, For the measurement of magnetic suscepti- we classify them as : bility c of a particular substance, we need to know the magnetisation M as well as the Diamagnetic : Magnetization is in oppo- external magnetic ?eld H. Although the ex- site direction to the applied ?eld . These kind ternal magnetic ?eld H can be found out of substances have paired electrons and hence using a Gauss probe, one cannot quantify no magnetic moment. However, on the ap- the magnetisation M within the material di- plication of magnetic ?eld, they get repelled. rectly. Thus, we need to characterize M by They have a negative susceptibility. some physical observable which varies with changing the H and hence can be measured. Paramagnetic : The direction of the in- We quantify this physical observable as the duced magnetic ?eld is parallel to the direc- change in mass of the sample as we vary the tion of applied ?eld The atoms of these sub- H, which will be described below. stances have unpaired electrons which gives 2PHY-320 Lab Report Figure 2: Force on a cylindrical specimen in a inho- Figure 3: Experimental Setup of Gouy’s Method mogenous magnetic ?eld The force dF on a differential volume ele- ment dV in upward direction along +Z axis due to a non-uniform magnetic ?eld H is : 1 ¶ 2 dF = dV(c\u0000 c ) H (2) a 2 ¶Z The energy density due to a magnetic ?eld 2 is given by B /2m. When we move the differ- ential volume element, the work done on it by the magnetic ?eld is equal to the decrease Sample holder : The sample holder is in the potential energy of the system. Thus, an air-tight cylindrical glass tube of known 2 in the expression for dF, we get ¶H /¶Z de- length. The sample is a ?ne powder of pendence. On integrating dF, we get the total CuSO\u0000 5H O or MnSO\u0000 H O tightly ?lled in- 4 2 4 2 force in the direction of increasing Z as : side the sample holder. The sample holder is suspended to the balance with the help of a 1 2 2 ?ne thread. The sample tube is adjusted in F = a(c\u0000 c )(H \u0000 H ) (3) a o 2 such a way that the lower edge of the tube comes near the center of the poles. where, a is the cross section area of the sam- ple, c is the susceptibility of the sample, c a is the susceptibility of air and H is the mag- o netic ?eld at the top of the sample tube. H o Balance : The balance attached to the is negligible compared to H. Also, c>> c . a sample holder registers a change in the Thus we get the total force F : weight of the sample as we vary the mag- netic ?eld. The sensitivity of the balance 1 2 F =\u0000 acH (4) \u0000 4 used should be very high (least count = 10 2 grams) as the variation of mass with magnetic However, this force which is in the negative ?eld is very weak(of the order of milligrams). Z direction is recorded in the balance as the change in mass of the substance. Thus, we have, 1 2 Electromagnets : By changing the cur- jFj = mg = acH (5) 2 rent passing through the coils wound around the magnet, the magnetic ?eld between the poles can be varied from 0 Gauss to 8000 III. Experiment Gauss. Due to the shape of the electromagnet We now describe the Gouy’s apparatus to as shown in the above ?gure, the magnetic measure the magnetic susceptibility. ?eld is inhomogenous at the edges. 3PHY-320 Lab Report Figure 4: Gauss pro?le between the two electromagnets V. Calculations I. Analytical Now, we calculate the magnetic susceptibility for every reading ofDm as follows, 1 2 jFj =Dmg = a(c \u0000 c )H (6) v a 2 M Using c \u0000 0 and a = , we have, the vol- a Lr ume susceptibility, 2DmgLr c = (7) v 2 MH (8) The speci?c susceptibility can be calculated from volume susceptibility according to the relation, c v c = (9) r Table 2: Specimen Parameters Description CuSO\u0000 5H O MnSO\u0000 H O 4 2 4 2 Length(L) 3.66 cm 3.58 cm Density(r) 2.286 gm/c.c. 2.950 gm/c.c. Mass(M) 0.8804 gm 0.9581 gm IV. Method Using the above information, we calculate average volume and speci?c susceptibilities \u0000 Set up the apparatus as shown in the and tabulate as follows: Figure. Table 3: Analytical Calculations \u0000 Measure the mass of the sample in the Susceptibility absense of any external magnetic ?eld. Substance Volume (c ) Speci?c (c) v \u0000 5 \u0000 5 CuSO\u0000 5H O 0.265\u0000 10 0.116\u0000 10 4 2 \u0000 5 \u0000 5 \u0000 Now increase the external magnetic MnSO\u0000 H O 10.1\u0000 10 3.44\u0000 10 4 2 ?eld by increasing the current in dis- crete steps. The errors in each of the above suscepti- bility values can be obtained by taking the standard deviation. \u0000 Note down the change in mass regis- tered by the balance in each step. II. Graphical \u0000 Calculate the volume susceptibility c v We plotted the graph ofDm versus the exter- and speci?c susceptibility c using the nal magnetic ?eld H for CuSO\u0000 5H O and 4 2 relations given in the following section. MnSO\u0000 H O. 4 2 4PHY-320 Lab Report 2 Figure 5: Graph ofDm versus H VI. Discussion 2 (Copper Sulfate) ?m Versus H Quadratic Relation : The graph of Dm 0.012 2 versus H is linear. The quadratic relation 0.01 between Dm and H is evident from the fol- 0.008 lowing graphs. 0.006 y = 2E-10x - 0.0003 R² = 0.9963 0.004 Figure 6: Graph ofDm versus H 0.002 0 0 1000 2000 3000 4000 5000 6000 7000 Mass Difference (?m) Versus Magnetic Field (H) x 10000 (Copper Sulfate)2 2 H in (Gauss) 0.012 0.01 2 (Manganese Sulfate) ?mVersus H 0.008 0.35 0.006 0.3 0.25 0.004 2 y = 2E-10x - 3E-07x + 0.0003 R² = 0.9983 0.2 0.002 0.15 y = 5E-09x - 0.0082 0 R² = 0.9822 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0.1 (H) in Gauss 0.05 0 0 1000 2000 3000 4000 5000 6000 x 10000 Mass Difference (?m) Versus Magnetic Field (H) 2 2 H in (Gauss) (Manganese Sulfate) 0.35 0.3 2 0.25 The graph of Dm versus H is linear as 0.2 expected. The magnetic susceptibility can be 0.15 calculated from the slope of the graph as fol- 0.1 2 lows: y = 7E-09x - 2E-05x + 0.014 0.05 R² = 0.9908 0 0 1000 2000 3000 4000 5000 6000 7000 8000 (H) in Gauss 2 Dm = slope\u0000 H (10) From Equation 7, we get, These graphs are in agreement with the 2gLr previously discussed quadratic relation be- c = slope\u0000 (11) v M tweenDm and H. Deviation in Graphs : From the graph Table 4: Slopes 2 ofDm versus H it can be seen that there is a deviation from the linear trend at higher mag- Substance Slope netic ?elds. This is possibly due to neglecting \u0000 10 the H term while calculating the total force CuSO\u0000 5H O 1.71\u0000 10 o 4 2 \u0000 10 F. At higher magnetic ?eld, we cannot ne- MnSO\u0000 H O 52.3\u0000 10 4 2 glect the magnetic ?eld H which lies outside o the magnetic pole pieces. Table 5: Graphical Calculations Paramagnetic Nature of CuSO\u0000 5H O 4 2 and MnSO\u0000 H O : From the graphs, we ob- 2 4 Susceptibility serve that theDm which is proportional to the force along -Z axis increases with the increase Substance Volume (c ) Speci?c (c) v in external magnetic ?eld H. This implies \u0000 5 \u0000 5 CuSO\u0000 5H O 0.318\u0000 10 0.139\u0000 10 4 2 that CuSO\u0000 5H O and MnSO\u0000 H O are para- 4 2 4 2 \u0000 5 \u0000 5 MnSO\u0000 H O 11.3\u0000 10 3.83\u0000 10 4 2 magnetic in nature. Further, from the suscep- 5 (?m) in grams (?m) in grams(?m) in grams (?m) in grams "