Magnetism and Matter Notes

Magnetism and Matter INTRODUCTION BASIC PROPERTIES OF MAGNETS In nature, first time iron ore was found which had a 1. property to attack small pieces of iron, nickel and cobalt etc. And this property of attracting small pieces of material such as iron, nickel and cobalt 2. etc. by a piece of material is called magnetism. There are two types of magnets. One which are found in nature are called natural magnets. These are of distorted shape. 3. Second which are artificially manufactured are called 4. artificial magnets. These are of defined shape and size i.e. bar magnets. 5. BAR MAGNET A system composed of two poles, equal in magnitude but opposite in polarity, placed at a small displacement apart is known as a bar magnet. 6. It is also known as a magnetic dipole. +mp –mp N S S (b) When a magnet is suspended with the help of unspum thread, its two end, which are called poles come to rest in north south direction. Poles exists always in pair and having equal strength i.e. magnetic monopoles do not exist. Like poles repel each other and unlike poles attract each other. North and south pole of magnet are a little inwards from geometrical end. The magnetic length of magnet is about 0.84 times the geometrical length. The force of attraction or repulsion between two magnetic poles is directly proportional to product of pole strengths (m1, m2) and inversely proportional to square of distance (r) between their centres F  (a) N A magnet attracts magnetic substances such as nickel, cobalt, iron etc. 0 4 m1m2 r 2 = 1 in CGS units  F = 40 m1m2 r2 = 107 wbA1m1 where 0 is absolute permeability of free space. 47 MAGNETIC DIPOLE AND It’s MOMENT It has two unlike poles of equal strength separated by a definite small distance. Unit of pole strength is ampere meter or Newton/ Magnetism and Matter (iv) Non-varying magnetic field: (a) The magnetic field which does not change with time is known as a constant magnetic field.  (b)The direction of magnetic field is that in which a Tesla. Magnetic dipole moment ( M ) is defined as the product of pole strength denoted by m and distance between the two poles, called magnetic length, force acts on a unit test pole. (c) It can be produced by moving charges, current  represented by 2  carrying loops, and variations in electric currents. N   M = m (2  ) It is a vector quantity and its direction is from south to north pole. Its unit is ampere meter2 or Joule/Tesla MAGNETIC FIELD (a) The space around a magnet in which a torque acts on a magnetic needle is known as magnetic field. (b) The space around a magnet in which a net force acts on a magnetic test pole is known as magnetic field. (c) The space around a magnet in which its effect is experienced is known as magnetic field. (d) There are four types of magnetic field : (i) Uniform magnetic field: (a) The magnetic field, in which the intensity of magnetic field is same at all points, is known as uniform magnetic field. (b)In such a magnetic field the magnetic lines of force are parallel and equidistant. e.g. the magnetic dines of force of earth’s magnetic field. (ii) Non-uniform magnetic field: (a) The magnetic field, in which the intensity of magnetic field at different points is different, is known as non-uniform magnetic field. (b)It is represented by non-parallel lines of force. (iii) Varying magnetic field: (a) The magnetic field, which keeps on changing with respect to time is known as a variable magnetic field. (b) Example :– B = B0 sin  t or B = B0 cos  t F F B B S MAGNETIC LINE OF FORCES AND THEIR PROPERTIES As electric field lines of forces we also have imaginary magnetic field lines to represent magnetic field in any region. These are closed lines when electric field lines are open lines. To draw magnetic lines we use a magnetic dipole i.e. small compose needle. The direction of magnetic field at a point is same as direction of tangent at the point. PROPERTIES : 1. No two magnetic lines can cut each other. 2. These are closed lines whose direction outside magnet is from north to south pole and inside from south to north pole. 3. Tangent at a point of magnetic line give direction of field at that point. 4. Crowding of magnetic lines of force represent stronger magnetic field. 5. There is always tension in magnetic field line and repulsion between two lines. 48 Magnetism and Matter MAGNETIC FIELD STRENGTH DUE TO MAGNETIC MOMENT (M) BAR MAGNET (a) If a magnet of length l and magnetic moment M is bent in the form of a semicircular are then its new (i) End-on position : In this position point lies on the axis of magnet. Magnetic field strength due to south pole BS = m 0 4 magnetic moment will be M’ = mp mp N S . (r   )2 along PS due to north pole. BN = 4 0 (r  ) 2 along NP l N  B1 = BN  BS along NP m 4 r 0 2Mr = 4 (r 2   2 )2 = 4 (r 2   2 ) along NP. For small magnet < < r.  0 2M  So BE = 4 3 along NP. r (ii) Broad - side - on - position : When point lies on equatorial line of magnet. Magnetic field strength at point P. m BN = 4  × (r 2   2 ) along NP Due to south pole 0 M = mpI (b) The magnetic moment of an electron due to its orbital motion is 1B whereas that due to its spin motion it is B . 2  eh   eh  i.e. Morbital = l  4 m    2m   l   l  and  eh   eh  (c) Other formulae of M: m (ii) M  eVr er 2  er 2 2f er 2     2 2 2 T e 2mJ = –mp S (d) Resultant magnetic moment : Net magnetic field (i) When two bar magnets m × ( 2  r 2 ) cos  along PS 0 m x 2 2 4  (r   2 ) 3 / 2 eh 4m (ii) B = 0.93 × 10–23 Amp-m2 (iv) M = nB =2  B Mspin = s  4m   s  2m   sB  2  s   s  Here B = Bohr magneton (iii) BB S (i) M = nir2 BS = 4 × ( 2  r 2 ) along PS 0 4 2r (i) The value of Bohr magneton B = Due to north pole 0 N S Net magnetic field strength at P 0 l = r l m  2M  = 0 M 2 4  (r   2 )3 / 2 are lying mutually perpendicular to each other, then l mp N S –mp l N mp M1 M  M12  M22  2mpl For small magnet < < r 0 M  BB = 4  3 along PQ r BE 2 Note B = 1 B M2 (ii) When two coils, each of radius r and carrying current i, are lying concentrically with their planes at right angles to each other, then 49 Magnetism and Matter (ii) For diamagnetic materials M  M12  M22  2ir 2 if M1 = M2 I M2 dia M r O r M1 (iii) For ferromagnetic materials INTENSITY OF MAGNETISATION (I) I (a) The magnetic moment per unit volume of a material is defined as intensity of magnetisation (I). i.e. I= Ferro M , when V V = 1m3 then I = M. (b) The unit of intensity of magnetisation is ampere/meter (i) I - H curve and its dimensions are M0L–1T0A1 (c) The pole strength per unit area of cross-section is defined as intensity of magnetisation. i.e. I  mp A Ferromagnetic I when A = 1m2 then A=1m2 then I = mp (d) It is a vector quantity whose direction is along the magnetic field. H Diamagnetic (e) In para- and ferro-magnetic materials its direction is in the direction of H and in dia-magnetic materials it (j) Its value depends on temperature. is opposite to that of H. (k) It is produced on account of induction in a material. (f) I is produced in materials due to spin motion of (l) For low magnetising field I  H. i.e. I  H or electrons. I = Ht (g) The value of I and its direction in a material depend (m)  is a dimensionless constant i.e. it carries no unit. on the nature of that material. (n) Other types of intensity of magnetisation: (h) I - M curve (i) Mass intensity of magnetisation (I )mass (i) For para magnetic materials Intensity of magnetisation 1 Imass or Im   density of magnetic material  I (ii) Molar intensity of magnetisation- Para Imolar = IMW = Mo. Wt.× mass intensity of magnetisation = MW × Im (iii) Molecular intensity of magnetisationImolecular  IM  M Molar int ensity of magnetisation IMW  Avogadro No. N 50 Magnetism and Matter MAGNETIC SUSCEPTIBILITY OR (OR K) (a) The ratio intensity of magnetisation (I) in a material and the magnetising field (H) is defined as magnetic susceptibility (). (j) – 1 : T  I H When H = 1 Oersted then  = I. i.e.   (b) The intensity of magnetisation induced in a material by unit magnetising field is defined as magnetic susceptibility. (a) T For paramagnetic substances  (c) It has no unit and no dimensions. (d) It is a measure of ease with which a material can be magnetised by a magnetised by a magnetising field (H). (e) Magnetic susceptibility of various materials(i) For diamagnetic materials –  = low and negative (ii) For paramagnetic materials –  = low but positive (iii)For ferromagnetic materials –  = high and positive (f) For paramagnetic substances it is inversely I proportional to temperature i.e.   T (g) For low magnetising field the value of  is constant. (h) Different types of magnetic susceptibility(i) Volume susceptibility I X H (ii) Mass or specific susceptibility m V  T– 1 For paramagnetic substances ABSOLUTE MAGNETIC PERMEABILITY () (a) The ratio of magnetic induction (B) to magnetising field (H) is defined as magnetic permeability (). (b) The extent to which magnetic permeability of that medium. (c) It is the characteristic property of a magnetic material because it represents the amplification of magnetising field in that material. (d) Its value is always positive and is different for different materials.  v Volume susceptibility   density of meterial (iii) Molar susceptibility MW (e) For materials its value can be greater or less than 0. MW = m × MW (i)  = 0r  = Specific susceptibility × Mol. Wt. (j) (i) For feeromagnetic materials  = high m  (f) Its value depends on H and T. (h)  = 0 [1 + ] (iv) Molecular susceptibility m (ii) For paramagnetic materials  = low m = Atomic wt. × specific susceptibility (iii)For diamagnetic materials  = very low = A m  (i) (b) –T curve AI H (k) The unit of magnetic permeability is Weber per ammere-meter or Henry per meter and its dimensions are M1L0T–2A–2. 51 RELATIVE PERMEABILITY (R) (a) The ratio of magnetic permeability of medium () to the magnetic permeabiliy of free space (0) is defined  as relative permeability (r). i.e. r   0 (b) r  (c) Number of lines of force pas sin g throughunit area in medium Number of lines of force pas sing throughunit area in vacuum r  magnetic flux density in medium Magnetic flux density in vacuum Magnetism and Matter Compare (2) with Gauss’s law in electrostatics electric flux through closed surface. S is given by   q    E.ds  -----(3) E0 S ‘q’ is electric charge enclosed by the surface Gauss’s law for magnetism is : The net magnetic flux through any closed surface zero. THE EARTH’S MAGNETISM The branch of physics which deals with the study of earth's magnetic field is called geomagnetism. (d) The limit unto which a magnetic field penetrates matter, is known as relative permeability of that material. (e) It has no unit and no dimensions. (f) r = 1 +  (g) Relative permeability of various substances(i) For diamagnetic substances the value of r is slightly less than one i.e. r < 1. (ii) For paramagnetic substances the value of r is slightly greater than one i.e. r > 1. (iii) For ferromagnetic substance the value of r is much 1 Important definitions : greater than one i.e. r >> 1. (a) Geographic axis : It is a straight line which passes MAGNETISM AND GAUSS’S LAW through the geographical poles of the earth. It is also According to Gauss’s law for magnetism, the net called axis of rotation or polar axis of the earth. magnetic flux ( s ) through any closed surface is (b) Geographic Meridian (GM) : It is a vertical plane always zero. which passes through geographical axis of the earth. The law implies that the number of magnetic field (c) Geographic equator : It is a great circle on the lines leaving any closed surface is always equal to surface of the earth, in a plane perpendicular to the the number of magnetic field lines entering it. geographic axis. All the points on the geographic equator are at equal distance from the geographic poles. A great plane which passes through geographic equator and perpendicular to the geographic axis called geographic equatorial plane. This plane According to Gauss’s law for magnetism. cuts the earth in two equal parts, a part has Magnetic flux geographic north called northen hemisphere   (NHS) and another part has geographic south called B   B   B.S  0 -----(1) all all southern hemi sphere (SHS). We can write (d) Magnetic axis : It is a straight line which passes   B   B.dS  0 through magnetic poles of the earth. It is inclined to -----(2) S the geographic axis at nearly 17°. 52 (e) Magnetic Meridian (MM) : (i) It is a vertical plane which passes through magnetic axis of the earth. Magnetism and Matter resultant magnetic field of the earth substends with respect to horizontal in magnetic meridian. (ii) At a given place it is an angle which the axis of freely suspended magnetic needle substends with respect to horizontal in magnetic meridian. (ii) At a particular place it is a vertical plane, which passes through axis of free suspended bar magnet Note :or magetic needle. In NHS, north pole of freely suspended magnetic needle will dip downwards i.e. towards the earth surface. In SHS, south pole of freely suspended magnetic needle will dip downwards i.e. towards the earth surface. (iii) At a particular place it is a vertical plane, which Dip circle : Angle of dip is measured by the contains all the field lines of earth's magnetic field. instrument called 'Dip-circle' in which a magnetic (f) Magnetic equator : It is a great circle on the surface needle is free to rotate in vertical plane about of the earth, in a plane perpendicular to the magnetic horizontal axis. The ends of the needle move over a axis. All the points on the magnetic equator are at vertical scale graduated in degree. equal distance from the magnetic poles. 2 Main elements of earth's magnetic field : 13.2.1 Angle of declination () :– At a given place the acute angle between geographic meridian and the magnetic meridian is called angle of declination, i.e. at a given place it is the angle between the geographical north-south direction and the direcation indicated by a magnetic compass needle in its equillibrium. Angle of dip () :– (i) At a given place it is an angle which the direction of Horizontal component of earth magnetic field (BH) :– At a given place horizontal component of earth magnetic field is the component of resultant magnetic field of the earth along the horizontal line in magnetic meridian. BH = Bcos and Bv = B sin  .......(1) so that tan  = .......(2) and B = 53 Magnetism and Matter Note 1 : * At magnetic poles   = 90° * At magnetic equator   = 0° Note 1 : * For a vertical plane other than magnetic meridian  > 0  cos  < 1  tan' > tan  ' > , so opparent angle of dip is always more than actual angle of dip at any place. Note 2 : * For a vertical plane which is perpendicular to magnetic meridian  = 90°  tan' = = Note 2 : ' = 90° , so in a plane which is perpendicular to  decides the plane in which magnetic field lies at magnetic meridian dip needle becomes just verticle. any place, () and () decides the direction of magnetic field and () and (BH) decides the magnitude Magnetic field of long Bar magnet : of the field. (i) At Axial position :– Magnetic field at point 'P' due to north pole :– 3 Apparent angle of dip (') : When the plane of vertical scale of dip circle is in the magnetic meridian, the needle stays in the direction of earth's magnetic field. The angle made by the needle with respect to horizontal is called true dip or actual dip. If the plane of vertical scale of dip circle not kept in magnetic meridian, then the needle will not indicate the correct direction of earth magnetic field. In this situation the angle made by the needle with respect to horizontal is called the apparent angle of dip. Suppose the dip circle is set at an angle  to the magnetic B1 =   ..... (1) (away from north pole)  Magnetic field at point 'P' due to south pole :– B2 =   ....... (2) (towards north pole)  Net magnetic field at point 'P' Baxis = B1 – B2 , ( B1 > B2) = LM N        OP Q =   LM N OP Q   , where M = m (2)  Note : If magnet is short r >> , then    (ii) At equatorial position : Magnetic field at point 'P' due to north pole :– B1=   meridian. Effective horizontal component in this plane will be BH cos  and vertical component BV remains same. App. angle of dip tan ' = FH  IK ..... (1) (along NP line)  tan ' = [from (2)] 54 Magnetism and Matter Magnetic field at point 'P' due to south pole :– B2=   FH  IK ..... (2) (along PS line) From equation (1) & (2) B1 = B2 =   ·  = B (Let) Net magnetic field at point 'P' Beq = 2 B cos = 2·    cos, where cos=     = 2·       , where M = m(2)  Note : If magnet is short r >> , then Ex.1 The length of a bar magnet is 10 cm and its pole strength is 103 Weber. It is placed in a magnetic field of induction 4  x 103 Tesla in a direction making an angle of 30º with the field direction. The value of torque acting on the magnet will be (A) 2 × 107 N-m (B) 2 × 105 N-m (C) 0.5 × 102 N-m (D) None of these Sol.  = MB sin  = m B sin  = 103 x 0.1 x 4 x 103 x 0.5 = 2 x 107 N-m  SOME TERMS RELATED TO MAGNETISM (A) Magnetic permeability : It is the ability of material to permit the passage of magnetic lines through it. If it denoted by . Its unit is weber/ammeter. Relative magnetic permeability (r), is a factor by which the magnetic field B is increased when a material is brought in the field. r is unitless.  r =  = 1 +  0 TORQUE ON A MAGNET IN where 0 is permeability of vaccum and  is MAGNETIC FIELD susceptibly of material. When a bar magnet of pole strength m and magnetic (B) Intensity of magnetization (I) : It is defined as length 2 is placed at an angle  with a uniform magnetic moment per unit volume.   magnetic field B . magnetic moment m  = I = m I= = V  Force on N-pole = mB along B  Force on S-pole = mB opposite B These forces are equal in magnitude and opposite in since so they will form couple. Torque  = mB x ON = MB 2 sin  = MB sin   = M x B The direction  is perpendicular to plane containing   M and B , and is given by right hand screw rule volume 2m  m A (2  ) = A V For bar magnet of pole strength m, length x and area of cross-section A. (C) Magnetic susceptibility : For paramagnetic and diamagnetic substances, the intensity of magnetization  ( I ) of a material is directly proportional to magnetic  in intensity ( H )   Thus I =  H The proportionality content  is called susceptibility. It is a dimensionless quantity. For vaccum  = 0. Note : r = 1 +  (in M.K.S. system) r = 1 + 4 (in C.G.S system) 55 Magnetism and Matter (D) Magnetic Intensity : The actual magnetic field Ex.5 A bar magnet made of steel has a magnetic moment of 2.5 A-m2 and a mass of 6.6 × 103 kg. If the inside the material is the sum of the applied magnetic density of steel is 7.9 × 109 kg/m3, find the intensity field and magnetic field due to magnetization.   of magnetization of the magnet. B B  Hence H =   I =  Sol. The volume of the bar magnet is 0  where B is the resultant magnetic field, and  permeability of the material and I intensity of  magnetization. The quantity H is called magnetic field intensity or magnetizing field intensity. Unit of it is ampere/meter. mass V = density = 6 .6 10  3 kg 7 .9  10 3 kg / m 3 = 8.3 x 107 m3. The intensity of magnetization is 2.5 A  m 2 M = = = 3.0 × 106 A/m V 8.3  10 7 m 2 MAGNETIC BEHAVIOUR OF ATOM Ex.2 The magnetic susceptibility of a paramagnetic substance is 3 × 104. It is placed in a magnetising field of 4 × 104 amp/m. The intensity of magnetisation will be (A) 3 × 108 A/m (B) 12 × 108 A/m (C) 12 A/m (D) 24 A/m Sol. I = XH = 3 × 10-4 × 4 x 103 = 12 A/m Ex.3 The horizontal component of flux density of earth’s magnetic field is 1.7 × 105 tesla. The value of horizontal component of intensity of earth’s magnetic field will be ? (A) 24.5 A/m (C) 0.135 a/m (B) 13.5 A/m (D) 1.35 A/m B 1.7 x 10 5 Wb / m 2 Sol. H =  = 4 x 10 7 Wb / A  m = 13.5 A/m 0 Hence the correct answer will be (B). If an electron having charge e is revoling in an orbit of radius r having uniform angular velocity . Then Magnetic moment of atom contributed by the electron M= eh 1 e r2 and magnetic field B = = 9.27 4 m 2 x 1024 . ampere meter2 If source as the unit of atomic magnetic dipole moment. Where h is plank’s constant. MOLECULAR / ATOMIC THEORY OF MAGNETIZATION According to this theory (i) Every molecule/atom of a magnetic substance has north pole as well as south pole of equal strength, so it is a magnetic dipole. (ii) In unmagnetized condition, atomic dipoles are randomly oriented and they cancel dipole moment of one another. Ex.4 A magnetising field of 2 × 103 amp/m produces a (iii) After applying magnetic field, atomic dipoles align in magnetic flux density of 8 Tesla in an iron rod. The its direction. relative permeability of the rod will be (iv) Extent of magnetization depends on extent of (A) 102 (B) 100 realignment of the molecular dipoles. (C) 104 (D) 101 (v) On heating magnetization reduces or vanish  B completely. Sol.  = = r 0 H 0 8 or r = 2 x 10 3 x 4 x 10 7 = 104 On the basis of this theory magnetic materials are classified into three categories 56 Magnetism and Matter (a) Paramagnetic substances : Every molecules / As we increase field intensity, magnetization point A atoms of these substances has its own dipole corresponds to saturation magnetization. As value moment. When we place such substances in of and direction of field intensity varies, magnetization magnetic field tries to align every individual molecule along path ACDEFGA is followed. / atom in its direction e.g. chromium, CuSO4 , crown Hence as field intensity increased or decreased, glass, aluminimium, platinum, manganese, solution magnetization does not return to its initial value. This of salts of iron and nickel are paramagnetic substances. fact is called hysteresis. The curve ACDEFGA is Permeability of paramagnetic substance is greater called hysteresis loop. This are of the hysteresis loop than unity. Susceptibility of paramagnetic substances is proportional to the thermal energy developed per varies inversely to the temperature of substance. unit volume of the material as it goes through the (b) Diamagnetic substances : Molecules / atoms of hysteresis cycle. this type of substances do not have dipole moment Hysteresis loop for soft iron is narrow and large, individually. When they are kept in any magnetic field, whereas hysteresis loop for steel is wide and short. magnetic moment induces but it is in opposite direction to the applied magnetic field. e.g. copper, gold, mercury , quart, antimony and Bismoth etc. Permeability of diamagnetic substances is always less Ex.6 The mass of a speciment of a ferromagnetic material than unity. is 0.6 kg. and its density is 7.8 x 103 kg/m3. Susceptibility of diamagnetic substances does not If the area of hysteresis loop of alternating change with temperature. magnetising field of frequency 50Hz is 0.722 MKS (c) Ferromagnetic substances : In these substance units then the hysteresis loss per second will be – each individual atom/molecule has magnetic moment. They have strong tendency to align themselves so (A) 277.7 × 105 Joule (B) 277.7 × 106 Joule they make domain. In a domain, a large no. of atoms/ (C) 277.7 × 104 Joule (D) 277.7 × 104 Joule molecules align in same direction. m = Aft Different domains have different directions of Sol. WH = VAft d magnetic moment hence, the material remains un 0.6 or WH = x 0.722 x 50 magnetized. 7.8  10 3 When a magnetic field is applied, the domains which = 277.7 x 105 Joule are aligned along the direction of the field grown in size and those opposite to it get reduced. iron nickel CURIE LAW and cobalt etc. are examples of such materials. According to this law, intensity of magnetization (I) of a magnetic material is directly proportional to magnetic induction (B) and inversely proportional to the temperature of magnetic material. HYSTERESIS As shown in adjacent curve, as we change field intensity, magnetization changes. In the beginning , the field is zero and sample has no magnetization, this is corresponding to point O. B T I 1 So  H T I we know B  H I C = = H T where  is susceptibility of material and C is a constant called curie constant. The variation of I with graph it is clear that saturation is reached when H/T exceeds a certain limit. Magnetic thermometers are based on curie law and filled with paramagnetism substance (e). 57