Magnetic Materials PDF

# CH. 11 Magnetic Materials [4/5]ap M ## Reduced syllabus: - 11.2 Torque acting on a magnetic dipole in uniform magnetic field - 11.5 Magnetic properties of materials - 11.6 Hysteresis - 11.7 Permanent magnet and electromagnet - 11.8 Magnetic shielding - 11.3 Origin of magnetism in materials ## Ob...

# CH. 11 Magnetic Materials [4/5]ap M ## Reduced syllabus: - 11.2 Torque acting on a magnetic dipole in uniform magnetic field - 11.5 Magnetic properties of materials - 11.6 Hysteresis - 11.7 Permanent magnet and electromagnet - 11.8 Magnetic shielding - 11.3 Origin of magnetism in materials ## Obtain an expression for orbital magnetic moment of an electron rotating about the nucles in an atom 1. Consider an electron of mass *m_e* and charge *e* revolving in circular orbit of radius *r* around the positive nucleus in clockwise direction, leading to an anticlockwise current. 2. If an electron travel distance (*2πr*) in time *T*, then its orbital speed, *V = 2πr/T* ---> (1) (*2πr* - circumference of circular orbit (or distance covered in time period *T*)) *Speed = distance/time* 3. Magnitude of circulating current is given by, *I = e/T* ---> *current = charge/time* But *T = 2πr/V* ---> from (1) *I = e/T* = *eV/2πr* ---> (2) 4. The orbital magnetic moment associated with orbital current loop is given by, *m_orb = IA* = *eV/2πr* x *πr²* ---> area of current loop (circular loop) = *πr²* *m_orb = evr/2* ---> (3) 5. The angular momentum of an electron due to its orbital motion is given by, *L = m_evr* ---> (4) 6. Multiplying and dividing RHS of eq? (3) by *m_e* *m_orb = e/2m_e* x *m_evr* ---> *m_e* = mass of electron *m_orb = e/2m_e* x *L* ---> (5) 7. eq? (5) shows that orbital magnetic moment (*m_orb*) is proportional to the angular momentum (*L*). But as electron b the orbital magnetic moment and orbital angular momentum are in opposite directions and perpendicular to the plane of orbit. In vector form, *m_orb = - (e/2m_e)* L ## Gyromagnetic Ratio : The ratio of orbital magnetic moment with angular momentum of revolving electron is called gyromagnetic ratio. Gyromagnetic ratio : *m_orb/L* = *e/2m_e* ---> from (5) = 1.6x10^-19/(2x9.1x10^-31) = 8.8 x 10¹⁰C kg^-1 ## Bohr magneton : 1. According to Bohr's theory, an electron in an atom can revolve only in certain stationary orbits in which angular momentum (*L*) of electron is an integral (*n*) multiple of *h/2π* where *h* is Planck's constant *L = m_e vr = nh/2π* ---> (1) 2) The orbital magnetic moment of an electron is given by, *m_orb = eL/2m_e* ---> (2) 3) Substituting eq? (1) in eq? (2) *m_orb = e/2m_e* x *nh/2π* *m_orb = n^-(eh/4πm_e)* ---> (3) 4) for first orbit, n=1 ... *m_orb = eh/4πm_e* The quantity *eh/4πm_e* is called as Bohr Magneton and its value is 9.274 x10^-24 Am² ## Magnetization: The ratio of magnetic moment to the volume of the material is called magnetization. It is denoted by *M*. If magnetic specimen of volume 'v' acquires net magnetic dipole moment 'M_net' due to the magnetising field, then *M = M_net/V* ---> V - Volume Magnetization is vector quantity SI unit → A/m or (Am^-1) Dimensions [M] = [M⁰L^-1T⁰I¹] ## Magnetic intensity (H): Definition: The ratio of strength of magnetising field to the permeability of free space (μ_o) is called as magnetic intensity (H) The strength of magnetic field at a point can be given in terms of vector quantity called as magnetic intensity (H) *H = B_o/μ_o* ⇒ *B_o = μ_oH* or *H = nI* SI unit → A/m or (Am^-1) for solenoid Dimensions – [M⁰L^-1T^-1I¹] ⇒ *B_o = μ_oHI = H* ... where n= N/L ## Magnetic susceptibility (X): The ratio of magnitude of magnetization to the magnetic intensity (H) is called as magnetic susceptibility (x). It is given by, *X = M/H* ## Relative permeability (μ_r): The ratio of magnetic permeability of material (μ) and magnetic permeability of free space (μ_o) is called relative permeability(μ_r) *μ_r = μ/μ_o* It has no unit and dimensions ## Derive the relation between magnetic field intensity (H) and magnetization (M) for a magnetic material placed in magnetizing field. 1. Consider a magnetic material (rod) placed in a magnetising field ( solenoid within turns per unit length and carrying current *I*) 2. The magnetic field inside the solenoid is given by, *B_o = μ_onI* ---> (1) ---> μ_o = permeability of free space 3. The magnetic field inside the rod is given as, *B_m = μ_oM* ---> (2)* ---> M = magnetization of material 4. The net magnetic field inside the rod is given by *B = B_o + B_m* ---> (3) *B = μ_onI + μ_oM* *B = μ_oH + μ_oM* ---> Where *H = nI* = magnetic intensity *B = μ_o(H+M)* ⇒ *H = B/μ_o - M* ---> (4) Eq? (4) shows that the magnetic field (B) induced in the material depends on magnetic field intensity (H) and magnetization (M)II ## Establish the relation between permeability and susceptibility of substance. Also express permeability in terms of relative permeability. 1. When magnetic material is placed in a magnetising field for its magnetization, the field inside the magnetic material is the resultant of the magnetising field *B_o* and the induced er field *B_m* *B = B_o + B_m* 2. Since *B_o = μ_oH* and *B_m = μ_oM* *B = μ_o(H+M)* *B = μ_o(H + XH)* ---> (M=XH) ---> *X* = magnetic susceptibility ... *B = μ_o(1+ X)H* μ_oH = μ (1+x) H ---> (B = μH) *μ = μ_o(1+x)* *μ = μ_oμ_r* ---> (μ_r = 1+x) *μ* = magnetic permeability *μ_r* = relative permeability This is the required relation. ## Magnetic properties of materials: ### Diamagnetism: Substances which are weakly repelled by a magnet are called as diamagnetic substances. e.g. copper, gold metal, bismuth, lead, silicon, glass, Water, wood, plastics. **Properties:** 1. If a thin rod of a diamagnetic material is freely Suspended in external uniform magnetic field, it comes to rest with its length perpendicular to the direction of the field. 2. When dia magnetic material is placed in non-uniform magnetic field, it moves from Stronger to weaker part of the field. 3. If diamagnetic liquid is filled in U-tube and one arm of U tube is filled placed in an external magnetic field the liquid is pushed in the arm which is outside the field. 4. When diamagnetic material is placed in extemal magnetic field, the induced magnetic field inside the material repels the magnetic lines of force resulting in reduction in magnetic field inside the material. 5. The magnetic susceptibility of diamagnetic material is small and negative... 6. In absence of extemal magnetic field the net magnetic moment of diamagnetic substance is zero. ### Paramagnetism : Substances which are weakly attracted by a magnet are called paramagnetic substances - e.g. magnesium, lithium, molybdenum, tantalum, MnSO₄, H₂O, oxygen.  a) Absence of external mag-field b) Weak external mag. field c) Strong extenal mag. field Consider a paramagnetic substance placed in external magnetic field **Case 1** a) In the absence of external magnetic field, the dipole moments of an atoms are randomly oriented and hence the net dipole moment of the substance is Zero. (fig. a) b) When weak field is applied some of the dipole moments tries to align with field (b) c) When paramagnetic substance is placed in strong magnetic field at low temperature, most of the magnetic dipoles align themselves in the direction of the applied field. i-e direction corresponding to less potential energy. As soon as the external magnetic field is removed the atomic magnets again get randomly oriented and substance loses its magnetism. **Properties:** i) When para magnetic material is placed in non-unifor magnetic field, it tends to move itself from weaker region to stronger region 2) If a magnetic field is applied to paramagnetic liquid in one arm of U-tube, the liquid levels rises in that am. 3) The net magnetic dipole moment is reduced to zero in the absence of magnetic field. 4) The magnetic susceptibility of paramagnetic material is small and positive. ## Curie law for paramagnetic material : Magnetization *M* in a paramagnetic material is directy proportional to applied magnetic field *B* and inversely proportional to absolute temperature *T* of material. This is known as Curie law. *M α B/T* ... *M = CB/T* ---> *C* = curie constant **Relation between magnetic susceptibility & temperature:** We have *M = CB/T* but *B = μ_oH* ... *M = Cμ_oH/T* *M/H = Cμ_o/T* ... *χ = Cμ_o/T* ---> *χ* = *M/H* *χ = μr - 1 = Cμ_o/T* ---> *χ* = *μr - 1* ... *χ α 1/T* ## Ferromagnetism : Substances which are strongly attracted by magnetic field are called ferromagnetic materials. They have positive and large magnetic susceptibility. e.g. iron, cobalt, nickel- ## Domain theory : 1) In ferromagnetic materials there is a strong interaction called as exchange coupling between neighbouring magnetic dipole moments - 2) Due to this interaction, small regions are formed in which all the atoms have their magnetic moments aligned in the same direction. Such a region is called a domain and the common direction of magnetic moment is called domain axis. 3) The boundary between adjacent domain with different orientation of magnetic moment is called domain wall.  Unmagnetised ferromagnetic material with domain B Magnetised ferromagnetic material 4) In unmagnetized state, as the domain axes of different domains being randomly oriented, the net magnetic moment of the whole material is zero. 5) When an external magnetic field is applied, domain rotation occurs. In this process domains try to allign themselves along the direction of magnetic field. 6) When sufficiently high magnetic field is applied all the domains coalesce together to form a giant domain as shownin fig (b). 7) When strong external magnetic field is completely removed, the domain boundaries doer are not set back to original position resulting in the net magnetic moment remaining non-zero. and ferromagnetic material is said to retain magnetization. ## Effect of temperature: It is observed that as the temperature of ferromagnetic material is increased, the domain structure starts distorting because of the exchange coupling between neighbouring atoms weakens. At a certain temperature, the domain structure collapses totally and the material behaves like para-magnetic material. The temperature at which par ferromagnetic material transforms into a paramagnetic substance is called Curie temperature (T_c) of that material- The relation bet? magnetic susceptibility and temperature is given by, *χ = C/(T-T_c)* for T>T_c ---> C = curie-constant. ## Hysteresis : Explain the behaviour of ferromagnetic material in an external magnetic field with the help of hysteresis cycle. 1) Consider an unmagnetized ferromagnetic material in the form of rod placed inside a solenoid  On passing the current through solenoed magnetic field is generated which magnerises the nod. 2) The hysteresis curve represents the relation between magnetic induction *B* of a ferromagnetic material With magnetic intensity (H). It represents behaviour of material as it is taken through eycle of magnetization 3) At point *o*, in the graph the material is in non-magnetized state. 4) As the strength of magnetic intensity *H* is increased *B* also increases non-linearly. 5) Near point *a*, the magnetic field is at its maximum value which is Saturation magnetization condition of the rod. This represents the complere alignment and merging of domains. If *H* is increased further (by increasing the current flowing through the solenoid ) there is no increase in *B*. 6) At this stage if the current in the solenoid is reduced the earlier path of graph is not retraced. i-e. the domain structure is not recovered. Thus the process is not reversible. 7) Next current through the solenoid is decreased to zero Hence magnetic intensity *H* reduces to zero. at point *b*, when *H=0*, *B≠0* 8) The value of magnetic induction *B* left in material when the magnetising force is reduced to zero is called as retentivity or remanence. (at point *b*) This shows some domain alignment is retained even when *H= 0* 9) The current in the solenoid is now reversed and increased The domain axes start getting randomly onented wrt each other still *B* reduces to zero at certain value of *H* (at point *c*). This value of magnetizing force (H) is called coercivity of the material. Mag field required to demagnetired the material 10) As the reverse current in the solenoid is increased, *B* increases and again reaches a saturation point. (point *d* in graph) At this point further increase in *H* does not increase *B*. 11) The solenoid current is now reduced. *H* reduces resulting in reduction of *B* along path *de* This means domain structure is present but direction of magnetization is reveresed. 12) Further increase in the current gives the curve *efa* On reaching point *a*, one loop is complered. This loop is called as hysteresis loop and taking magnetic material through the loop once is called hysteresis cucle. What does the area inside the curve *B-H* (hysteresis curve) indicate ? Area with the *BH* loop represents energy dissipated per unit volume in the material. ## Electromagnets: Soft iron (μ> 1000) is highly suitable for making electromagnet. Electromagnet consists of a solenoid in which a soft iron is kept. Due to current flowing through the coil of the solenoid a strong magnetic field is produced along the axis of the solenoid. The iron core is strongly magnetised. When the current through the solenoid is switched off, the associated magnetic field effectively becomes Zero. The electromagner material should have low retentivity and low coercivity, and low hysteresis loss - Uses: Electromagnets are used in electric bells, loud Speakers, circuit breakers and also in research laboratories, cranes. ## Superconducting magnets are used in NMR (Nuclear magnetic resonance) ## Permanent magnets: Permanent magnets are prepared by using a hard ferromagnetic rod. Hard magnetic materials have high coercivity, high retentivity and large hysteresis loss. * When the current is switched on, magnetic freld of solenoid magnetises the rod. * due to the property of retentivity, the material remaing magnetised even after switching off the current through the solenoid. ## Magnetic sheilding: When a ferromagnetic material is kept in a uniform magnetic field, large number of magnetic lines crowd up inside the material leaving a few outside.  Magnetic sheilding For a closed structure of this material spherical shell of iron kept in magnetic field, very few lines of force pass through the enclosed space. Most of the lines will be crowded into the iron shell. This effect is known as magnetic sheilding. The instrument to be protected from magnetic field is completely surrounded by a soft ferromagnetic substance. Uses: 1) In spaceships 2) Some scientific experiments require the experiment to be protected from magnetic field. In a laboratory. A case made up of soft ferromagnetic material helps in sheilding the high. magnetic fields of magnet ## Derive an expression for the potential energy of bar magnet placed in uniform magnetic field.  Magnet kept in uniform magnetic field 1. Consider a bar magnet of moment *m* held at an angle *θ* with direction of a uniform magnetic field *B*. The magnitude of torque acting on dipole is, *T = m x B* *T = MB Sin θ* ---> (1) 2. Due to torque the bar magnet will undergo rotational motion. This torque tends to align the dipole in the direction of the field. Work has to be done in rotating the dipole against the action of torque. This workdone is stored as P.E of magnetic dipole. 3. If *dw* is amound of Workdone in rotating a dipole, then *U_m = ∫dw* = *∫(τ) dθ* = *∫MB Sin θ dθ* *U_m = - MB cos θ* ---> *∫Sin θ = - cos θ* **Special cases:** **Case 1** When *θ = 0*, cos *θ* = 1 ... *U_m = - MB* This is the position when *m* & *B* are parallel. and bar magnet possess minimum P. E. and it is in most Stable state. **Case 2:** When *θ = 180*, cos *180* = -1 ... *U_m = (-1) MB* = * - MB* This is the position when *m* & *B* are antiparallel and bar magnet possesses maximum P.E. and thus in most unstable State. **Case 3:** When *θ = 90°*, cos *90* = 0 → *U_m = 0* This is the position when bar magnet aligned perpendicular to the direction of magnetic field. P. E as a function of *θ* is shown:  *P. E Versus angular position of magnet* ## Derive an expression for the time period of angular acceleration of a bar magnet. • Consider a bar magnet of magnetic moment *m* and moment of inertia *I*, suspended freely using an extensible string. The restoring torque in the string acts opposite to deflecting torque. Thus the magnet rotates through an angle and gradually becomes stationary. • At equillibrium both torque balances, *T = I d²θ/dt²* ---> (1) ---> *d²θ/dt²* - angular acceleration of magnet. • Let the magnet be rotated in direction opposite to the direction in which torque is acting on magnet. The restoring torque is acting in opposite direction. *T = - MB sin θ* ---> (2) ---> *m*(2l) = *S_N* *B_oN* *M*- mag- dipole moment *m*- pole strength Comparing eq?s (1) and (2) *I d²θ/dt²* = - *MB sin θ* ---> (3) for small values of angular displacement *θ*, *sin θ = θ* … *I d²θ/dt²* = - *MB θ* … *d²θ/dt²* = - (*MB/I*) θ ---> (4) The differential eq? of linear SHM is given as, *d²θ/dt²* = - ω²θ ---> (5) comparing (4) and eq? (5) ω² = *MB/I* ω = √*MB/I* ---> (6) The time period of angular oscillations of bar magnet is, *T = 2π/ω* *T = 2π√I/MB*