5_magnetism.docx

**Basic properties of magnets** 1. **Attractive property** A magnet attracts small pieces of iron, cobalt, nickel, etc. 2. **Directive property** When a magnet is suspended or pivoted freely, it aligns itself in the geographical north-south direction. 3. **Like poles and unlike p...

**Basic properties of magnets** 1. **Attractive property** A magnet attracts small pieces of iron, cobalt, nickel, etc. 2. **Directive property** When a magnet is suspended or pivoted freely, it aligns itself in the geographical north-south direction. 3. **Like poles and unlike poles attract** 4. **Magnetic poles always exist in pairs** 5. **Magnetic induction** A magnet induces magnetism in a magnetic substance placed near it. This phenomenon is called magnetic induction **Some important definitions connected with magnetism** 1. **Magnetic field** The space around a magnet within which its influence can be experienced is called its magnetic field. 2. **Uniform magnetic field** A magnetic field in a region is said to be uniform if it has same magnitude and direction at all points of that region. 3. **Magnetic poles** These are the region of apparently concentrated magnetic strength in a magnet where the magnetic attraction is maximum. 4. **Magnetic axis** The line passing through the poles of a magnet is called the magnetic axis of the magnet. 5. **Magnetic equator** The line passing through the center of the magnet and at right angles to the magnetic axis is called the magnetic equator of the magnet. 6. **Magnetic length** The distance between the two poles of a magnet is called the magnetic length of the magnet. It is slight less than the geometrical length of the magnet. **Coulomb's law of magnetic force** This law states that the force of attraction or repulsion between two magnetic poles is directly proportional to the product of their pole strengths and inversely proportional to the square of the distance between them. If and are the pole strengths of the two magnetic poles which are distance **r** apart, then the force between them is given by Or Where **k** is proportionality constant which depends on the nature of the medium as well as on the system of units chosen. For SI units and for poles in vacuum, Where **μ~0~** is the permeability of free space and is equal to **4π × 10^-7^** henry/meter. If , then. Hence a unit magnetic pole may be defined as that pole which when placed in vacuum at a distance of one meter from an identical pole repels it with a force of **10^-7^** Newton. **Magnetic dipole** An arrangement of two equal and opposite magnetic poles separated by a small distance is called a magnetic dipole. **Magnetic dipole moment** The magnetic dipole moment of a magnetic dipole is defined as the product of its pole strength and magnetic length. It is a vector quantity, directed from S -- pole to N -- pole. The SI unit of magnetic dipole moment is **ampere metre^2^ (Am^2^)** or **joule per tesla (JT^-1^)**. **Magnetic field lines** A magnetic line of force may be defined as the path along which a unit north pole would tend to move if free to do so in a magnetic field. **Properties of lined of force** 1. Magnetic lines of force are closed curves which start in air from the N -- pole and end at the S -- pole and then return to the N -- pole through the interior of the magnet. 2. The lines of force never cross each other. If they do so, that would mean there are two directions of the magnetic field at the point of intersection, which is impossible. 3. They start from and end on the surface of the magnet normally. 4. The relative closeness of the lines of force gives a measure of the strength of the magnetic field which is maximum at the poles. 5. The lines of force have a tendency to contract length wise and expand side wise. This explains attraction between unlike poles and repulsion between like poles. **Magnetic field of a bar magnet at an axial point (end -- on position)** Let **NS** be a bar magnet of length 2l and of pole strength **q~m~**. Suppose the magnetic field is to be determined at a point P which lies on the axis of the magnet at a distance r from its center. Imagine a unit north pole placed at point **P**. Then from Coulomb's law of magnetic force, the force exerted by the N -- pole of strength **q~m~** on unit north pole will be , along Similarly, the force exerted by S -- pole on unit north pole is , along Therefore, the strength of the magnetic field at point P is **B~axial~** = Force experienced by a unit north -- pole at point P \] But **q~m~**. **2l = m,** is the magnetic dipole moment, so For a short bar magnet, **l \< \< r**, therefore, we have , along...(1) Clearly, the magnetic field at any axial point of magnetic dipole is in the same direction as that of its magnetic dipole moment i.e., from S -- pole to N -- pole, so we can write **Magnetic field of a bar magnet at an equatorial point (broadside -- on position)** Consider a bar magnet NS of length 2l and of pole strength **q~m~**. Suppose the magnetic field is to be determined at a point P lying on the equatorial line of the magnet NS at a distance r from its center. Imagine a unit north -- pole placed at point **P**. Then from Coulomb's law of magnetic force, the force exerted by the N -- pole of the magnet on unit north -- pole is , along **NP** Similarly, the force exerted by the S -- pole of the magnet on unit north pole is , along **PS** As the magnitudes of F~N~ and F~S~ are equal, so their vertical components get cancelled while the horizontal components add up along PR. Hence the magnetic field at the equatorial point P is B~eq~ = Net force on a unit N -- pole placed at point P Or Where **m = q~m~. 2l**, is the magnetic dipole moment. Again for a short magnet, **l \< \< r**, so we have , along **PR**...(2) Clearly, the magnetic field at any equatorial point of a magnetic dipole is in the direction opposite to that of its magnetic dipole moment i.e., from N -- pole to S -- pole. So we can write On comparing equations (1) and (2), we note that the magnetic field at a point at a certain distance on the axial line of a short magnet is twice of the same distance on its equatorial line. **Torque on a magnetic dipole in a uniform magnetic field** Consider a bar magnet NS of length 2l placed in a uniform magnetic field. Let q~m~ be the pole strength of its each pole. Let the magnetic axis of the bar magnet make an angle θ with the field. Force on N -- pole = q~m~ B; along Force on S -- pole q~m~ B; opposite to The force on the two poles are equal and opposite. They form a couple. Moment of couple or torque is given by **τ =** Force × perpendicular distance Or...(1) Where **m = q~m~ × 2l,** is the magnetic dipole moment of the bar magnet. In vector notation,...(2) The direction of the torque is given by the right hand screw rule if in Eq. (1), **B = 1, θ = 90°,** then **τ = m** Hence the magnetic dipole moment may be defined as the torque acting on a magnetic dipole perpendicular to a uniform magnetic field of unit strength. SI unit of magnetic moment. As Therefore SI unit of **= NmT^-1^ or JT^-1^ or Am^2^.** **Special cases** 1. When the magnet lies along the direction of the magnetic field, **θ =0°, sin θ = 0, τ = 0,** Thus the torque is minimum. 2. When the magnet lies perpendicular to the direction of the field, **θ =90°, sin θ = 1, τ = mB,** Thus the torque is maximum. **τ~max~ = mB** **Potential energy of a magnetic dipole** When a magnetic dipole is placed in a uniform magnetic field at angle θ with it, it experiences a torque **τ = mB sin θ** This torque tends to align the dipole in the direction of. If the dipole is rotated against the action of this torque, work has to be done. This work is stored as potential energy of the dipole. The work done in turning the dipole through a small angle **dθ** is **dW = τ dθ = mB sin θ dθ** If the dipole is rotated from an initial position **θ = θ~1~** to the final position **θ = θ~1~,** then the total work done will be This work done is stored as the potential energy U of the dipole. The potential energy of the dipole is zero when. So potential energy of the dipole in any orientation **θ** can be obtained by putting **θ~1~ = 90°** and **θ~2~ = θ** in the above equation. Or **Special cases** 1. When **θ =0°, U = -mB cos 0° = - mB.** Thus the potential energy of a dipole is minimum when is parallel to. In this state, the magnetic dipole is in stable equilibrium. 2. When **θ =90°, U = -mB cos 90° = 0.** 3. When **θ =180°, U = -mB cos 180° = + mB.** Thus the potential energy of a dipole is maximum when is anti parallel to. In this state, the magnetic dipole is in unstable equilibrium. **Current loop as a magnetic dipole** We know that the magnetic field produced at a large distance r from the center of a circular loop (of radius a) long its axis is given by Or...(1) Where I is the current in the loop and **A = πa^2^** is its area. On the other hand, the electric field of an electric dipole at an axial point lying far away from it is given by...(2) Where p is the electric dipole moment of the electric dipole. On comparing equation (1) and (2), we note that both B and E have same distance dependence. Moreover, they have same direction at any far away point, not just on the axis. This suggests that a circular current loop behaves as a magnetic dipole of magnetic moment, In vector notation, Thus the magnetic dipole moment of any current loop is equal to the product of the current and its loop area. Its direction is given by right hand thumb rule. If a current carrying coil consists of N turns, then The factor NI is called amperes turns of current loop. So, magnetic dipole moment of current loop = Ampere turns × loop area **Magnetic dipole moment of a revolving electron** According to Bohr model of hydrogen -- like atoms, negatively charged electron revolves around the positively charged nucleus. This uniform circular motion of the electron is equivalent to a current loop which possesses a magnetic dipole moment = IA. Consider an electron revolving anti clockwise around a nucleus in an orbit of radius r with speed ѵ and time period T. Equivalent current, Area of the current loop, **A = πr^2^** Therefore, the orbital magnetic moment (magnetic moment due to orbital motion) of the electron is Or...(1) Also, the angular momentum of the electron due to its orbital motion is **l = m~e~ ѵr**...(2) The direction of is normal to the plane of the electron orbital and in the upward direction, as shown in fig. Dividing equation (1) by (2), we get The above ratio is a constant called gyro magnetic ratio. Its value is **8.8 × 10^10^ C kg^-1^**. So Vectorially, The negative sing shows that the direction of is opposite to that of. According to Bohr's quantization condition, the angular momentum of an electron in any permissible orbit is integral multiple of **h/2π**, where h is Planck's constant i.e., , where n = 1, 2, 3,... Therefore This equation gives orbital magnetic moment of an electron revolving in n th orbit. **Bohr magneton** It is defined as the magnetic moment associated with an electron due to its orbital motion in the first orbit of hydrogen atom. It is the minimum value of which can be obtained by putting n = 1 in the above equation. Thus Bohr magneton is given by An electron also has spin angular momentum due to its spinning motion. The magnetic moment possessed by an electron due to its spinning motion is called intrinsic magnetic moment or spin magnetic moment. It is given by The total magnetic moment of the electron is the vector sum of these two moment a. It is given by **Gauss's law in magnetism** It states that the surface integral of a magnetic field over a closed surface is always zero, the net magnetic flux through a closed surface is zero. **Consequences of Gauss's law** 1. Gauss's law indicates that there are no sources or sinks of magnetic field inside a closed surface. Hence isolated magnetic poles (also called monopoles) do not exist. 2. The magnetic poles always exist as unlike pairs of equal strengths. 3. If a number of magnetic lines of force enter a closed surface, then an equal number of lines of force must leave that surface. **Earth's magnetism** 1. **Geographic axis** The straight line passing through the geographical north and south poles of the earth is called its geographic axis. It is the axis of rotation of the earth. 2. **Magnetic axis** The straight line passing through the magnetic north and south poles of the earth is called its magnetic axis. 3. **Magnetic equator** It is the great circle on the earth perpendicular to the magnetic axis. 4. **Magnetic meridian** The vertical plane passing through the magnetic axis of a freely suspended small magnet is called magnetic meridian. The earth's magnetic field acts in the direction of the magnetic meridian. 5. **Geographic meridian** The vertical plane passing through the geographic north and south poles is called geographic meridian. **Elements of earth's magnetic filed** The earth's magnetic field at a place can be completely described by three parameters which are called elements of earth's field. They are declination, dip and horizontal component of earth's magnetic field. 1. **Magnetic declination** The angle between the geographical meridian and the magnetic meridian at a place is called the magnetic declination (α) at that place. 2. **Angle of dip or magnetic inclination** The angle made by the earth's total magnetic field with the horizontal direction in the magnetic meridian is called angle of dip (δ) at any place. 3. **Horizontal component of earth's magnetic field** It is the component of the earth's total magnetic field in the horizontal direction in the magnetic meridian. **B~H~ = B cos δ** At the magnetic equator, **δ = 0°, B~H~ = B cos 0° = B** At the magnetic place, **δ = 90°, B~H~ = B cos 90° = 0** **Relations between elements of earth's magnetic field** If **δ** is the angle of dip at any place, then the horizontal and vertical components of earth's magnetic field at that place will be **B~H~ = B cos δ**...(1) And BV = B cos δ Or...(2) Also Or...(3) Equations (1), (2) and (3) are the different relations between the elements of earth's magnetic field. By knowing the three elements, we can determine the magnitude and direction of the earth's magnetic field at any place. **Neutral point** It is the point where the magnetic field due to a magnet is equal and opposite to the horizontal component of earth's magnetic field. The resultant magnetic field at the neutral point is zero. If a compass needle is placed at such a point, it can stay in any position. 1. **Magnet placed in the magnetic meridian with its north -- pole pointing north** Fig. shows the magnetic lines of force of a bar magnet placed in the magnetic meridian with its north -- pole pointing towards the geographic north of the earth. The fields due to the magnet and the earth are in same directions at points on the axial line and are in opposite directions at points on the equatorial line. The two neutral points **P** and **Q** lie on the equatorial line. Let **r** = distance of each neutral point from the center of the magnet **2l** = length of the magnet **m** = dipole moment of the magnet Then magnetic field strength at each neutral point is For a short magnet, **l \< \< r**, therefore, At the neutral point, the field of the magnet is balanced by the horizontal component **B~H~** of the earth's magnetic field so that Knowing **r** and **B~H~**, the value of the magnetic dipole moment m can be determined. 2. **Magnet placed in the magnetic meridian with its south -- pole pointing north** Fig. shows the magnetic lines of force of a bar magnet placed in the magnetic meridian with its south -- pole pointing towards the geographic north of the earth. Hence the fields due to the magnet and the earth are in the same direction at points on the equatorial line and are in opposite directions at points on the axial line of the magnet. So the resultant field is weaker at axial points and is stronger at equatorial points. In this case the two neutral points P and Q lie on the axial line near the ends of the magnet. Suppose r be the distance of each neutral point from the center of the magnet. Let 2l be the length of the magnet. Then magnitude of the magnetic field at either of the neutral points will be For a short magnet, **l \< \< r**, therefore Again, at the neutral point, the field of the magnet is balanced by the horizontal component **B~H~** of the earth's magnetic field, so we have Knowing the values of **r** and **B~H~**, the magnetic dipole moment **m** of the magnet can be determined. **Magnetizing field** The magnetic field that exists in vacuum and induces magnetism is called magnetizing field. Consider a toroidal solenoid carrying current I and placed in vacuum. If the solenoid has n turns per unit length, then the magnetic field set up in the solenoid is given by **Magnetic induction** Suppose the toroidal solenoid is wound round a ring of magnetic material. Under the influence of field the magnetic moments of the atomic current loops of the magnetic material tend to align themselves with or against the magnetizing field. This gives rise to a net current on the surface of the material and is called magnetization surface current **I~M~**. This current induces magnetic field inside the material which is given by The total magnetic field inside a magnetic material is the sum of the external magnetizing field and the additional magnetic field produced due to magnetization of the material and is called magnetic induction. **Magnetizing field intensity** The ability of magnetizing field to magnetise a material medium is expressed by a vector , called magnetizing field intensity or magnetic intensity. Its magnitude may be defined as the number of ampere -- turns (**n I**) flowing round the unit length of the solenoid required to produce the given magnetizing field. Thus **H = n I** or The dimensions of magnetic intensity are \[**L^-1^ A**\]. Its SI unit is ampere meter^-1^ (**A m^-1^**) which is equivalent to **N m^-2^ T^-1^** or **J m^-1^ W b^-1^**. **Intensity of magnetization** The magnetic moment developed per unit volume of a material when placed in magnetizing field is of magnetization or simply magnetization. Thus If I~M~ is the surface magnetization current set up in a solenoid of cross -- sectional area A and having n turns per unit length, then magnetic moment developed per unit length of the solenoid is **n I~M~ A**. Therefore, magnetic moment developed per unit volume or the magnetization is given by Hence Again, consider a bar of magnetic material having cross -- sectional area ***a*** and length ***2l***. Its volume is ***V = a × 2l*** Suppose the bar develops pole strength ***q~m~*** when placed in a magnetizing field, then its magnetic moment, ***m = q~m~ × 2l*** Hence intensity of magnetization may also be defined as the pole strength developed per unit cross -- section area of a material. As the total magnetic field or the magnetic induction inside a magnetic material is the resultant of the magnetizing field and produced due to the magnetization of the material, therefore, Or Clearly, both **H** and **M** have the same units, namely **A m^-1^**. **Magnetic permeability** The magnetic permeability of a material may be defined as the ratio of its magnetic induction **B** to the magnetic intensity **H**. SI unit of **μ** Is tesla meter ampere^-1^ or T m A^-1^ **Relative permeability** It is defined as the ratio of the permeability of the medium to the permeability of free space. Thus, For vacuum **μ~r~ = 1**, for air it is **1.0000004** and for iron, the value of may exceed **1000**. **Magnetic susceptibility** It is defined as the ratio of the intensity of magnetization **M** to the magnetizing field intensity **H**. it is denoted by **X~m~**. It has no units. **Relation between magnetic permeability and magnetic susceptibility** If a linear magnetic material subjected to the action of a magnetizing field intensity **H**, develops magnetization **M** of magnetic induction **B**; then But **B = μ H** Or or Or or. **Classification of magnetic materials** 1. **Diamagnetic substances:** Diamagnetic substances are those which develop feeble magnetization in the opposite direction of the magnetizing field. Such substances are feebly repelled by magnets and tend to move from stronger to weaker parts of a magnetic field. Examples: Bismuth, copper, lead, zinc, tin, gold, silicon, nitrogen (at STP), water, sodium chloride, etc. 2. **Paramagnetic substances:** Paramagnetic substances are those which develop feeble magnetization in the direction of the magnetizing field. Such substances are feebly attracted by magnets and tend to move from weaker to stronger parts of a magnetic field. Examples: Manganese, aluminium, chromium, platinum, sodium, copper chloride, oxygen (at STP), etc. 3. **Ferromagnetic substances:** Ferromagnetic substances are those which develop strong magnetization in the direction of the magnetizing field. They are strongly attracted by magnets and tend to move from weaker to stronger parts of a magnetic field. Examples: Iron, cobalt, nickel, gadolinium and alloys like alnico. **Curie's law** From experiments, it is found that the intensity of magnetization (M) of a paramagnetic material is \(i) directly proportional to the magnetizing field intensity H, because the latter tends to align the atomic dipole moments. \(ii) inversely proportional to the absolute temperature T, because the latter tends to oppose the alignment of the atomic dipole moments. Therefore at low H/T values. We have M ∝ Or M = C. Or = or χ~m~ = Here C is curie constant and χ~m~ is the susceptibility of the material. The above relation is called Curie's law. This law states that far away from saturation, the susceptibility of a paramagnetic material is inversely proportional to the absolute temperature. The graph shows the variation of intensity of magnetization M as a function of H/T. Beyond the saturation value M~s~, Curie's law is not valid. The temperature at which a ferromagnetic substance becomes paramagnetic is called Curie temperature or Curie point T~c~. **Modified Curie's law for ferromagnetic substances** This law states that the susceptibility of a ferromagnetic substance above its Curie temperature is inversely proportional to the excess of temperature above the Curie temperature. χ~m~ = where C' is a constant. This is modified Curie's law for a ferromagnetic material above the Curie temperature. It is also known as Curie-Weiss law. **Comparative study of the properties of dia-, para- and ferromagnetic substances** **Property** **Diamagnetic substances** **Paramagnetic substances** **Ferromagnetic substances** ---------------------------------------- ---------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------- **Effect of magnets** They are feebly repelled by magnets. They are feebly attracted by magnets. They are strongly attracted by magnets. **In external magnetic field** Acquire feeble magnetization in the opposite direction of the magnetizing field. Acquire feeble magnetization in the direction of the magnetizing field. Acquire strong magnetization in the direction of the magnetizing field. **In a non-uniform magnetic field** Tend to move slowly from stronger to weaker parts of the field. Tend to move slowly from weaker to stronger parts of the field. Tend to move quickly from weaker to stronger parts of the field. **In a uniform magnetic field** A freely suspended diamagnetic rod aligns itself perpendicular to the field. A freely suspended paramagnetic rod aligns itself parallel to the field. A freely suspended ferromagnetic rod aligns itself parallel to the field. **Susceptibility value (χ~m~)** Susceptibility is small and negative. -1 ≤ χ~m~ \< 0 Susceptibility is small and positive. -1 ≤ χ~m~ \< ε, where ε is a small number. Susceptibility is very large and positive. χ~m~ \> 1000 **Relative permeability value (μ~r~)** Slightly less than 1. 0 ≤ μ~r~ \< 1 Slightly greater than 1. 1 \< μ~r~ \< 1 + ε Of the order of thousands μ~r~ \> 1000 **Permeability value (μ~r~)** μ \< μ~0~ μ \> μ~0~ μ \>\> μ~0~ **Effect of temperature** Susceptibility is independent of temperature. Susceptibility varies inversely as temperature: χ~m~ ∝ Susceptibility decreases with temperature in a complex manner. χ~m~ ∝ (T \> T~c~) **Removal of magnetizing field** Magnetization lasts as long as the magnetizing field is applied. As soon as the magnetizing field is removed, magnetization is lost. Magnetization is retained even after the magnetizing field is removed. **Variation of M with H** M changes linearly with H M changes linearly with H and attains saturation at low temperature and in very strong fields. M changes with H non- linearly and ultimately attains saturation. **Hysteresis effect** B-vector shows no hysteresis B-vector shows no hysteresis B- vector shows hysteresis **Physical state of the material** Solid, liquid or gas Solid, liquid or gas Normally solids only **Examples** Bi, Cu, Pb, Si, N~2~ (at STP), H~2~O, NaCl Al, Na, Ca, O~2~ (at STP), CuCl~2~ Fe, Ni, Co, Gd, Fe~2~O~3~, Alnico