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Magnetic properties of materials- I Magnetic properties of materials- I Magnetic dipoles Field of a magnetic dipole Force on a dipole in a non uniform field Induced magnetic dipole moment Circular current loop B p dB B r z 0 I 2 R R 2 2 z 2 3/ 2 Circular current loop B 0 I 2 R R 2 z If z >>R B 0 I R 2 z 2 2 3 0 I R B 3 2 z 2 2 3/ 2 0 B 3 2z Where is = 2 R I Known as Magnetic dipole moment • and B are in the same direction. • B is produced by . • In general, = I A, true for any shape of the loop. • A is a vector area, with direction assign by right hand rule. Magnetic dipole in an external Magnetic field Magnetic dipole experiences a torque Torque of a current loop B B0 zˆ z n h y e x g f Torque on a current loop z B B0 zˆ n h y e g a b x f Plane of the loop makes angle with the field n̂ B z e b y f Fef IbB sin( 90 ) ( xˆ ) n̂ B z h b y g Fgh IbB sin( 90 ) ( xˆ ) z n Forces on a current loop h e y g b x a f F fg IaB Fhe IaB ( y direction ) ( y direction ) y Forces on a current loop Fef IaB sin( 90 ) Fgh IaB sin( 90 ) ( x direction ) ( x direction ) F fg IaB Fhe IaB ( y direction ) ( y direction ) Contribute to torque r F n̂ b R IaB sin xˆ 2 b L IaB sin xˆ 2 z Fhe e y f Ffg Torque on a current loop IabB sin xˆ IAB sin xˆ NIA (nˆ B) Torque of a magnetic dipole IA nˆ B B Torque tends to rotate so that it lines up with B. Potential energy for the dipole • B makes an angle with the dipole U d U B U has smallest value when and B are parallel - B Largest when anti-parallel B Find the magnetic dipole moment of the circuit. iA i a b 2 2 2 Find the magnetic dipole moment of the loop. All sides have equal length and it carries a current I. z a a y x a Ia yˆ Ia zˆ 2 2 Direction is along the line y = z. Important to note • In general, potential energy (PE) can not be defined for a Magnetic field alone. • Since torque on the dipole depends upon the its position with respect to the field, PE can be defined for magnetic dipole in the field. • This PE corresponds to any change in the rotational configuration. Comparison with electric dipole P U = - B cos -PE cos Monopole does not exist Independent of choice of origin =xB Single charge exists Only when total charge vanishes. =PxE The field of a magnetic dipole Electric field of an electric dipole Magnetic field lines of a magnetic dipole Bar magnet can also be considered to be a magnetic dipole. Field lines do not start or end but continue through the interior of the magnets, forming close loops. Similarities Electric and magnetic dipole fields vary as r-3 when we are far from the dipoles. Force on a dipole in a nonuniform field In a uniform field total force on the dipole (electric as well as magnetic) is zero. There is only torque but no net motion In a non uniform field net force is not zero. Dipole may move. dF i2 ds B1 Net force will be downward on the loop. U 2 2B1 U 2 2 z B1z U 2 2 z B1z dU F21 dz d F21 2 z B1z dz dB1z F21 2 z dz If, 2z 0 dB1z 0 dz F21< 0 Force on the loop 2 due to 1 is attractive (downward). Induced magnetic dipole moments Induced magnetic dipole moments • An applied magnetic field can induce dipole moments I-Iind I+Iind Downward Induced magnetic dipole moment. Important points • In a non uniform magnetic field, permanent dipoles are attracted towards the source of the field • Induced dipoles are repelled from the source of the field. • Similar effect is observed in materials that lack permanent magnetic dipole moments. An idea about magnetic moments System Nucleus Magnetic dipole Moment (J/T) ~ 10-28 Electron ~ 10-23 Bar Magnet 5 Earth 1022 Atomic and nuclear magnetism • Magnetic properties depend upon magnetic properties of the individual atoms. • Magnetic material is consists of atomic dipoles dipole moment associated with circulation of electron. Calculation • We consider magnetic materials to be composed of a collection of atomic dipoles. • These dipoles might align when an external electric field is applied. • An electron circulating about the nucleus can be considered as a current loop of radius r and speed v. Calculation • Current in the loop = q qv i T 2r ev 2 evr iA r 2r 2 • Bohr’s model • Orbital magnetic dipole moment nh mvr 2 el l 2m Bohr’s magneton • This is a basic unit of atomic magnetic dipole moment ev 2 evr iA r 2r 2 nh mvr 2 eh 24 B 9.27 10 J / T 4m Nuclear magnetism Nuclear magnetic moments Orbital part Intrinsic part (spin)