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Chapter-6 : Magnetic Fields in Matter

Chapter-6 : Magnetic Fields in Matter. Magnetization Dia-magnets, Paramagnets, Ferro-magnets At microscopic level, electrons in atom revolve around the nucleus as well have spin. This will form tiny current loop and hence dipole moment.

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Chapter-6 : Magnetic Fields in Matter

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  1. Chapter-6 : Magnetic Fields in Matter • Magnetization • Dia-magnets, Paramagnets, Ferro-magnets • At microscopic level, electrons in atom revolve around the nucleus as well have spin. This will form tiny current loop and hence dipole moment. • Ordinarily the dipole moments in matter cancel each other due to random orientation of atoms. Dr. Rakesh Choubisa, BITS, Pilani

  2. What happens when a material is subjected to external magnetic field. • Unlike electric field, magnetic dipoles aligns either along field (para-magnetism) or opposite to it (Diamagnetism) . • Some materials retain magnetic property even after switching off the magnetic field (Ferromagnetism). Dr. Rakesh Choubisa, BITS, Pilani

  3. Torque & Force on Magnetic Dipoles Z Z m m F θ θ I Y θ Y θ b a F X Dr. Rakesh Choubisa, BITS, Pilani

  4. Torque & Force on Magnetic Dipoles • In a uniform magnetic field; net force is zero. • While in non-uniform magnetic field, it is non-zero and can be expressed as (Pr. 6.3); Similar to the electrostatics case. Dr. RakeshChoubisa, BITS, Pilani

  5. Pr. 6.1: Calculate the torque exerted on the square loop due to the circular loop. • Torque will be Dr. Rakesh Choubisa, BITS, Pilani

  6. Effect of a magnetic field on atomic orbit • Electrons in atom revolves around nucleus and contribute steady current of period T. Z B R v e- m Dr. Rakesh Choubisa, BITS, Pilani

  7. Effect of a magnetic field on atomic orbit • Electron will speed up by the amount; • Hence change in magnetic dipole; Dipole always aligns opposite to filed Dr. Rakesh Choubisa, BITS, Pilani

  8. Magnetization • In the presence of external magnetic field; • We have a net polarization either • Along the field (Para-magnetism related to spin of unpaired electrons) • Opposite to the field (Dia-magnetism related to the orbital motion of electrons) • For this we define magnetization as magnetic dipoles per unit volume. Dr. Rakesh Choubisa, BITS, Pilani

  9. The Field of a Magnetized Object rs m dΓ’ Dr. Rakesh Choubisa, BITS, Pilani

  10. The Field of a Magnetized Object • Using product rule 7, we have; Volume bound current Jb Surface bound current Kb Dr. RakeshChoubisa, BITS, Pilani

  11. Problem 6.8 A long cylinder of radius R carries a magnetizationFind the magnetic field. Dr. Rakesh Choubisa, BITS, Pilani

  12. Physical interpretation of bound currents • When we have uniform magnetization, we have, on average, surface bound current due to cancellation of currents of all the internal sides of tiny current loops within the material. • In a non-uniform magnetization, we have also volume bound current due to net current from the internal tiny current loops. • In both cases; current is due to motion of bound charges attached to each atom. Dr. Rakesh Choubisa, BITS, Pilani

  13. The Auxiliary Field H • Ampere’s law in Magnetized Materials: • Total current through the material is (bound + free currents); • Using Ampere’s differential law; Dr. Rakesh Choubisa, BITS, Pilani Dr. Rakesh Choubisa, BITS, Pilani

  14. The Auxiliary Field H • We get differential form of Ampere’s law for H • Also, we have the integral form; Dr. Rakesh Choubisa, BITS, Pilani

  15. Pr. 6.12: An infinite long cylinder, of radius R, carries a frozen-in magnetization, parallel to the axis, , and there is no free current anywhere. Find the magnetic field. • It can also be solved using H. Dr. Rakesh Choubisa, BITS, Pilani

  16. A Deceptive Parallel • H and B are not like even if we have similar from of Ampere’s law. • As field is found by both curl and div. of vector, div. of B is always zero however div. of H in general is not zero (as, div. M is not zero) & hence both are not like. • However, when Div. of M is zero, the parallel between B and µ0H is faithful. Dr. Rakesh Choubisa, BITS, Pilani

  17. Boundary Conditions Dr. RakeshChoubisa, BITS, Pilani

  18. Linear and Nonlinear Media • When field is weak enough, we should write; • But customary it is written in terms of H; Magnetic susceptibility Dr. Rakesh Choubisa, BITS, Pilani Dr. Rakesh Choubisa, BITS, Pilani

  19. Linear and Nonlinear Media • Materials which obey this relation is called linear media • Where; Permeability of the material Dr. Rakesh Choubisa, BITS, Pilani Dr. Rakesh Choubisa, BITS, Pilani

  20. Pr. 6.17: A current I flows down a long straight wire, made of linear material with susceptibility , of radius a. Find the magnetic field, all bound currents and the net bound current flowing down the wire. The current I is distributed uniformly. Dr. Rakesh Choubisa, BITS, Pilani

  21. Pr. 6.25: A toy consists of donut-shaped permanent magnet. (a) If you put two back-to-back magnets on the rod. At what height (z) does the upper one float?(b) If you now add a third magnet, write the equation of motion for the floating condition for the upper magnets. y z x Dr. Rakesh Choubisa, BITS, Pilani

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