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 DipolesZ 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. Rakesh Choubisa, BITS, Pilani

5. Dr. Rakesh Choubisa, BITS, Pilani
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. Dr. Rakesh Choubisa, BITS, Pilani
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 Objectrs
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. Rakesh Choubisa, BITS, Pilani

11. Dr. Rakesh Choubisa, BITS, Pilani
Problem 6.8 A long cylinder of radius R carries a magnetization Find 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. Dr. Rakesh Choubisa, BITS, Pilani
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. Dr. Rakesh Choubisa, BITS, Pilani
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. Dr. Rakesh Choubisa, BITS, Pilani
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 µ0 H is faithful.
Dr. Rakesh Choubisa, BITS, Pilani

17. Dr. Rakesh Choubisa, BITS, Pilani
Boundary Conditions
Dr. Rakesh Choubisa, 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. Dr. Rakesh Choubisa, BITS, Pilani
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. Dr. Rakesh Choubisa, BITS, Pilani
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. x y z
Dr. Rakesh Choubisa, BITS, Pilani