Matter and Magnetism

 1. Permanent magnets of different shapes may be prepared from iron, steel, nickel, cobalt, and their alloys.

2. A bar magnet or a magnetic dipole is the simplest type of a permanent magnet.

3. A bar magnet is characterized by:  Its the directional property,  Attracting magnetic materials towards it,  Including magnetism in other magnetic materials etc.

4. Magnetic monopoles are not found in nature.

5. The magnetic effects in magnetic material are due to atomic magnetic dipole in the materials. These dipoles result from the effective current loops of electrons in atomic orbits.

6. All magnetic phenomena can be explained in terms of circulating currents. A current loop of area A carrying a current I is equivalent to a magnetic dipole of dipole moment $ \vec{m} = I\vec{ A} $ along the axis of the loop. If there are N number of turns in a coil then $ \vec{m} = NI\vec{ A} $. SI unit of magnetic dipole moment is ampere-meter².

7. From the resemblance of magnetic field lines for a bar magnet and a solenoid, we may consider a bar magnet as a large number of circulating current like a solenoid. In fact, a bar magnet and a solenoid produce similar magnetic fields. The magnetic moment of a bar magnet is thus equal to the magnetic moment of an equivalent solenoid that produces the same magnetic field.

8. Magnetic field lines are a visual and intuitive realization of the magnetic field. A magnetic field is a smooth curve in a magnetic field, tangent to which at any point gives the direction of the magnetic field at that point.

9. In free space around a magnetic dipole the magnetic field lines start from N-pole and end at S-pole. However, inside the magnet, they travel from S-pole to N-pole. Thus, magnetic field lines of a magnetic or a solenoid form continuously closed curves.

10. The magnetic field lines do not intersect one another. It is so since the direction of the magnetic field would not be unique at the point of intersection.

11. The larger the number of magnetic field lines crossing per unit normal area in a given region, the stronger is the magnetic field B there.

12. The magnetic field B at the point on the axial line of a bar magnet (magnetic dipole) of dipole moment m at a distance r from the mid-point of magnet is given by \[\vec{B} = \frac{\mu_0}{4\pi}\frac{2\vec{m}r}{(r^2-l^2)^2}\]for short dipole or where r >> l, \[\vec{B} = \frac{\mu_0}{4\pi}\frac{2\vec{m}}{r^3}\]Direction of $ \vec{B}$ is same as of $\vec{m}$. 

13. The magnetic field B at a point on the equatorial line of a magnetic dipole of magnetic moment m at a distance r from the mid-point dipole is \[\vec{B} = -\frac{\mu_0}{4\pi}\frac{\vec{m}}{(r^+l^2)^(3/2)}\]and for short dipole or when r >> l, \[\vec{B} = -\frac{\mu_0}{4\pi}\frac{\vec{m}}{r^3}\]Here -ve sign means that direction of magnetic field is opposite to the direction of dipole moment.

14.Torque acting on a magnetic dipole of moment M placed in a uniform magnetic field B is given by\[\vec{\tau} = \vec{M} \times \vec{B}\]and \[{\tau} = MBsin(\theta)\],where $\theta $ is is the angle between the magnetic axis of dipole and the magnetic field. The torque tends to align the magnetic dipole along the direction of magnetic field.

15. The potential energy of a magnetic dipole placed at angle $\theta $ with the magnetic field B is \[U = -\vec{m}.\vec{B} = -mBcos\theta\]where we choose the zero of energy at the orientation when m is perpendicular to B. For $ \theta $ = 0⁰, potential energy of a magnetic dipole is -mB and it corresponds to stable equilibrium state of magnetic dipole in a magnetic field. However, for $ \theta = \pi $, U = mB and it corresponds to unstable equilibrium of magnetic dipole.

16. A magnetic dipole freely suspend in a uniform magnetic field B, if once twisted by a small angle $\theta$ and then released, executes simple harmonic oscillations. The time period of oscillation is given by \[T = 2\pi \sqrt{\frac{I}{mB}}\]Where I = moment of inertia of magnetic dipole about the suspension axis and m = magnetic dipole moment.

17. According to Gauss’ law for magnetism, “ the net magnetic flux through any closed surface is zero” i.e., \[\Phi_B = \oint \vec{B}.\vec{dS} = 0\]It is so because in magnetism isolated monopoles do not exist. There are no source/ sink of magnetic field B. the simplest magnetic element is a dipole or a current loop.

18. Our earth has a magnetic field of its own. The earth’s magnetic field resembles that of a (hypothetical ) giant magnetic dipole which is aligned making a small angle with the rotational axis of the earth. Its magnetic north pole N_m is near the geographic south pole S_g and its magnetic south pole S_m is near the geographic north pole N_g. the earth’s magnetic field may be approximated by a dipole with magnetic moment 8.0 x 10²² A-m².

19. The strength of earth’s magnetic field varies from place to place on the earth’s surface. The magnitude of the field is of the order of 4 x10^-5 T.

20. Magnetic element of a place are three quantities needed to specify the magnetic field of the earth at the given place. The three magnetic element are (i) the magnetic declination, the magnetic dip, and (iii) the horizontal component of earth’s magnetic field.

21. Magnetic declination (D) at a place are the angle which magnetic meridian at that place subtends from the geographic meridian. Effectively, it is the angle between the true geographic north and the north shown by a compass needle.

22. Magnetic dip $\delta $ or angle of inclination is the angle in which direction of earth magnetic field at a place subtends from the horizontal direction along the magnetic meridian.

23. If $B_E$ be the magnetic field of earth at a given place and $\delta $ be the magnetic dip then horizontal component of earth magnetic field is $ B_H = B_Ecos\delta $ and the vertical component of earth field is $B_V = B_Esin\delta $. \[B_E^2 = B_H^2+B_V^2 \implies B_E = \sqrt{B_H^2+B_V^2}\] \[tan\delta = \frac{B_V}{B_H}\]

23. Earth magnetic field is thought to arise due to electrical produced by convective motion of metallic fluids (consisting mostly of molten iron and nickel) in the outer core of the earth. It is known as the ‘dynamo effect’.

24. Magnetic equator is the axis, at all points of earth’s magnetic field is directed horizontally i.e. $ B_E = B_V $ and angle of dip, as well as vertical component of earth’s magnetic field $B_V $, have zero value.  

25. At magnetic poles of earth $\delta = \frac{\pi}{2}$, $B_H= 0$ and $B_V=B_H $ value of dip angle gradually increases as one goes from equatorial region towards the poles of earth.

26. At the magnetic poles a compass needle may point along any direction. However a dip needle will point straight down at the magnetic poles.

27. In free space if magnetic field at a given place be $ \vec{B_0 } $ then we define a term known as” magnetic intensity” H as \[\vec{H} = \frac{\vec{B_0}}{\mu_0}\] where $ \mu_0 $ is the magnetic permeability of free space.

28. When a magnetic material is placed in a magnetic field $ B_0 $ the field changes to B on account of magnetization of that material. the net magnetic moment developed in the given material per unit volume is known as “magnetization” (or intensity of magnetization) M of that material. Thus \[\large \vec{M} = \frac{\vec{m_{net}}}{V}\]SI unit of magnetization $\vec{M}$ is A-m^-1.

29. In the presence of a magnetic material, the magnetic field change from $ \vec{B_0} $ to $ \vec{B}$ where \[\large \vec{B} = \mu_r \vec{B_0}\]and $ \mu_r $ is known as relative magnetic permeability of given material and is a unitless and dimensionless quantity.

30.  It is observed that \[\large \vec{B} = \vec{B_0} + \vec{B_m} = \mu_0\vec{H} + \mu_0\vec{M} = \mu_0(\vec{H} + \vec{M})\]

31. Magnetic susceptibility of a magnetic material $ \chi $ is defined as per relation\[\large \vec{M} = \chi \vec{H}\]It is a measure of how a magnetic material responds to an external magnetic field. Magnetic susceptibility is a unitless and dimensionless quantity. It is found that  \[\large \mu_r=1+ \chi\]

32. \[\large \mu_r . \mu_0 = \mu\]is the absolute magnetic permeability of given material. Units and dimensions of $ \mu$ are same as of $ \mu_r $

33. Diamagnetic materials are those which experience a feeble force of repulsion when placed in a strong external magnetic field. Diamagnetic substances tend to move from stronger to weaker part of the external magnetic field. The field lines are repelled or expelled and the field inside a diamagnetic material is reduced. The individual atoms of a  diamagnetic material do not possess a permanent magnetic dipole moment of their own but a small dipole moment in the opposite direction is developed in them when placed in an external magnetic field. Bismuth, copper, lead, nitrogen, water, etc., are diamagnetic in nature. For diamagnetic material $ -1 \leq  \chi $ <0, $ 0 \leq \mu_r < 1  $ and $\mu < \mu_0 $. A superconductor is a perfect diamagnetic for which $\chi = 0, \mu_r = 0, \mu_0 = 0 $.

34. Paramagnetic materials are those which experience a weak force of attraction when placed in an external magnetic field. Paramagnetic substances are weakly magnetized when placed in an external magnetic field. Field lines are attracted and the field inside a paramagnetic material is increased. They have a tendency to move from weaker to stronger regions of magnetic field.  The individual atoms possess a permanent dipole moment and this dipole moment tries to align itself in the direction of external field B₀. Aluminum, sodium, calcium, oxygen, etc., are paramagnetic. For paramagnetic materials, $ \chi $ is small positive, $\mu_r $ is greater than 1. 

35. Ferromagnetic materials are those which are strongly attracted by an external magnetic field and which can themselves be magnetized. Iron, nickel, cobalt and some of their alloys are ferromagnetic. For ferromagnetic materials $\chi >> 1, \mu_r >>1, \mu >> \mu_0    $.

36. The individual atoms in a ferromagnetic material possess a permanent dipole moment. These atomic dipoles interact with one another so as to form domains. Ordinarily, the magnetization varies randomly from domain to domain and net magnetization is zero. Under the influence of an external magnetic field the domains are aligned accordingly and the sample acquires magnetization.

37. Ferromagnetic materials are said to be hard if magnetization persists even after the removal of external magnetic field. Ferromagnetic materials are called soft it magnetization disappears on removal of external field.

38. According to Curie’s law magnetization $\vec{M} $ of a paramagnetic material is directly proportional to applied magnetic field B₀ and inversely proportional to the absolute temperature T. Thus, \[\large \vec{M} = \frac{C\vec{B_0}}{T}\]where C is known as the Curie’s constant. In terms of susceptibility, we have $ \chi = C \frac{\mu_0}{T} $.

39. The ferromagnetic property of a material gradually decrease as the temperature is raised. Above a certain “temperature is transition” (also known as Curie temperature) a ferromagnetic material begins to behave as a paramagnetic substance. The susceptibility above the Curie’s temperature is described by: \[\large \chi = \frac{C}{T- T_C}\]

40. Relation between B and H in ferromagnetic material is complex and represented by a hysteresis curve. The word hysteresis means lagging behind of B w.r.t. H. 

41. The residual magnetization of a ferromagnetic substance undergoing an hysteresis cycle must be subjected in order to demagnetize it completely, is known as ‘coercive force’ or ‘coercivity’. 

42. During a complete magnetization cycle of a material some energy is dissipated, which appears as heat. Area of B-H hysteresis loop gives the energy dissipation per unit volume per cycle. Steel has a wide hysteresis loop but soft iron has a narrow hysteresis curve. 

43. The hysteresis curve allows us to select suitable materials for a magnet. Material for a permanent magnet should have high retentivity ,high coercivity and a high permeability. Steel is a favoured  choice for permanent magnet. Material for an electromagnet should  have high permeability, low retentivity and a narrow hysteresis curve soft iron is therefore preferred for making an electromagnet.