Chapter 5 - Magnetism and Matter

Bar magnet has two poles North N and South S, like electric dipole has positive and negative charges. But unlike electric charges, magnetic poles cannot be isolated - you cannot get single north pole monopole. If you cut bar magnet into two, each piece again has N and S. Magnetic monopoles do not exist.

Magnetic field lines pattern: Outside magnet field lines emerge from N pole and enter S pole, inside magnet from S to N forming closed loops continuous. Unlike electric field lines which start on positive charge and end on negative charge and are not closed, magnetic field lines are always closed loops.

Properties: Field lines never intersect because at intersection two directions of B would exist impossible, density of lines represents strength crowded strong sparse weak, tangent at point gives B direction, closed loops continuous.

Magnetic dipole moment m of bar magnet = pole strength q_m × separation 2a vector from S to N. SI unit Am² (ampere meter²) or J/T. Direction S to N similar to electric dipole moment p from - to +. Bar magnet in uniform magnetic field experiences torque τ=m×B and potential energy U=-m·B same formulas as current loop.

Bar magnet as magnetic dipole length 2a magnetic moment m=q_m×2a S to N. Field at point distance r from center:

On axial line (along dipole axis): B_axial = μ0/4π × 2m/r³ for far distance r>>a direction along m (from S to N). At point on N side field away from N, on S side towards S but along m direction.

On equatorial line (perpendicular bisector): B_equatorial = μ0/4π × m/r³ opposite to m direction, magnitude half of axial. So B_axial=2×B_equatorial magnitude similar to electric dipole but constants different.

Comparison with electric dipole: Electric axial E=1/4πϵ0 2p/r³ along p, equatorial E=1/4πϵ0 p/r³ opposite p. Magnetic axial B=μ0/4π 2m/r³ along m, equatorial B=μ0/4π m/r³ opposite m. So formulas similar replacing 1/4πϵ0 with μ0/4π and p with m. Important difference: Electric field lines start + end - not closed, magnetic closed loops.

Torque on magnetic dipole in uniform B: τ=m×B magnitude mB sinθ tends to align m with B stable equilibrium θ=0 m parallel B. Potential energy U=-m·B=-mB cosθ minimum 0° stable maximum 180° unstable.

We have seen in last chapter current loop N turns area A current I has magnetic moment m=N I A vector normal by right hand rule curl fingers along current thumb gives m direction. This loop behaves exactly like bar magnet. Solenoid with many loops also behaves like bar magnet. Polarity rule: If looking at loop current anticlockwise, that face is N pole, clockwise face is S pole. Right hand rule: Curl fingers along current thumb gives N pole.

Gauss law for magnetism: Net magnetic flux through any closed surface is zero ∮ B·dA=0 because magnetic monopoles do not exist. Number of field lines entering closed surface equals number leaving, field lines closed loops. Compare Gauss law for electricity ∮ E·dA=Q/ε0 net electric flux through closed surface = enclosed charge/ε0 non-zero because electric charges exist monopole exists. But magnetic Gauss zero because no magnetic charge monopole. If you place bar magnet inside closed surface, flux entering = leaving net zero.

This law is one of Maxwell equations fundamental.

When magnetic material placed in external magnetic field, atomic dipoles align and material gets magnetized. Magnetization M defined as net magnetic moment per unit volume M=m_net/V. Unit A/m ampere per meter.

Magnetic intensity H defined as H = B/μ0 - M. Unit A/m. H is due to free currents conduction currents not bound currents magnetization. B is total magnetic field due to both free currents and bound currents magnetization. Relation B=μ0(H+M). Total field = μ0 times (free current field + magnetization).

Magnetic susceptibility χ = M/H dimensionless ratio magnetization per unit intensity how easily material magnetized. Magnetic permeability μ = B/H = μ0(1+χ) ratio B per H how easily field passes. Relative permeability μr=μ/μ0=1+χ dimensionless.

For vacuum M=0 so B=μ0 H, χ=0 μr=1. For diamagnetic χ small negative ~-1e-5, μr slightly less than 1 <1. For paramagnetic χ small positive ~1e-5 to 1e-3, μr slightly >1. For ferromagnetic χ huge positive 1e3 to 1e5, μr huge >>1.

Classification based on atomic structure:

Diamagnetic: Materials where atoms have paired electrons net magnetic moment zero without external field. When external B applied, orbital motion of electrons changes Lenz law induced moment opposite to B small negative magnetization. So diamagnetic materials are weakly repelled by magnet, χ small negative ~-1e-5 almost independent of temperature. Examples bismuth Bi, copper Cu, gold Au, silver Ag, water H2O, helium He, silicon Si, quartz. Superconductor is perfect diamagnet χ=-1 B=0 inside Meissner effect expels magnetic field completely. Diamagnetism is universal but weak masked by para ferro if unpaired electrons exist.

Paramagnetic: Materials where atoms have unpaired electrons small net magnetic moment random orientation without field net M=0 due to thermal agitation. When external B applied, moments partially align along B attracted weakly, χ small positive 1e-5 to 1e-3. Susceptibility decreases with temperature Curie law χ=C/T C Curie constant because thermal agitation randomizes alignment more at high T. Examples aluminium Al, sodium Na, calcium Ca, oxygen O2, copper chloride CuCl2.

Ferromagnetic: Materials where atoms have unpaired electrons strong interaction domains formed groups of atoms with aligned moments even without external field spontaneous magnetization domains randomly oriented net M=0. When external B applied, domains grow aligned along B strongly attracted χ huge positive 1e3 to 1e5 μr huge. Above Curie temperature Tc domains break ferromagnetic becomes paramagnetic. Examples iron Fe, cobalt Co, nickel Ni, gadolinium Gd. Curie temperatures Fe 770°C Co 1131°C Ni 358°C. Ferromagnetism explained by Weiss molecular field theory and quantum exchange interaction.

Ferromagnetic materials exhibit hysteresis memory effect. Hysteresis loop is B-H or M-H curve when H cycled.

Process: Starting H=0 M=0 domains random. Increase H M increases non-linear initially slowly then rapidly domains aligning then saturates at Ms saturation magnetization all domains aligned further H no increase.

Now decrease H to zero M does not go to zero retains some magnetization retentivity or remanence Mr residual magnetism at H=0. This is how permanent magnets made - material retains magnetism after external field removed.

To demagnetize need apply reverse H coercivity or coercive force Hc reverse H required to make M zero. Further increasing reverse H saturates opposite direction. Then increasing H forward again forms loop closed.

Loop area represents energy loss per cycle as heat when material repeatedly magnetized demagnetized AC. For transformer core energy loss undesirable want small area.

Soft iron: Small retentivity small Mr, small coercivity small Hc, narrow loop small area low energy loss easy to magnetize and demagnetize good for electromagnets transformer cores.

Hard steel: Large retentivity large Mr, large coercivity large Hc, wide loop large area high energy loss hard to demagnetize retains magnetization permanent magnet good for permanent magnets.

Earth behaves like bar magnet with magnetic field approximately dipole field. Magnetic north pole near geographic south pole, magnetic south pole near geographic north pole. Field lines emerge near geographic south magnetic north and enter near geographic north magnetic south. Field approximately dipole at center tilted about 11.5° from rotation axis.

Magnetic elements three quantities defining Earth's magnetic field at a place:

1. Declination: Angle between geographic north and magnetic north in horizontal plane east or west. Compass shows magnetic north not geographic north, correction needed for navigation. Declination varies place to place.

2. Inclination or Dip angle: Angle between horizontal and Earth's magnetic field B. At equator dip 0° field horizontal, at poles dip 90° field vertical, in northern hemisphere dip downward north pole down, southern hemisphere upward. Measured by dip needle compass free to rotate vertical plane.

3. Horizontal component BH: Earth's field B resolved horizontal BH = B cosδ and vertical BV = B sinδ where δ dip angle. BH maximum at equator where dip 0° BH=B, zero at poles where dip 90° BH=0. BH useful for experiments like tangent galvanometer BH provides field.

Magnetic meridian is vertical plane containing Earth's magnetic field B at a place and vertical, also contains magnetic north-south. Geographic meridian is vertical plane containing geographic north-south. Angle between them is declination.