Magnetism: Fundamentals and History

• Magnetic phenomena are universal, permeating vast galaxies, tiny atoms, humans, and beasts.
• Earth's magnetism predates human evolution.
• The word "magnet" originates from Magnesia, a Greek island where magnetic ore deposits were found as early as 600 BC.
• Oersted, Ampere, Biot, and Savart discovered that moving charges or electric currents produce magnetic fields in the early 19th century.

Magnetism Basics

• Earth behaves as a magnet with a magnetic field pointing approximately from the geographic south to the north.
• A freely suspended bar magnet aligns in the north-south direction.
• The tip pointing to the geographic north is the north pole, and the tip pointing to the geographic south is the south pole.
• Like poles repel each other, while unlike poles attract.
• It's impossible to isolate a single magnetic pole (north or south).
• Breaking a bar magnet results in two smaller bar magnets, not isolated poles.
• Isolated magnetic north and south poles, called magnetic monopoles, are not known to exist.
• Magnets can be created from iron and its alloys.

Bar Magnets

• Iron filings sprinkled around a bar magnet align in a pattern resembling an electric dipole, with distinct north and south poles.
• The poles of a freely suspended bar magnet point towards the geographic north and south, respectively.
• A similar iron filing pattern is observed around a current-carrying solenoid.

Magnetic Field Lines

• Magnetic field lines are a visual representation of the magnetic field.
• Magnetic field lines form continuous closed loops, unlike electric dipole field lines that start on a positive charge and end on a negative charge or extend to infinity.
• The tangent to a field line at a point indicates the direction of the net magnetic field B at that point.
• A higher density of field lines indicates a stronger magnetic field.
• Magnetic field lines do not intersect because the direction of the magnetic field would be non-unique at the intersection.
• Plotting field lines can be done using a small compass needle.

Bar Magnets and Solenoids

• Magnetic field lines resemble those of a solenoid, suggesting a bar magnet may be thought of as a large number of circulating currents.
• Cutting a bar magnet in half is analogous to cutting a solenoid, resulting in two smaller solenoids.
• Moving a compass needle near a bar magnet and a current-carrying solenoid produces similar deflections.
• The axial field of a finite solenoid resembles that of a bar magnet at large distances.
• The magnitude of the field at point P due to the solenoid is expressed by B = (μ₀ / 4π) * (2m / r³).
• Magnetic moment of a bar magnet equals that of an equivalent solenoid.

Dipoles in Uniform Magnetic Fields

• The torque (τ) on a compass needle in a magnetic field is given by τ = m × B, or in magnitude, τ = mBsinθ.
• Magnetic potential energy (Um) is given by Um = -m ⋅ B = -mBcosθ.
• The zero of potential energy is typically set at θ = 90°.
• Potential energy is at a minimum (-mB) when θ = 0° (stable position) and maximum (+mB) when θ = 180° (unstable position).

Electrostatic Analogy

• Magnetic field equations are analogous to those for electric dipoles.
• By making certain replacements, such as E → B and p → m, magnetic field equations can be derived from electric field equations.
• The equatorial field (B₁) of a bar magnet (r >> l) is: B₁ = (μ₀ / 4π) * (m / r³).
• The axial field (B^) of a bar magnet (r >> l) is: B^ = (μ₀ / 4π) * (2m / r³).

The Dipole Analogy Table

• Electrostatics - Dipole moment 1/ε₀ * p, Equatorial Field for a short dipole -p/4πε₀³ , Axial Field for a short dipole 2p/4πε₀³ , External Field: torque p x E, External Field: Energy -p-E.
• Magnetism - Dipole moment μοm, Equatorial Field for a short dipole μο m / 4π r³, Axial Field for a short dipole μο 2m / 4π r³, External Field: torque m x B, External Field: Energy -m-B.

Guass's Law

• Gauss's law for magnetism states that the net magnetic flux through any closed surface is zero.
• This is represented mathematically as ΦB = Σ B⋅ΔS = 0 (summed over all area elements ΔS).
• The difference between Gauss's law for magnetism and electrostatics reflects the non-existence of isolated magnetic poles whereas they do exist in electric charge.

Magnetization and Magnetic Intensity

• Magnetization (M) is defined as the net magnetic moment per unit volume: M = mnet / V.
• A long solenoid with n turns per unit length and current I has a magnetic field in its interior given by Βο = μο nI.
• If the interior of the solenoid is filled with a material with non-zero magnetization, the net B field inside is B = Bo + Bm.
• The field contributed by the material core (Bm) is proportional to the magnetization M: Bm = μοΜ.
• Magnetic intensity (H) is defined as H = (B / μo) - M.
• The total magnetic field B can be written as Β = μο (Η + Μ).
• The influence of external factors on M is mathematically expressed as M = XH, where X is the magnetic susceptibility.
• X is small and positive for paramagnetic materials and small and negative for diamagnetic materials.
• Relative magnetic permeability (μ₁) is defined as μ₁= 1 + x.
• Magnetic permeability (μ) is defined as μ = μὅμ₁ = μο (1+x).
• The quantities χ, μ₁, and u are interrelated, and knowing one allows for the determination of the others.

Magnetic Properties of Materials

• Diamagnetism: Weakly repelled by magnetic fields. Susceptibility (χ) is negative and small (-1 ≤ x < 0). Relative permeability (μᵣ) is less than 1.
• Paramagnetism: Weakly attracted by magnetic fields. Susceptibility (χ) is positive and small (0 < x < ε). Relative permeability (μᵣ) is greater than 1 but only to a certain extent.
• Ferromagnetism: Strongly attracted by magnetic fields. Susceptibility (χ) is large and positive (x >> 1). Relative permeability (μᵣ) is much greater than 1 (μ >> 1).
• Diamagnetic substances tend to move from stronger to weaker parts of an external magnetic field.
• Field lines are repelled from a diamagnetic material.
• In superconductors, the field lines are completely expelled (Meissner effect).

Atomic Properties

• Electrons orbiting around a nucleus possess orbital angular momentum, creating an orbital magnetic moment.
• Diamagnetic substances: Atoms have with resultant magnetic moment of zero. They develop a net magnetic moment opposite to the applied field.
• Examples of diamagnetic materials: bismuth, copper, lead, silicon, nitrogen (at STP), water, and sodium chloride.

Paramagnetism

• Paramagnetic substances weakly magnetized when placed in an external magnetic field.
• They move from weak to strong magnetic field regions.
• Atoms, ions, or molecules possess permanent magnetic dipole moments and tend to align with external fields.
• Enhancement depends on field strength and temperature. Lowered temperatures or stronger fields increase magnetization.
• Common paramagnetic materials include aluminum, sodium, calcium, oxygen (at STP), and copper chloride.

Ferromagnetism

• Ferromagnetic substances strongly magnetized when placed in an external magnetic field.
• Individual atoms have dipole moments that align spontaneously in domains.
• Domains align with external fields and grow in size.
• In non-uniform fields, they move towards the high field region.
• Hard magnetic materials remain magnetized after external field removal
• Soft magnetic materials loose magnetization when external field is removed.
• Elements include: iron, cobalt, nickel, gadolinium. A high relative magnetic permeability of >1000!
• Loss of magnetisation depends on temperature