Magnetism: Field Lines, Dipole Moment & Torque

Magnetism and Matter

• Chapter 5 discusses magnetism and matter in detail.

Magnetic Field Lines

• Imaginary lines represent the magnetic field.

Properties of Magnetic Field Lines

• Outside a magnet, magnetic field lines go from the North Pole to the South Pole, and through the magnet from South Pole to North Pole.
• Magnetic field lines form closed loops.
• A tangent drawn to a magnetic field line at any point indicates the direction of the magnetic field at that point.
• Magnetic field lines never intersect because if they did, it would imply two directions of the field at the intersection, which is impossible.
• Higher density of field lines means a stronger magnetic field B.
• In a uniform magnetic field, magnetic field lines are parallel and equally spaced.

Magnetic Lines of Force

• Magnetic lines of force form continuous closed loops, due to the dipole nature of magnets.

Magnetic Dipole Moment

• Denoted as m, it equals NIA
  • N = number of turns
  • I = current
  • A = area
• Magnetic dipole moment is the product of pole strength and the distance between the poles.
• The magnetic dipole moment m is a vector pointed from the South Pole to the North Pole of the magnet.
• Pole strength equals qm.
• 2a represents the length of the magnet.

Dipole Moment of Cut Bar Magnet

• For a bar magnet cut into two equal pieces transversely, pole strength is halved.
• Consequently, the dipole moment becomes m/2.

Electrostatic Analog

• Analogy between electrostatics and magnetism.
• Electrostatic dipole moment = p
• Magnetic dipole moment = m
• Equatorial field for a short dipole = -p/(4πε₀r³) in electrostatics, μ₀m/(4πr³) in magnetism.
• Axial field for a short dipole = 2p/(4πε₀r³) in electrostatics, μ₀2m/(4πr³) in magnetism.

Bar Magnet as Equivalent Solenoid

• A solenoid and a bar magnet produce similar magnetic effects and the bar magnet can be considered as a solenoid.

Dipole in a Uniform Magnetic Field

• A magnetic needle in a magnetic field experiences torque and oscillates in simple harmonic motion.
• Torque on the needle τ = mBsinθ = m x B.
• Time period of oscillations is T=2π√(I/mB),
  • I = moment of inertia.
• Potential energy of a magnet in a magnetic field: U = -m⋅B = -mBcosθ.

Bar Magnet in a Uniform Magnetic Field

• For a bar magnet with magnetic dipole moment M in a uniform magnetic field B, when its initial position is parallel to the field, the torque and force acting on it are both zero.
• In a non-uniform magnetic field, it experiences both force and torque, resulting in rotation and translation.
• In a uniform magnetic field the angle between B&m=0, so work done = change in PE =2mB and answer equals 0.09 J

Gauss's Law in Magnetism

• The surface integral of magnetic flux over a closed surface is zero.
• Magnetic monopoles do not exist.

Modification of Gauss's Law

• If magnetic monopoles existed, Gauss law of magnetism would change to § B⋅ds = μ₀qm.

Terms Related to Magnetism

• Magnetization is the process of bringing up magnetism in a material
• Intensity of magnetization (M) = m/V, with m as dipole moment and V as volume.
• Magnetic field intensity (B) or Magnetic flux density measures overall magnetic field within a substance (B = μH)
• Magnetic intensity (H) or Magnetizing field relates to external field applied. (H=nI)
• Magnetic susceptibility (χ) indicates how a material responds to a magnetic field.
  • Represents how easily a material can be magnetized.
  • χm=M/H.
• Magnetic permeability (μ) is the degree of magnetization a material obtains in response to an applied magnetic field.
  • μ=B/H

Relationships Between B, H, and Material Properties

• B = μ₀[H + M]
• μr = 1 + χm

Magnetic Properties of Materials

• Magnetic substances are divided into weakly and strongly magnetic substances.
• According to magnet properties magnetic materials are classified into three: Diamagnetic, Paramagnetic, and Ferromagnetic. Diamagnetic materials include Au, Ag, Cu, H₂O, Hg, and Zn. Paramagnetic materials include Al, Na, Mg, Li, Ti, Pt, and liquid oxygen. Ferromagnetic materials include Iron, Cobalt and Nickel

Diamagnetic Materials

• Weakly magnetized in the opposite direction of the applied field.
• Have a small and negative magnetic susceptibility (χm), typically -1 <= χm <= 0.
• Relative permeability (μr) is less than 1.
• Magnetic field lines are expelled from the material.
• Tend to move from stronger to weaker field regions in a non-uniform magnetic field.

Paramagnetic Materials

• Weakly magnetized in the direction of the applied field.
• Have a small and positive magnetic susceptibility (χm), typically 0 < χm< 1.
• Relative permeability (μr) is slightly greater than 1 but less than 2. 1 μr <2
• Low concentration of magnetic field lines inside the material.
• Move from weaker to stronger field regions in a non-uniform magnetic field.

Ferromagnetic Materials

• Strongly magnetized in the direction of the applied field.
• Have very high magnetic susceptibility (χm) and relative permeability.
• Magnetic field lines have a high concentration inside the material.
• Move from weaker to stronger field regions in a non-uniform magnetic field.
• Ferromagnetic materials consist of tiny regions known as domains which are spontaneously magnetized to saturation.

Superconductors

• Metals cooled to very low temperatures.
• Exhibits perfect conductivity and perfect diamagnetism.
• Repel magnetic fields
• Magnetic field lines are completely expelled.
• Magnetic susceptibility χ = -1 and relative permeability μr = 0. This is a property known as the Meissner Effect

Atoms

• The chapter deals with different types of atom models, structure of atom, their energy states.

Geiger-Marsden Experiment

• Involves scattering of alpha particles.
• 5. 5 MeV alpha-particles from a source impact a thin gold foil with a thickness of 2. 1 x 10⁻⁷ m.
• Alpha particles, after scattering, are absorbed by a rotatable zinc sulphide screen and observed using a microscope.
• Alpha particles produce light flashes; the number of alpha particles can be counted at different angles.

Observations from Scattering Experiment

• Most alpha particles pass straight through the foil with a very small deviation
• Approximately 0.14% of incident alpha-particles are scattered by more than 1°.
• About 1 in 8000 alpha particles are deflected through 90° or more.
• A very small fraction of incident alpha particles retraced its path.
• Plotted data of number of scattered particles vs scattering angle, n∝1/sin⁴(θ/2).

Conclusions from Scattering

• Most of the alpha particles passed through with no deflection, concluding that atoms are mostly empty..
• Few alpha particles suffer large angle deviations, and a very small fraction retraced its path.
• Since alpha particle itself is 7350 times heavier than the electron, it implies a concentrated positive charge is responsible for the large angle deviation.
• The electrostatic force of repulsion causes its large angle deviation.
• The central positive core is named as nucleus.
• Very rarely does an alpha particle travels head on towards the nucleus, so the the size of the nucleus is very small

Impact Parameter

• Impact parameter is the perpendicular distance of the initial velocity vector of the alpha particles from the centre of the nucleus.
• Smaller impact parameter means greater scattering.
• Larger impact parameter results in particles that get scattered through smaller angles.
• For b = 0, θ = 180°.

Distance of Closest Approach

• Involves alpha particles of mass m and velocity v approaching a nucleus of charge Ze.
• As alpha-particle approaches, its velocity decreases due to electrostatic repulsion from the nucleus.
• At distance d, the alpha particle stops momentarily and retraces its path.
• At this distance d, kinetic energy converts to electrostatic potential energy.
• The distance 'd' represents the distance of closest approach is given by d= 1/(4πε₀) *(2e)(Ze)/KE

Rutherford's Nuclear Model

• Based on the alpha particle scattering experiment in 1911.
• Model is also known as the planetary atom model. Atoms are seen as a sphere of size 10⁻¹⁰m.
• Most of the mass and positive charge is concentrated in the central core called nucleus.
• Electrons revolve around the nucleus, Centripetal force is provided by electrostatic force.
• The atom is electrically neutral.

Drawbacks of Rutherford's Model

• If electrons revolve around the nucleus in circular orbits, centripetal acceleration would mean energy loss.
• Accelerated electric charge should emit electromagnetic radiation, causing the electron to spiral into the nucleus.
• Rutherford’s model cannot explain how electrons are arranged, nor can it explain hydrogen spectrum.

Bohr Atom Model

• Electrons can only revolve through stationary orbit is when the angular momentum of the revolving electron is an integer multiple of h/2ㅠ.
• The angular momentum of the revolving electron is an integer multiple of h/2π. L = mvr = nh/2π, n = 1, 2, 3,…
• The electron will not radiate energy when in stationary orbits. If an electron jumps from an outer orbit of higher energy Ei to an inner orbit of lower energy Ef, radiation of energy takes place.
• Energy of the emitted photon: E = Ei - Ef. Frequency of the emitted photon: ν = (Ei - Ef)/ h

Line Spectra of Hydrogen Atom

• The line spectra of hydrogen atom forms when an electron transitions from an outer orbit ni of energy Ei to inner orbit nf of energy Ef, a photon of energy is emitted
• For Lyman series, nf=1, ni= 2, 3, 4, ... producing ultraviolet (UV) spectra.
• For Balmer series, nf=2, ni= 3, 4, 5, ..., producing the visible region.
• For Paschen series, nf=3, ni= 4, 5, 6..., producing infrared (IR) spectra.
• For Brackett series, nf=4, ni= 5, 6, 7, producing infrared (IR) spectra.

Radius of nth Orbit

• The radius of nth orbit is r = (n²h²ε₀) / (πme²), implying r ∝ n². The radii of the stationary orbit are in the ratio 1:4:9:16.

Velocity

• The valocity is equal to e²/(2ε₀nh) showing that V ∝ 1/n
• This equals -13.6/n² eV

Energy of nth Orbit

• En = -13.6/n² eV, where n is an integer relating to radius of orbit
• In the case of first orbit which is the ground state equals -13.6 ev
• Kinetic Energy equals -TE meaning K.E.=-E
  • The potential energy P.E= 2(TE)

De Broglie's Explanation of Bohr's Postulate

• Electrons are seen as particles, also as waves
• The explanation being in the form of L= mur = nh/(2π ) and the electrons in their permitted orbits behave in the form of a particle wave.