Christopher Crawford PHY 417 2015-01-16 §5.1: Magnetic Field Christopher Crawford PHY 417 2015-01-16

Outline Magnetic field and torque William Gilbert – field lines of permanent magnets Hans Christian Ørsted – magnetic field of a current Magnetostatics André-Marie Ampère – force on two straight wires Current element – continuity equation

Static force laws Electrostatics – Coulomb’s law Magnetostatics – Biot-Savart law Geometrical Gauss -> Ampere’s law U interpretation as boundary currents Statement in terms of boundary conditions Technique for calculating coils B.C.’s: Flux lines bounded by charge Flux lines continuous Flow sheets continuous (equipotentials) Flow sheets bounded by current

History of magnetism The magnetic force was known in antiquity Magnetism more predominant in nature but more difficult to quantify: Permanent magnets (magnetization), not electric currents No magnetic (point) charge (monopole) –> dipole effect (N,S poles) 1-d currents instead of 0-d charges –> can’t split a wire! Static electricity produced in the lab long before steady currents Timeline (from “A Ridiculous Brief History of Electricity and Magnetism”) 600 BC Thales of Miletus discovers lodestone’s attraction to iron 1200 AD Chinese use lodestone compass for navigation 1259 AD Petrus Peregrinus (Italy) discovers the same thing 1600 AD William Gilbert discovers that the Earth is a giant magnet 1742 AD Thomas LeSeur shows inverse cube law for magnets 1820 AD Hans Christian Ørsted discovers that current twists magnets Andre Marie Ampere shows that parallel currents attract/repel Jean-Baptiste Biot & Felix Savart show inverse square law

Early magnetism Wu Ching Tsung Yao (1040) Petrus Perigrinus (1269) https://www.tcd.ie/Physics/Magnetism/Guide/compass.php Wu Ching Tsung Yao (1040) First mention of the magnetic compass Petrus Perigrinus (1269) “Epostolia de Magnete” – rediscovered it William Gilbert (1600) “De Magnete” – the earth is a magnet René Descartes effluvia of "threaded parts”

Magnetic fields In magnetism it is more natural to start with the concept of “Magnetic field” than the actual force law! (dipole) Compass aligns with B-field Iron filings line up along magnetic field lines Magnetic field lines look like an electric dipole (in fact the magnetic dipole was discovered first!)

Iron Filings

Iron Filings

Difference between E, B dipoles Same as the differences between Flux and Flow! Charge = sources of flux Conservative flow [potential] Example: Amber (electric) Rub to charge 2 charges (+/–) “monopole fluids” Exerts force on charges Continuous field lines [flux] Rotational (source of flow?) Example: Lodestone (magnet) Always charged 2 poles (N/S) “inseparable dipole” Exerts torque on other magnets

First connection to currents: from Wittaker, “A history of the theories of Aether and Electricity”

The current connection There is no magnetic monopole – N/S cannot be separated The source of all magnetism is electric current A dipole is just a current loop Hans Christian Ørsted Current produces B-field Quantified by Biot & Savart

Hans Christian Ørsted: from Wittaker, “A history of the theories of Aether and Electricity”

Electrodynamics André-Marie Ampère showed that parallel currents attract and antiparallel currents repel The magnetic force is the basis for electromagnetic units

Ampère’s generalization Experimental laws: Force formula Problem The force between two current elements does NOT point along the line from the one to the other!

Peer criticism MAXWELL HEAVISIDE

Current elements Analogous to charge elements – different dimensions Relations between charge / current and different dimensions – analogy: multi-lane highway – current flux

Continuity equation Local conservation of charge Current I is a flux; current density J = flux density 4-vector current