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

2. 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

3. 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

4. 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

5. Early magnetism Wu Ching Tsung Yao (1040) Petrus Perigrinus (1269)
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”

6. 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!)

7. Iron Filings

8. Iron Filings

9. 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

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

11. 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

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

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

14. 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!

15. Peer criticism
MAXWELL
HEAVISIDE

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

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