Sources of Magnetic Fields Moving point charge: Bits of current: I also Permeability constant Biot-Savart Law The magnetic field “circulates” around the wire. http://falstad.com/vector3dm/

(or for L>>R)

Ampere’s Law in Magnetostatics Biot-Savart’s Law can be used to derive another relation: Ampere’s Law The path integral of the dot product of magnetic field and unit vector along a closed loop, Amperian loop, is proportional to the net current encircled by the loop, Choosing a direction of integration. A current is positive if it flows along the RHR normal direction of the Amperian loop, as defined by the direction of integration.

Example: Magnetic field of a long wire outside the wire inside the wire

Example: A Non-Uniform Current Distribution Long, hollow cylindrical current of current density: Insider the cylinder, the total current encircled by the Amperian loop is

Ampere’s Law applied to a solenoid // to axis Long solenoid (a<<L): B inside solenoid B outside solenoid nearly zero (not very close to the ends or wires) Ampere’s Law: n windings per unit length

Limitations of Ampere’s Law q -q Not enough symmetry Ampere’s Law needs correction!

Warm-up quiz Three currents I1, I2, and I3 are directed perpendicular to the plane of this page as shown. The value of the Ampere’s Law line integral of B∙ dl counterclockwise around the circular path is +0I1. What’s the currents in I2 and I3? a. I2=0, I3 can be any value b. I2=0, I3 can only be zero c. I2=I1, I3 can be any value d. I2=2I1, I3 can be any value e. I2=0.5I1, I3 can be any value I1 I3 I2

Dipole Moments in Applied Fields Magnetic Dipole Electric dipole External fields tend to align dipoles. B increases at center E decreases at center

Magnetization and “Bound” Current in Matter Strong externally applied field Bapp aligns the magnetic moments in matter.  Magnetization Ampere: Aligned magnetic moments in magnetized matter arise due to microscopic current loops inside the material.  Bound current magnetic moment dµ due to Amperian current di current /length Equivalent to a solenoid of nI=M

Magnetism Exhibited by Materials Diamagnetism: (small) magnetic moment opposite to the external magnetic field Bapp is induced Any material – but shows only if non-paramagnetic repelled from region of large B Paramagnetism: magnetic moment of individual atoms become aligned parallel to the applied magnetic field Bapp transition element, rare earth, ... attracted toward region of large B Ferromagnetism: magnetic moment of individual atoms are already (partially) aligned in some direction even if Bapp=0 Fe, Ni,... Antiferromagnetism: like ferromagnetism except that alternating moments are (partially) aligned opposite to each other (when B=0) Mn, Cr,...

Magnetic Susceptibility Magnetic susceptibility m paramagnet diamagnet Relative permeability Km permeability

Hysteresis for a Ferromagnet Lack of retraceability shown is called hysteresis. Memory in magnetic disk and tape Alignment of magnetic domains retained in rock (cf. lodestones) Area enclosed in hysteresis loop Energy loss per unit volume hard magnet: broad hysteresis loop (hard to demagnetize, large energy loss, highe memory) soft magnet: narrow hysteresis loop (easy to demagnetize,…)

Quiz A Three currents I, 2I, and 3I are directed perpendicular to the plane of this page as shown. What is the value of the Ampere’s Law line integral of B∙ dl clockwise around the circular path shown? a. 5μ0I b. −μ0I c. μ0I d. 6μ0I e. zero I 2I 3I

Quiz B Three currents I, 2I, and 3I are directed perpendicular to the plane of this page as shown. What is the value of the Ampere’s Law line integral of B∙ dl counterclockwise around the circular path shown? 2I a. 40I b. 20I c. 20I d. 60I e. zero 3I I

Quiz C Three currents I, 2I, and 3I are directed perpendicular to the plane of this page as shown. What is the value of the Ampere’s Law line integral of B∙ dl clockwise around the circular path shown? I a. 50I b. 0I c. 0I d. 60I e. zero 3I 2I