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

2. (or for L>>R)

3. 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.

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

5. 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

6. 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

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

8. 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

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

10. 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

11. 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,...

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

13. 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,…)

14. 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

15. 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

16. 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