Sources of Magnetic Fields

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

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

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

7. E q -q Limitations of Ampere’s Law 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. magnetic moment dµ due to Amperian current di current /length Equivalent to a solenoid of nI=M 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

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

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

13. 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 Hysteresis for a Ferromagnet Lack of retraceability shown is called hysteresis. • 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