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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/. (or for L>>R ). Ampere’s Law in Magnetostatics.

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sources of magnetic fields

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/

ampere s law in magnetostatics
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.
example magnetic field of a long wire
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:

Example: A Non-Uniform Current Distribution

Insider the cylinder, the total current encircled by the Amperian loop is

ampere s law applied to a solenoid

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

limitations of ampere s law

E

q

-q

Limitations of Ampere’s Law

Not enough symmetry

Ampere’s Law needs correction!

slide8

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
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

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
magnetism exhibited by materials

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)
magnetic susceptibility

paramagnet

diamagnet

Relative permeability Km

permeability

Magnetic Susceptibility
  • Magnetic susceptibility m
hysteresis for a ferromagnet

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,…)
slide14

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

slide15

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

slide16

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