Ampere’s Law. General Statement Magnetic fields add as vectors, currents – as scalars.

0

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Circulation of around a closed loop is times

the total current through the surface bounded by the loop

(2 )2

Id l B dl B dl r I

r

B

B

Ampere’s Law

0 01 2 1 2

1 2

( ) ( ) ( ) 02 2

b d

a c

I Id l B dl B dl B dl r r

r r

B

General Statement

0 (Ampere's Law)encld l I B

Magnetic fields add as vectors, currents – as scalars

Just as with the integral form of Gauss’s law, the integral form of Ampere’s law is powerful to use in symmetric situations

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2)2( :1path For

irestraight w a inside and around field Magnetic

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rIrB

r

IBIrB

Magnetic Field of a Solenoid

0 0

Wire wound around a long cylinder

produces uniform longitudinal field in

the interior and almost no field outside

For the path in an ideal solenoid:

( turns of the coil per unit length

BL nIL B nI

n

)

Field of a toroidal solenoid

wireof loops for total 2

)2(

:insidepath For the

zero iscurrent total theoutside,path any For

: toroida of field Magnetic

0

0

Nr

NIB

NIrB

0

0

The field is parallel to the plane

(still perpendicular to the current)

For the path: 2

for current per unit length2

Independent of distance from the plane

just as the electric field of

s

ss

Bl J l

JB J

the charged sheet

Magnetic Field of a Sheet of Current

The field of a magnetic “capacitor”

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0R s

P s

B J

B B

Magnetic materials

When materials are placed in a magnetic field, they get

magnetized.

In majority of materials, the magnetic effects are small. Some however show strong responses.

The small magnetism is of two kinds:

• Diamagnetics are repelled from magnetic fields

• Paramagnetics are attracted towards magnetic fields

This is unlike the electric effect in matter, which always causes dielectrics to be attracted.

The Bohr Magnetron

2

Magnetic effects have to do with microscopic currents

(magnetic moments) at the atomic level such as the

orbital motion of electrons:

Current 2

Magnetic moment ( ) ( )2 2

The ang

e evI

T re e

μ I r mvr Lm m

-34

24B

ular momentum is

; integer number2

h=6.626 10 Planck's constant

Fundamental unit of magnetic moment

= Bohr magnetron2 2 4

9.274 10 /

quantized

hL n n

J s

e h eh

m m

J T

spin B

There is also magnetic moment associated with

electron spin: =4

eh

m

0

Magnetization of a substance is its magnetic moment per unit volume

(similar to polarization in case of dielectrics in electric fields)

Total magnetic field at a point is a sum

total

V

M

M

B B 0

0 m 0All equations can be adapted by replacing K

Small magnetic effects are linear:

1

0 for diamagneticsMagnetic susceptibility

0 for paramagnetics

m mK

M

Magnetization

• Diamagnetism occurs in substances where magnetic moments inside atoms all cancel out, the net magnetic moment of the atom is zero. The induced magnetic moment is directed opposite to the applied field. Diamagnetism is weakly dependent on T.

• Diamagnetic (induced atomic moment) effect is overcome in paramagnetic materials, whose atoms have uncompensated magnetic moments. These moments align with the applied field to enhance the latter. Temperature T wants to destroy alignment, hence a strong (1/T) dependence.

Magnetic effects are a completely quantum-mechanical phenomenon, although some classical physics arguments can be made.

BM=C Curie's Law

T

Example: Magnetic dipoles in a paramagnetic material

Nitric oxide (NO) is a paramagnetic compound. Its molecules have maximum magneticmoment of ~ . In a magnetic field B=1.5 Tesla, compare the interaction energy of themagnetic moments with the field to the average translational kinetic energy of the moleculesat T=300 K.

23 5max

21

1.4 10 8.7 10

36.2 10 0.039

2

BU B J eV

K kT J eV

Ferromagnetism

Alignment of magnetic domains in applied field

• In ferromagnetic materials, in addition to atoms having uncompensated magnetic moments, these moments strongly interact between

themselves.

• Strongly nonlinear behavior with remnant

magnetization left when the applied field is lifted.

Permeability Km is much larger, ~1,000 to 100,000

Hysteresis and Permanent Magnets

Magnetization value depends on the “history” of applied magnetic field

Magnetization curve for soft iron showing

hysteresis

Example: A ferromagnetic materialA permanent magnet is made of a ferromagnetic material with a M~106 A/mThe magnet is in the shape of a cube of side 2 cm. Find magnetic dipole moment of a magnet. Estimate the magnetic field at a point 10 cm away on the axis

2

303

8

~ 10 102

total

total

MV A m

B T Gx

Experiments leading to Faraday’s Law

Electromagnetic Induction – Time-varying magnetic field creates electric field

Changing Magnetic Flux

No current in the electromagnet – B=0 - galvanometer shows no current.

When magnet is turned on – momentarily current appears as B increases.

When B reaches steady value – current disappears no matterhow strong B field is.

If we squeeze the coil as to change its area – current appearsbut only while we are deforming the coil.

If we rotate the coil, current appears but only while we arerotating it.

If we start displacing the coil out of the magnetic field – current appears while the coil is in motion.

If we decrease/increase the number of loops in the coil – current appears during winding/unwinding of the turns.

If we turn off the magnet – current appears while the magnetic field is being disappearing

The faster we carry out all those changes- the greater the current is.

Faraday’s Law quantified

effect theproduce will

flux magnetic changing Anything

cos

coil loop-Nan for

coil loop-single afor

BA

dt

dN

dt

d

B

B

B