I

B

I

I

B

Magnetic field similar to a bar magnet

For a very long solenoid, the magnetic field can be considered to be confined to the region inside the coils.

Magnetic field from current loop

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2a

z

x

2a

x

z

a

zr

dB

dB

cos sinz xdB dB dB dB

x components cancel

2 2 2r a z

z >> a

cosz

r

a

r

i

B

m

At a point along the axis z >> a

032zB i A

z

current loop in xy-plane

m i A

magnetic dipole momentright hand screw rule

H

B

Diamagnetic material

m < 0 (small)

B = o (1+ m ) H

Permeability

= o (1+ m ) =slope of B-H line

Ideal magnetic material or paramagnetic material

m > 0 (small)

B = o r H = H

= constant = slope of B-H curve

B

H

L1L111.5 .5 : : MagnetizationMagnetizationIf H is large or substance strongly magnetic (e.g. ferromagnetic), as H increases, the magnetization M (and hence B) may increase nonlinearly:

Measure from thegraph

So r varies with H. Could also use “differential permeability”

High field region where slope decreases is called "saturation" region.

L1L111.6 .6 : : MagnetizationMagnetizationHysteresis

Ferromagnetic materials also show a “hysteresis” effect, where decreasing the applied magnetic field, or H, doesn’t produce the reverse effect of increasing the field:

Br = “remanence” or“residual magnetism”

Hc = “coercivity”

L1L111.7 Magnetization.7 Magnetization“hard” magnetic materials: Hc is high, area of the loop is

large, used for permanent magnets.“soft” magnetic materials: Hc is small, area of loop is

small, used for transformer cores & electromagnets.

Material can be demagnetized by strikingor heating it, or go round the hysteresis loop,gradually reducing its size. "Degaussing"

LL99.1 .1 : : Magnetic fields due to Magnetic fields due to currents currents

Magnetic fields are produced by currents.

Biot-Savart law

Ampere’s law

Example:

so

LL99.2 .2 : : Magnetic fields due to Magnetic fields due to currents currents

A solenoid:

(n is number of turns/length)

Therefore (inside)

LL99.3 .3 : : Magnetic fields due to Magnetic fields due to currents currents

Use the Biot-Savart law to derive the magnetic field on the axis of a current loop:

and

Therefore

LL99.4 .4 : : Magnetic fields due to Magnetic fields due to currents currents

Magnetic field of the Earth

LL99.5 Magnetic fields due to .5 Magnetic fields due to currents currents

The magnetic field of a magnetic dipole:

This magnetic field has the same shape as the electric field of anelectric dipole: do the exercise in the Exercise Set.

(I, A0)

w

tI

B

-

+ + + + + + +

- - - - - - - - -

charge carriers are electrons for copperRight hand rule electrons are deflected down bottom of probe is negative

Z

X

Y

w

tI

B

Z

X

Y

Hysteresis Curve for an Iron sample

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-100 -50 0 50 100

H (A.m-1)

B

(T

)

M H

Saturation of M

coercivity

retentivity (remanence)

retentivity (remanence)

Saturation of M

Area enclosed = energy dissipated in a cycle in reversing the magnetic domains

A B

CD

F

J

E

K

H

I

LI

G