5. Magnetostatics

The force is generatedby the magnetic field B.

Early evidence for the magnetic field.

Magnetic Field

Bar magnet Magnetic field lines

Magnetic Field of the Earth

Direction measuredby a compass.

Oersted, in 1820 What he saw.

Qersted’s experiment, modern version

Biot and Savert found that the force on a compass encirclesa wire. Formulated the Biot-Savert law.

Ampere stated that all magnetic field are generated by currents.The elementary magnetic dipoles are current loops.

The force is generatedby the magnetic field B.

How does the field acton the moving charge?

How is the field generatedby the moving charges?

)( BvEF Q

Total force on a charge:

Lorentz’s law:

Right – hand rule

Magnetic force on a moving charge:

)( BvF Qmag

Example 5.2 – Cyclotron Motion

Cloud Chamber

1.6 MeV proton

7 MeV alpha

The Cylcotron

O. E. Lawrence M. S. Livingston 1934

Magnetic Bottle

Van Allen Belts

Protons and electrons from the solar wind are trapped bythe earth’s magnetic field.

Currents

Line current I 1 A = 1 C/s

)( BlF dImag

Magnetic forces do not work!

0)( dtQddW magmag vBvlF

Example 5.3

B points into the page.

Surface current density

vK

dadamag )()( BKBvF

Volume current density

vJ

ddmag )()( BJBvF

t

Jcontinuity equation:

Example 5.4

Biot-Savart Law

Magnetostatics:

Steady currents 00

JJ

t

22

ˆ'

4'

ˆ

4)(

r

r

r

r lIrB

dIdl oo

Magnetic field of a steady line current

27 /104 ANo

Permeability of free space – defines the unit Ampere

Example 5.5

Example 5.6?

Superposition principle for the magnetic field gives the differentversions of the Biot-Savart law.

'ˆ

4)( 2 dlo

r

rIrB

line current

'ˆ

4)( 2 dao

r

rKrB

surface current

'ˆ

4)( 2

do

r

rJrBvolume current

cvqo

for only

)ˆ(

4)( 2

r

rvrB

Moving point charge