Simple Nature - Light and Matter

is related by

Φ = 4πkqin

to the charge enclosed within the surface.

Chapter 11, Electromagnetism, page 673

Relativity implies that there must be an interaction between moving charges and other moving

charges. Thismagneticinteraction is in addition to the usual electrical one. The magnetic field

can be defined in terms of the magnetic force exerted on a test charge,

F=qv×B,

or, alternatively, in terms of the torque on a magnetic test dipole,

|B|=

τ

|mt|sinθ

,

whereθis the angle between the dipole vector and the field. The magnetic dipole momentmof
a loop of current has magnitudem=IA, and is in the (right-handed) direction perpendicular
to the loop.
The magnetic field has no sources or sinks. Gauss’ law for magnetism is

ΦB= 0.

The external magnetic field of a long, straight wire is

B=

2 kI

c^2 R

,

forming a right-handed circular pattern around the wire.

The energy of the magnetic field is

dUm=

c^2

8 πk

B^2 dv.

The magnetic field resulting from a set of currents can be computed by finding a set of

dipoles that combine to give those currents. The field of a dipole is

Bz=

km

c^2

(

3 cos^2 θ− 1

)

r−^3

BR=

km

c^2

(3 sinθcosθ)r−^3 ,

which reduces toBz=km/c^2 r^3 in the plane perpendicular to the dipole moment. By construct-
ing a current loop out of dipoles, one can prove theBiot-Savart law,

dB=

kId`×r

c^2 r^3

,

which gives the field when we integrate over a closed current loop. All of this is valid only for
static magnetic fields.

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