Tensors for Physics

128 8 Integration of Fields

The line integral is over the circle with radiusR. For symmetry reasons, theH-field

is tangential, i.e. parallel to dr, and the magnitudeHof the field is constant around

the circle. Thus one has

∮

H·dr=HR

∫ 2 π

0

dφ= 2 πRH.

Consequently, the strength of the magnetic field, at the distancerfrom the center of

the wire, with the previousRnow calledr,is

H=

I

2 πr

. (8.41)

Clearly, outside of the wire, the field strengthHdecreases with increasing distance
rlike 1/r. The situation is different inside the wire. There the integral over a circular
disc with radiusrincreases like its area, viz. liker^2 , provided that the electric current
density is homogeneous. The constant electric currentI,in(8.41) is replaced by a
term proportional tor^2. As a consequence,Hincreases linearly withR. Notice,
however, that the charge density within a metal wire is not homogeneous, but rather
confined to a surface layer.
Furthermore, the present considerations apply to long wires, end-effects are not
taken into account.

8.2.8 Application: Faraday Induction.

Consider an almost closed ring-like electrically conducting wire, placed into a mag-

neticBfield, cf. Fig.8.13.

Integration of the Faraday law

εμνλ∇νEλ=−

∂

∂t

Bμ,

Fig. 8.13Schematic view of

the Faraday induction

experiment