ABSTRACT
The area projection divided by 2r defines the solid angle subtended by that area at P. Then is represented by the solid angle which the band-like surface, corresponding to the displacement of the whole current loop, intercepts at the point P, given by
Ωd
2 2 ( ) r rdr dl e dl e dl rd dr d
r rΓ Γ
− × ⋅ × 3r r ×Ω = = − ⋅ = − ⋅∫ ∫
G G ∫ GG G G GG G (13.1)
In Cartesian coordinates, the solid angle Ω can be expressed as a function of space coordinates , and ,x y z so that we formally have
rdrdgraddz z
dy y
dx x
d GG ⋅Ω∇=⋅Ω=∂ Ω∂+∂
Ω∂+∂ Ω∂=Ω
and Eq.(13.1) becomes
3 dl r
rΓ
×∇Ω = −∫ G G
(13.2)
Comparison with the integral form (12.21) shows that the Biot-Savart law reads
0 0 4 4
I IB µ µ φπ π Ω⎛ ⎞= − ∇Ω = −∇ = −∇⎜ ⎟⎝ ⎠
G (13.3)
and gives the magnetic induction produced by a current loop at a given point in terms of the space rates of change of the solid angle which the surface of the loop intercepts at that point. It is now convenient to interpret this result in the sense of Eq.(12.28), that is the magnetic induction B
G due to a current loop can be derived from a scalar
magnetostatic potential defined as
0 4 Iµφ π Ω= (13.4)
Assuming a small loop, as given in Figure 13.1, of area S, so that ,rS << Eq.(13.4) becomes ( )0 0
4 4
IS rI S r r
µ µθφ π π ⋅⎛ ⎞= =⎜ ⎟⎝ ⎠
G G
This expression should be compared with Eq.(11.12) for the electric dipole, to which it becomes identical, if 0/ εpG is replaced by 0 .ISµ
G Thus a small current loop behaves as a
dipole source for the magnetic field, given by a magnetic dipole moment µG of the form ISµ = GG (13.5) which is defined as a vector whose magnitude is the product of the current and the area of the loop. Hence we have
0 34 r
r µ µφ π
⋅= G G
(13.6)
and this is the magnetostatic analogue of Eq.(11.12).
