(628) |

Because the geometry is cylindrically symmetric, we can, without loss of generality, choose the observation point to lie in the - plane (i.e., ). It follows from Equation (621) that

(629) | ||

(630) | ||

(631) |

where . It is clear that the integral for averages to zero. Hence, only , which corresponds to , is non-zero, and we can write

where

(633) |

which reduces to

(634) |

The previous integral can be expressed in terms of complete elliptic integrals,

(635) |

Here, represents the lesser of and , whereas represents the greater of and . Hence,

where now represents the lesser of and , whereas represents the greater of and . It follows from Equation (309) that

(637) |

Thus, Equation (637) yields

(638) |

However, according to Equation (309),

(639) | ||

(640) |

Hence, we obtain

(641) |

where we have made use of the fact that when is even.

(642) |

for , and

(643) |

for .

Now, according to Equations (620) and (633),

(644) | ||

(645) | ||

(646) |

Given that

(647) |

we find that

(648) | ||

(649) |

in the region . In particular, because and , we obtain

(650) | ||

(651) |

The previous two equations can be combined to give

(652) |

Of course, this result can be obtained in a more straightforward fashion via the direct application of the Biot-Savart law. We also have

(653) | ||

(654) |

in the region . A long way from the current loop (i.e., ), we obtain

where .

Now, a small planar current loop of area
, carrying a current
, constitutes a *magnetic dipole* of moment

Here, is a unit normal to the loop in the sense determined by the right-hand circulation rule (with the current determining the sense of circulation). It follows that in the limit and the current loop considered previously constitutes a magnetic dipole of moment . Moreover, Equations (656)-(658) specify the non-zero components of the vector potential and the magnetic field generated by the dipole. It is easily seen from Equation (656) that

(659) |

Taking the curl of this expression, we obtain

(660) |

which is consistent with Equations (657) and (658).