We wish to calculate the
magnetic field of a long solenoid with *N*
turns per unit
length carrying a current
which has the same value at every point of the solenoid as in
Jackson problem 6.24. We will solve for the case where the
current is
*ReIe*^{ - it}, and
use this result to calculate the field the of triangular
wave current, which is periodic and has the value.

+ k^{ 2} = - 4( - )
| (1) |

G(,,',') = iJ_{m}(k)H_{m}^{(1)}(k)e^{ im( - ')} .
| (2) |

A_{} = J_{1}(k)H_{1}^{(1)}(k)
| (3) |

B_{z} = (A_{}) .
| (4) |

We can use the identify for Bessel functions *C*_{1} , that

[xC_{1}(x)] = C_{0}(x)
| (5) |

B_{z} =
| (6) |

I(t)
| = | ||

= |
Re
| (7) |

B_{z}(,t) = Re e^{ - int}
| (8) |

Defining *x* = /*a* ,
= *a*/*c* ,
= *t* , and
*B*_{0} = 4*NI*_{0}/*c*
this can be written

B_{z}(,t) =
| (9) |