A plane coil, with a spiral winding, placed in air, has the exterior
(maximal) radius *a*, *N* turns and carries the current
*i*. Calculate the magnetic flux density in points placed on
the coil's symmetry axis.

**Solution**

We choose a Cartesian coordinate system, such that the coil is placed in the yOz plane, with the center in the origin. The field will be calculated along the Ox axis.

The turns of the coil are considered to be uniformly distributed, such
that a segment of unit length of the coil's radius contains *n
= N/a* turns.

The elementary magnetic field produced by a circular element of radius
*r* and of width *dr * has, according to the
solution of problem 4, the expression:

*dH*_{z} : = [(*i* *r*^{2} *n* *dr*)/(2 *R*^{3})].,

in which
*R* = Ö{*r*^{2} + *x*^{2}}. The field produced by the whole coil is obtained by integration of
the elementary field, for *r* from 0 to *a.*

>
`restart:
`

>
`H:=n*i/2 *
int(r^2/(r^2+x^2)^(3/2),r=0..a);
`

*H* : =
1

2
*n* *i* (
- *a* + ln(*a* +

Ö

*a*^{2} + *x*^{2}

)

Ö

*a*^{2} + *x*^{2}

Ö

*a*^{2} + *x*^{2}

- ln( *x*))

**Numerical application:** N = 100, a = 10cm, i = 5A

>
`plot(subs(a=0.1,n=100/0.1,i=5,
H),x=0..0.2);
`

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On 16 Feb 2000, 02:04.