A cylindrical coil, of negligible thickness, of length *l* and
radius *a*, has *N* turns and carries the current
*i*. The coil is placed in a medium of permeability
m. Calculate the magnetic flux density in the points on the coil's
symmetry axis.

**Solution**

The elementary magnetic field produced by a portion of length
*dx * which contains *N/l dx* turns has, according to
the solution of problem 4, the expression:

*dH*_{z} : = [(*i* *a*^{2} *n* *dx*)/(2 *R*^{3})]

in which *n = N/l * represents the per-unit-length turn
density.

The field produced by the whole coil is obtained by integration from 0 to 1 of the elementary field.

>
`restart: with(linalg): with(plots):
`

>
`Hz := Int(n*i*a^2/(2*sqrt(a^2+(x-z)^2)^3),
z=0..l);
`

*
Warning, new definition for norm
*

*
Warning, new definition for trace
*

*Hz* : =
ó

õ
*l*

0
1

2
*n* *i* *a*^{2}

(*a*^{2} + *x*^{2} - 2 *x* *z* + *z*^{2})^{3/2}
*dz*

>
`Hz:=value(");
`

*Hz* : =
1

2
( *l* - *x*) *n* *i*

Ö

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

+
1

2
*x* *n* *i*

Ö

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

>
`subs(x=l/2, Hz); # Magnetic field in the
center of the coil
`

1

2
*l* *n* *i*

Ö

*a*^{2} + [1/4] *l*^{2}

>
`subs(x=0, Hz); # Magnetic field at the end
of the coil
`

1

2
*l* *n* *i*

Ö

*a*^{2} + *l*^{2}

>
`limit(",l=infinity); # Magnetic field
produced by the infinitely long solenoid
`

1

2
*n* *i*

**Numerical application:** N = 1000 sp, l = 30 cm, a = 3cm, i
= 8A, the coil is placed in air.

>
`plot(subs(n=1000/0.3, l = 0.3, a = 0.03, i =
8, 4*Pi*10^(-7)*Hz), x=-0.3..0.6); # Magnetic flux density variation
on the coil's axis
`

File translated fromT

On 16 Feb 2000, 02:04.