Symbolic Analysis of Magnetic Field with Symbolic Analysis of Magnetic Field with the Biot-Savart-Laplace Method

Daniel Ioan and Irina Munteanu

9.6  Problem 6 - The short, thin coil

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:

dHz : = [(i a2 n dx)/(2 R3)]

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 : =
l
0 
1
2
  n i a2
(a2 + x2 -x z + z2)3/2
 dz

> Hz:=value(");


Hz : = 1
2
  (l - xn i



a2 + x2 -x l + l2
+ 1
2
  x n i



a2 + x2

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


1
2
  l n i



a2 + [1/4]  l2

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


1
2
  l n i



a2 + l2

> 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

pictures/bsl_en26.gif
>


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On 16 Feb 2000, 02:04.