Magnetic field of the coil with a current.
Calculation of an optimum ratio of the diameter to the lenght of the coil.
![f](/evolution/images/fieldcoil/coil01.gif)
The task consists in finding such L and H at which a magnetic field on an axis z, at a point Z = 0, would be maximum, at other identical parametres.
Formula for the magnetic field on Z axis of coil with a current looks so:
![f](/evolution/images/fieldcoil/mag_fi1.gif)
where h - is thikness of a wire, resistance of the wire:
![f](/evolution/images/fieldcoil/mag_fi2.gif)
length of wire:
current:
here P - is dissipated power, field at a point z = 0:
![f](/evolution/images/fieldcoil/mag_fi7.gif)
having entered dimensionless parameters , we have:
![f](/evolution/images/fieldcoil/mag_fi8.gif)
then the formula for a magnetic field at a point z = 0 takes form:
![f](/evolution/images/fieldcoil/mag_fi9.gif)
![f](/evolution/images/fieldcoil/mag_fi10.gif)
![f](/evolution/images/fieldcoil/mag_fi11.gif)
![f](/evolution/images/fieldcoil/mag_fi12.gif)
Let:
![f](/evolution/images/fieldcoil/mag_fi13.gif)
- dimensionless function of two parametres which has a maximum, according to a reasoning:
![f](/evolution/images/fieldcoil/mag_fi14.gif)
from this it follows that there exist optimal values :
![f](/evolution/images/fieldcoil/mag_fi15.gif)
Thus we have found that at L = 1.2 R0, and H = 0.5 R0 a magnetic field at the centre of coil maximum.
![f](/evolution/images/fieldcoil/coil2.gif)
The figure shows two coils of different sizes, but have equal proportions with the optimum ratio of coil length to height of the wound wire.
© 1992 - 2014 Alexander Kucherenko