Solenoid

A solenoid is a device made up of an electric wire regularly rolled up in propeller in order to form a winds long. Traversed by a running, it produces a Magnetic field in its vicinity, and more particularly inside the propeller where this field is almost uniform. The advantage of the solenoid lies in this uniformity which is sometimes necessary in certain experiments of physics. But it presents also disadvantages: it is more cumbersome than the Reels of Helmholtz and cannot produce a magnetic field raised without expensive material and cooling system.

Theory

Magnetic field on the axis

The solenoid is modelled by a series of NR whorls of rays R , of the same axis, traversed by the same current I and regularly laid out over a length 2a . One notes O the center of the solenoid, and has and B its ends.

One knows the magnetic field created by a whorl of current on his axis. One can then deduce from it the field created by the solenoid on his axis:

$B\; (Z)\; =\; \backslash \; mu\_0\; NOR\; \backslash \; frac\; \{\backslash \; Omega\_A-\; \backslash \; Omega\_B\}\; \{4\; \backslash \; pi\}$,

where $\backslash \; Omega\_A$ and $\backslash \; Omega\_B$ is the solid angles under which one respectively sees the face A' and the face B since the distance Z compared to O , and $\backslash \; mu\_0$ is the magnetic Perméabilité of the vacuum.

In the center of the solenoid, i.e. in z=0 , this formula becomes:

$B\; (0)\; =\; \backslash \; mu\_0\; NOR\; \backslash \; frac\; \{has\}\; \{\backslash \; sqrt\; \{a^2+\; R^2\}\}$

One can thus notice that the magnetic field created in the center increases if one adds whorls or current, but which it decreases if the diameter of the solenoid is increased.

Magnetic field out of the axis

One can show that it is possible to determine the magnetic field in all space ($\backslash \; vec\; B\; (R,\; Z)\; =B\_z\; (R,\; Z)\; \backslash \; vec\; u\_z+B\_r\; (R,\; Z)\; \backslash \; vec\; u\_r$) starting from the magnetic field on the axis ($B\; (0,\; Z)$ noted $F\; (Z)$) thanks to the following relations:

$B\_z\; (R,\; Z)\; =\; F\; (Z)\; -\; \backslash \; frac\; \{1\}\; \{4\}\; r^2\; F$ (Z) and

$B\_r\; (R,\; Z)\; =\; -\; \backslash \; frac\; \{1\}\; \{2\}\; R\; F\; “(Z)\; +\; \backslash \; frac\; \{1\}\; \{16\}\; r^3\; F”$ (Z) .

One realizes whereas this field is quasi-homogeneous in all the volume delimited by the solenoid. That corresponds to lines of field quasi-parallels between them. Outside the solenoid, the field is similar to that of a Aimant: it presents a north pole and a south pole. It is however very weak.

Magnetic monopoly, cord of Dirac

If one considers an infinitely long half-solenoid of very small ray, the magnetic field in all space, except the interior of the solenoid which constitutes a singularity called cord of Dirac , of a magnetic Monopôle is that.

This strange object is unrealizable in practice, but it has a certain quantum interest in electrodynamic.

See too

• Magnetostatic
• magnetic Dipole of a sphere
• Electromagnet

References

• Goldhaber & Trower: resource letter, magnetic monopolies, Am.J.Phys 1990

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Bert Bell | December 6th | Novion-Porcian | Courvite | My small company | Hillsboro (Missouri) | Brown

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