Saturday, September 24, 2011

solenoids

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Solenoid
A solenoid is a coil wound into a tightly packed helix. In physics, the term solenoid refers to a long, thin loop of wire, often wrapped around a metallic core, which produces a magnetic field when an electric current is passed through it. Solenoids are important because they can create controlled magnetic fields and can be used as electromagnets. The term solenoid refers specifically to a magnet designed to produce a uniform magnetic field in a volume of space (where some experiment might be carried out).

In short: the magnetic field inside an infinitely long solenoid is homogeneous and its strength does not depend on the distance from the axis, nor on the solenoid cross-sectional area.
This is a derivation of the magnetic flux density around a solenoid that is long enough so that fringe effects can be ignored. In the diagram to the right, we immediately know that the flux density vector points in the positive z direction inside the solenoid, and in the negative z direction outside the solenoid. We see this by applying the right hand grip rule for the field around a wire. If we wrap our right hand around a wire with the thumb pointing in the direction of the current, the curl of the fingers shows how the field behaves. Since we are dealing with a long solenoid, all of the components of the magnetic field not pointing upwards cancel out by symmetry. Outside, a similar cancellation occurs, and the field is only pointing downwards.
Now consider the imaginary loop c that is located inside the solenoid. By Ampère's law, we know that the line integral of B (the magnetic flux density vector) around this loop is zero, since it encloses no electrical currents (it can be also assumed that the circuital electric field passing through the loop is constant under such conditions: a constant or constantly changing current through the solenoid). We have shown above that the field is pointing upwards inside the solenoid, so the horizontal portions of loop c doesn't contribute anything to the integral. Thus the integral of the up side 1 is equal to the integral of the down side 2. Since we can arbitrarily change the dimensions of the loop and get the same result, the only physical explanation is that the integrands are actually equal, that is, the magnetic field inside the solenoid is radially uniform. Note, though, that nothing prohibits it from varying longitudinally, which in fact it does.
In engineering, the term solenoid may also refer to a variety of transducer devices that convert energy into linear motion. The term is also often used to refer to a solenoid valve, which is an integrated device containing an electromechanical solenoid which actuates either a pneumatic or hydraulic valve, or a solenoid switch, which is a specific type of relay that internally uses an electromechanical solenoid to operate an electrical switch; for example, an automobile starter solenoid, or a linear solenoid, which is an electromechanical solenoid.

Magnetic field and vector potential for finite continuous solenoid

Magnetic field line and density created by a solenoid with surface current density
A finite solenoid is a solenoid with finite length. Continuous means that the solenoid is not formed by discrete coils by a sheet of conductive material. We assume the current is uniformly distributed on the surface of it, and it has surface current density K. In cylindrical coordinates:
\vec{K}= \frac{I}{L} \hat{\phi}
The magnetic field can be found by vector potential. the vector potential for a finite solenoid with radius a, length L in cylindrical coordinates is \vec{r} = (\rho, \phi, z) is:
A_\phi = \frac{\mu_0 I}{4\pi } \frac{1}{L} \sqrt{\frac{a}{\rho}} \left[ \zeta k \left( \frac{k^2+h^2-h^2k^2}{h^2k^2}K(k^2)-\frac{1}{k^2}E(k^2) +\frac{h^2-1}{h^2} \Pi(h^2,k^2) \right) \right]_{\zeta_-}^{\zeta_+}
where
\zeta_{\pm}=z\pm \frac{L}{2}
h^2=\frac{4a\rho}{(a+\rho)^2}
k^2=\frac{4a\rho}{(a+\rho)^2+\zeta^2}
K(m)=\int_0^{\pi/2}{\frac{1}{\sqrt{1-m sin^2 \theta }}} d\theta
E(m)=\int_0^{\pi/2}{\sqrt{1-m sin^2 \theta} } d\theta
\Pi(n,m)=\int_0^{\pi/2}{\frac{1}{(1-n sin^2 \theta)\sqrt{1-m sin^2 \theta }}} d\theta
The K(m), E(m), and Π(n,m) are complete elliptic integral of first, second, and third kind.
By using
\vec{B} = \nabla \times \vec{A}
the magnetic flux density is:
B_\rho = \frac{\mu_0 I}{4\pi} \frac{1}{L} \sqrt{\frac{a}{\rho}} \left[ \frac{k^2-2}{k}K(k^2) + \frac{2}{k} E(k^2)\right]_{\zeta_-}^{\zeta_+}
B_z =-\frac{\mu_0 I}{4\pi} \frac{1}{L} \frac{1}{2 \sqrt{a \rho}} \left[ \zeta k \left(K(k^2) + \frac{a-\rho}{a+\rho} \Pi(h^2,k^2)\right)\right]_{\zeta_-}^{\zeta_+}
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