Solenoid

File:Solenoid-1.png
 File:VFPt Solenoid correct2.svg
A **solenoid** is a coil wound into a tightly packed helix. The term was invented in 1823 by André-Marie Ampère to designate a helical coil.

As a technical term in the study of electromagnetism, a solenoid is a coil that is "pipe-like" in the sense that its length is substantially greater than its diameter. In practice, the coil is often wrapped around a metallic core, which produces a uniform magnetic field in a volume of space (where some experiment might be carried out) when an electric current is passed through it. A solenoid is a type of electromagnet the purpose of which is to generate a controlled magnetic field. If the purpose of the solenoid is instead to impede changes in the electric current, a solenoid can be more specifically classified as an inductor rather than an electromagnet.
The solenoid is not necessarily straight, for example, William Sturgeon's electromagnet of 1824 consisted of a solenoid bent into a horseshoe shape.

In engineering, the term 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. Solenoid bolts, a type of electronic-mechanical locking mechanism, also exist.

## Infinite continuous solenoid

An infinite solenoid is a solenoid with infinite length but finite diameter. Continuous means that the solenoid is not formed by discrete finite-width coils but by infinitely many infinitely-thin coils with no space between them; in this abstraction, the solenoid is often viewed as a cylindrical sheet of conductive material.

### Inside

 File:Solenoid with 3 loops (1).png
The magnetic field inside an infinitely long solenoid is homogeneous and its strength neither depends on the distance from the axis, nor on the solenoid's 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 Figure 1, 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 confirm 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* do not 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 uniform.

### Outside

A similar argument can be applied to the loop *a* to conclude that the field outside the solenoid is constant.

An intuitive argument can also be used to show that the flux density outside the solenoid is actually zero. Magnetic field lines only exist as loops, they cannot diverge from or converge to a point like electric field lines can (see Gauss's law for magnetism). The magnetic field lines follow the longitudinal path of the solenoid inside, so they must go in the opposite direction outside of the solenoid so that the lines can form a loop. However, the volume outside the solenoid is much greater than the volume inside, so the density of magnetic field lines outside is greatly reduced. Now recall that the field outside is constant. In order for the total number of field lines to be conserved, the field outside must go to zero as the solenoid gets longer.

### Quantitative description

 File:Solenoid and Ampere Law - 2.png
Applying Ampère's Law to the solenoid (see the right figure) gives us

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle B l= \mu\_0 N I,}**

where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle B}**
 is the magnetic flux density, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle l}**
 is the length of the solenoid, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \mu\_0}**
 is the magnetic constant, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle N}**
 the number of turns, and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle I}**
 the current. From this we get

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle B = \mu\_0 \frac{N I}{l}.}**

This equation is valid for a solenoid in free space, which means the permeability of the magnetic path is the same as permeability of free space, μ0.

If the solenoid is immersed in a material with relative permeability μr, then the field is increased by that amount:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle B = \mu\_0 \mu\_{\mathrm{r}} \frac{N I}{l}.}**

In most solenoids, the solenoid is not immersed in a higher permeability material, but rather some portion of the space around the solenoid has the higher permeability material and some is just air (which behaves much like free space). In that scenario, the full effect of the high permeability material is not seen, but there will be an effective (or apparent) permeability *μ*eff such that 1 = *μ*eff = *μ*r.

The inclusion of a ferromagnetic core, such as iron, increases the magnitude of the magnetic flux density in the solenoid and raises the effective permeability of the magnetic path. This is expressed by the formula

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle B = \mu\_0 \mu\_{\mathrm{eff}} \frac{N I}{l} = \mu \frac{N I}{l},}**

where *µ*eff is the effective or apparent permeability of the core. The effective permeability is a function of the geometric properties of the core and its relative permeability. The terms relative permeability (a property of just the material) and effective permeability (a property of the whole structure) are often confused; they can differ by many orders of magnitude.

For an open magnetic structure, the relationship between the effective permeability and relative permeability is given as follows:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \mu\_\mathrm{eff} = \frac{\mu\_r}{1+k(\mu\_r -1)},}**

where *k* is the demagnetization factor of the core.

## Inductance

See also: Inductance with physical symmetry
As shown above, the magnetic flux density **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle B}**
 within the coil is practically constant and given by

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle B = \mu\_0 \frac{NI}{l},}**

where *µ*0 is the magnetic constant, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle N}**
 the number of turns, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle I}**
 the current and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle l}**
 the length of the coil. Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux density **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle B}**
 by the cross-section area **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle A}**
:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \Phi = \mu\_0 \frac{NIA}{l}.}**

Combining this with the definition of inductance

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle L = \frac{N \Phi}{I},}**

the inductance of a solenoid follows as

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle L = \mu\_0 \frac{N^2A}{l}.}**

A table of inductance for short solenoids of various diameter to length ratios has been calculated by Dellinger, Whittmore, and Ould.

This, and the inductance of more complicated shapes, can be derived from Maxwell's equations. For rigid air-core coils, inductance is a function of coil geometry and number of turns, and is independent of current.

Similar analysis applies to a solenoid with a magnetic core, but only if the length of the coil is much greater than the product of the relative permeability of the magnetic core and the diameter. That limits the simple analysis to low-permeability cores, or extremely long thin solenoids. The presence of a core can be taken into account in the above equations by replacing the magnetic constant *µ0* with *µ* or *µ0µr*, where *µ* represents permeability and *µr* relative permeability. Note that since the permeability of ferromagnetic materials changes with applied magnetic flux, the inductance of a coil with a ferromagnetic core will generally vary with current.