How To Calculate A Solenoid

A solenoid is a coil of wire that is substantially longer than its diameter that generates a magnetic field when a current passes through it. In practice, this coil is wrapped around a metallic core and the strength of the magnetic field depends on the coil density, the current passing through the coil and the magnetic properties of the core.

This makes a solenoid a type of electromagnet, the purpose of which is to generate a controlled magnetic field. This field can be used for various purposes depending on the device, from being used to generate a magnetic field as an electromagnet, to impede current changes as an inductor, or to convert the energy stored in the magnetic field to kinetic energy as an electric motor.

The magnetic field of a solenoid derivation can be found using ​Ampère's Law​. We get

\(Bl=\mu_0 NI\)

where ​B​ is the magnetic flux density, ​l​ is the length of the solenoid, μ₀ is the magnetic constant or the magnetic permeability in a vacuum, ​N​ is the number of turns in the coil, and ​I​ is the current through the coil.

Dividing throughout by ​l​, we get

\(B=\mu_0(N/l)I\)

where ​N/l​ is the ​turns density​ or the number of turns per unit length. This equation applies for solenoids without magnetic cores or in free space. The magnetic constant is 1.257 × 10^-6 H/m.

The ​magnetic permeability​ of a material is its ability to support the formation of a magnetic field. Some materials are better than others, so the permeability is the degree of magnetization a material experiences in response to a magnetic field. The relative permeability ​μ​_r tells us how much this increases with respect to free space or the vacuum.

\(\mu = \mu_r \mu_0\)

where ​μ​ is the magnetic permeability and ​μ​_r is the relativity. This tells us how much the magnetic field increases if the solenoid has a material core going through it. If we placed a magnetic material, e.g., an iron bar, and the solenoid is wrapped around it, the iron bar will concentrate the magnetic field and increase the magnetic flux density ​B​. For a solenoid with a material core, we get the solenoid formula

\(B=\mu (N/l)I\)

One of the primary purposes of solenoids in electrical circuits is to impede changes in electrical circuits. As an electric current flows through a coil or solenoid, it creates a magnetic field that grows in strength over time. This changing magnetic field induces an electromotive force across the coil that opposes the current flow. This phenomenon is known as electromagnetic induction.

The inductance, ​L​, is the ratio between the induced voltage ​v​, and the rate of change in the current ​I​.

\(L=-v\bigg(\frac{dI}{dt}\bigg)^{-1}\)

Solving for ​v​ this becomes

\(v=-L\frac{dI}{dt}\)

Deriving the Inductance of a Solenoid

​Faraday's Law​ tells us the strength of the induced EMF in response to a changing magnetic field

\(v=-nA\frac{dB}{dt}\)

where n is the number of turns in the coil and ​A​ is the cross sectional area of the coil. Differentiating the solenoid equation with respect to time, we get

\(\frac{dB}{dt}=\mu (N/l)\frac{dI}{dt}\)

Substituting this into Faraday's Law, we get the induced EMF for a long solenoid,

\(v=-\bigg(\frac{\mu N^2 A}{l}\bigg)\bigg(\frac{dI}{dt}\bigg)\)

Substituting this into ​v = −L(​d​I​/d​t)​ we get

\(L=\frac{\mu N^2 A}{l}\)

We see the inductance ​L​ depends on the geometry of the coil – the turns density and the cross sectional area – and the magnetic permeability of the coil material.