# Formula Long Solenoid (Coil) Magnetic field Coil length Electric current Number of turns Relative permeability

## Magnetic field

`$$ \class{violet}{B} $$`Unit

`$$ \mathrm{T} = \frac{\mathrm{kg}}{\mathrm{A} \, \mathrm{s}^2} $$`

Magnetic flux density inside the long coil. Inside the coil \(\class{violet}{B}\) is homogeneous (constant) and exactly this constant value \(\class{violet}{B}\) inside the coil is referred to by the formula.

Note that the magnetic flux density of a coil is not homogeneous outside.

## Coil length

`$$ l $$`Unit

`$$ \mathrm{m} $$`

Length from one end to the other end of the coil. The formula is precise only if the coil length is significantly larger than the radius \( r\) of the coil.

## Electric current

`$$ I $$`Unit

`$$ \mathrm{A} $$`

Electric current is the amount of charge \(Q\) that flows through the coil per unit time \(t\). If you double the current, the magnetic field also doubles.

## Number of turns

`$$ N $$`Unit

`$$ - $$`

Number of turns of the coil ("number of spirals"). The more turns a coil has, the larger the magnetic field generated by the coil.

## Relative permeability

`$$ \mu_{\text r} $$`Unit

`$$ - $$`

It is possible to significantly amplify the magnetic field \(\class{violet}{B}\) inside the coil by pushing a certain material into the coil interior. This material is characterized by the relative permeability.

If there is vacuum (or air) inside the coil, then \( \mu_{\text r} ~=~ 1 \). If you push an iron core into the coil, then the relative permeability can be 300 up to 10000. Thus the magnetic field would be amplified by the factor 300 to 10000.

## Vacuum permeability

`$$ \mu_0 $$`Unit

`$$ \frac{\mathrm{Vs}}{\mathrm{Am}} = \frac{ \mathrm{kg} \, \mathrm{m} }{ \mathrm{A}^2 \, \mathrm{s}^2 } $$`

Magnetic field constant is a natural constant and occurs whenever magnetic fields are involved. It has the value:

`\[ \mu_0 ~=~ 4\pi \cdot 10^{-7} \, \frac{ \text{N} }{ \text{A}^2 } \]`