A solenoid is tightly coiled helical arrangement of a wire that maximises the Magnetic Field that goes through it’s loops when a current is flowing through it. They operate in a similar principle to Capacitors*, in the sense that they generate a uniform, straight field.
*See Inductors for the magnetic equivalent to capacitors.
Magnetic Field
Note Magnetic Field of a Solenoid
\vec{B} = \mu_{0} nI = \mu_{0} \dfrac{NI}{l}
>>[!terms]-
>>* $\vec{B}$ = [Magnetic Field](Magnetic%20Field.md) (in $\text{T}$)
>>* $n$ = Turns per unit length ($\text{1/m}$)
>>* $N$ = Total turns of wire ($\text{unitless}$)
>>* $l$ = Length of solenoid (in $\text{m}$)
>>* $I$ = [Current](Current.md) flowing through solenoid (in $\text{A}$)
### Relation to a [Magnet](Magnet)
### Derivation (using [Ampère's Law](Ampère's%20Law.md))
Assume we have a solenoid with a current $I$ flowing through each of the loops inside it, with $N$ loops and a total length of $L$. Also assume the magnetic field is only flowing between the solenoid (due to superposition) #TODO
Using [Ampère's Law](Ampère's%20Law.md):
\oint \vec{B} \cdot d\vec{l} = I_{enc} \mu_{0}
We know in the amperian loop, certain sections (in red) are perpendicular to the magnetic field, so $d\vec{l} \cdot \vec{B} = 0$ (i.e. does not contribute).
We have only two sections of the loop parallel to the field, but in one side (outside the solenoid), $\vec{B} = 0$, so:
\oint \vec{B} \cdot d\vec{l} = \oint B \ \text{d}{l} = B l
The total current, $I_{enc}$, flowing through the amperian loop is given by the number of loops. Because through each loop, the current flowing is the *same*, and not divided up:
I_{enc} = N_{loops} \times I
Bl = NI \mu_{0}
B = \mu_{0}I \times \dfrac{N}{l}
If we define $n$ to be the ratio of number of wire loops to the length of the solenoid, we get:
B = \mu_{0} I n