Solenoid

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 §

Formula §

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

$$Combiningthetwo:$$

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

$$Whichgivesourformula$$