Similar to how to Capacitors can store energy through an Electric Field, an inductor is a device designed to store energy through a Magnetic Field. A capacitor stores Charge, while an inductor ‘stores’ Magnetic Flux. Inductors are only really useful for non-constant Currents.
An inductor is defined as a coil of wire used for Inductance. The most common inductor is a Solenoid, a helical loop of wire.
L = \dfrac{\mu_{0}N^2A}{l}
##### Terms
* $L$ = [Inductance](Inductance.md) (in $\text{H}$)
* $\mu_{0}$ = Permeability of free space $\approx 1.26 \times 10^{-6}$ (in $\text{N/A}^2$)
* $N$ = Number of turns in the solenoid ($\text{unitless}$)
* $A$ = Area of the circle enclosed by the solenoid (in $\text{m}^2$)
* $l$ = Length of solenoid (in $\text{m}$)
Notice how, just like the capacitance of a capacitor can be given purely geometrically (without voltage or charge), the inductance of an inductor can be given purely geometrically (without current or flux)!
### [Induced EMF](Induced%20EMF.md)
##### Formula
>[!formula] Induced EMF in a Solenoid
>
>$$
>\mathcal{E} = -L\frac{ \text{d}I }{ \text{d}t }
>$$
##### Terms
* $\mathcal{E}$ = [Static-Induced EMF](Static-Induced%20EMF.md), as the conductor does not move, but the magnetic fields that are generated from the electric fields fluctuate. (in $\text{V}$)
* $L$ = [Inductance](Inductance.md) (in $\text{H}$)
* $\frac{ \text{d}I }{ \text{d}t }$ = Change in [Current](Current.md) (in $\text{A/s}$)
### Derivations
##### Inductance
We can derive the inductance of a solenoid easily. Let's start with the magnetic field of a [Solenoid](solenoid.md):
we know that [Magnetic Flux](magnetic%20flux.md) is simply the [Dot Product](Dot%20Product.md) of the magnetic field and area:
In a *single loop of the solenoid* , $\vec{A}$ is the circle enclosed by the loop and is always parallel to $\vec{B}$:
\Phi_{loop} = \vec{B}\cdot \vec{A} = BA
\Phi_{loop} = \dfrac{\mu_{0}NIA}{l}
Note that this one a *single* loop, even though we have an $N$ factor, it increases the magnetic field through the loop, through superposition.
Hence the total magnetic flux of the solenoid is:
\Phi_{net} = N\Phi_{loop} = \dfrac{\mu_{0}N^2IA}{l}
Andnowwejustdividethecurrent:
L = \dfrac{\Phi_{B}}{I} = \dfrac{\mu_{0}N^2A}{l}
##### Induced EMF
From [Faraday's Law of Induction](Faraday's%20Law%20of%20Induction.md):
\Phi_{B} = LI
Iftheinductanceremainsconstant,butthecurrentchanges,then:
\text{d}{\Phi_{B}} = L \text{d}{I}
\dfrac{\text{d}{\Phi_{B}}}{\text{d}{t}} = L \dfrac{\text{d}{I}}{\text{d}{t}}
\mathcal{E} = -L\dfrac{\text{d}{I}}{\text{d}{t}}
### Applications
* Spark plugs
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