Editing Inductor (section)

=== Energy stored in an inductor ===
One intuitive explanation as to why a potential difference is induced on a change of current in an inductor goes as follows:

When there is a change in current through an inductor there is a change in the strength of the magnetic field.  For example, if the current is increased, the magnetic field increases.  This, however, does not come without a price.  The magnetic field contains [[potential energy]], and increasing the field strength requires more energy to be stored in the field.  This energy comes from the electric current through the inductor.  The increase in the magnetic potential energy of the field is provided by a corresponding drop in the electric potential energy of the charges flowing through the windings.  This appears as a voltage drop across the windings as long as the current increases.  Once the current is no longer increased and is held constant, the energy in the magnetic field is constant and no additional energy must be supplied, so the voltage drop across the windings disappears.

Similarly, if the current through the inductor decreases, the magnetic field strength decreases, and the energy in the magnetic field decreases.  This energy is returned to the circuit in the form of an increase in the electrical potential energy of the moving charges, causing a voltage rise across the windings.

====Derivation====
The [[Work (physics)|work]] done per unit charge on the charges passing through the inductor is <math>-\mathcal{E}</math>. The negative sign indicates that the work is done ''against'' the emf, and is not done ''by'' the emf.  The current <math>I</math> is the charge per unit time passing through the inductor.  Therefore, the rate of work <math>W</math> done by the charges against the emf, that is the rate of change of energy of the current, is given by
:<math>\frac{dW}{dt} = -\mathcal{E}I </math>
From the constitutive equation for the inductor, <math>-\mathcal{E} = L\frac{dI}{dt}</math> so
:<math>\frac{dW}{dt}= L\frac{dI}{dt} \cdot I = LI \cdot \frac{dI}{dt}</math>

:<math>dW = L I \cdot dI</math>

In a ferromagnetic core inductor, when the magnetic field approaches the level at which the core saturates, the inductance will begin to change, it will be a function of the current <math>L(I)</math>. Neglecting losses, the [[energy]] <math>W</math> stored by an inductor with a current <math>I_0</math> passing through it is equal to the amount of work required to establish the current through the inductor.

This is given by:
<math>W = \int_0^{I_0} L_d(I) \, I \, dI</math>, where <math>L_d(I)</math> is the so-called "differential inductance" and is defined as: <math>L_d = \frac{d\Phi_{\mathbf{B}}}{dI}</math>. 
In an air core inductor or a ferromagnetic core inductor below saturation, the inductance is constant (and equal to the differential inductance), so the stored energy is
:<math>W = L\int_0^{I_0} I \, dI</math>
{{Equation box 1 |indent = |cellpadding = 0 |border = 2 |border colour = black |background colour = transparent
|equation = <math>\quad W = \frac{1}{2}L {I_0}^2\quad</math>
}}  
For inductors with magnetic cores, the above equation is only valid for [[linear circuit|linear]] regions of the magnetic flux, at currents below the [[magnetic saturation|saturation]] level of the inductor, where the inductance is approximately constant. Where this is not the case, the integral form must be used with <math>L_d</math> variable.