Editing Inductance (section)

== Self-inductance and magnetic energy ==
If the current through a conductor with inductance is increasing, a voltage <math>v(t)</math> is induced across the conductor with a polarity that opposes the current—in addition to any voltage drop caused by the conductor's resistance. The charges flowing through the circuit lose potential energy. The energy from the external circuit required to overcome this "potential hill" is stored in the increased magnetic field around the conductor. Therefore, an inductor stores energy in its magnetic field. At any given time <math>t</math> the power <math>p(t)</math> flowing into the magnetic field, which is equal to the rate of change of the stored energy {{nowrap|<math>U</math>,}} is the product of the current <math>i(t)</math> and voltage <math>v(t)</math> across the conductor<ref name="Serway">{{cite book
 | last1  = Serway
 | first1 = Raymond A.
 | last2  = Jewett
 | first2 =  John W.
 | title  = Principles of Physics: A Calculus-Based Text, 5th Ed.
 | publisher = Cengage Learning 
 | year   = 2012
 | pages  = 801–802
 | url    = https://books.google.com/books?id=egmU-OumDAgC&pg=PA802
 | isbn   = 978-1133104261
 }}</ref><ref name="Ida">{{cite book
 | last1  = Ida
 | first1 = Nathan 
 | title  = Engineering Electromagnetics, 2nd Ed.
 | publisher = Springer Science and Business Media
 | year   = 2007
 | page  = 572
 | url    = https://books.google.com/books?id=2CbvXE4o5swC&pg=PA572
 | isbn   = 978-0387201566
 }}</ref><ref name="Purcell2">{{cite book
 | last1  = Purcell
 | first1 = Edward 
 | title  = Electricity and Magnetism, 2nd Ed.
 | publisher = Cambridge University Press
 | date   = 2011
 | page  = 285
 | url    = https://books.google.com/books?id=Z3bkNh6h4WEC&pg=PA285
 | isbn   = 978-1139503556
 }}</ref>

<math display=block>p(t) = \frac{\text{d}U}{\text{d}t} = v(t)\,i(t)</math>

From (1) above

<math display=block>\begin{align}
\frac{\text{d}U}{\text{d}t} &= L(i)\,i\,\frac{\text{d}i}{\text{d}t} \\[3pt]
                  \text{d}U &= L(i)\,i\,\text{d}i
\end{align}</math>


When there is no current, there is no magnetic field and the stored energy is zero.  Neglecting resistive losses, the [[energy]] <math>U</math> (measured in [[joule]]s, in [[SI]]) stored by an inductance with a current <math>I</math> through it is equal to the amount of work required to establish the current through the inductance from zero, and therefore the magnetic field. This is given by:

<math display=block>U = \int_{0}^{I} L(i)\,i\,\text{d} i\,</math>

If the inductance <math>L(i)</math> is constant over the current range, the stored energy is<ref name="Serway" /><ref name="Ida" /><ref name="Purcell2" />

<math display=block>\begin{align}
U &= L\int_{0}^{I}\,i\,\text{d} i \\[3pt]
  &= \tfrac{1}{2} L\,I^2
\end{align}</math>

Inductance is therefore also proportional to the energy stored in the magnetic field for a given current. This energy is stored as long as the current remains constant. If the current decreases, the magnetic field decreases, inducing a voltage in the conductor in the opposite direction, negative at the end through which current enters and positive at the end through which it leaves. This returns stored magnetic energy to the external circuit.

If [[ferromagnetic]] materials are located near the conductor, such as in an inductor with a [[magnetic core]], the constant inductance equation above is only valid for [[linear circuit|linear]] regions of the magnetic flux, at currents below the level at which the ferromagnetic material [[magnetic saturation|saturates]], where the inductance is approximately constant. If the magnetic field in the inductor approaches the level at which the core saturates, the inductance begins to change with current, and the integral equation must be used.