Energy stored by an inductor

Properties of inductance

The equation relating inductance and flux linkages can be rearranged as follows:

$\Phi =Li\,$

Taking the time derivative of both sides of the equation yields:

${\frac {d\Phi }{dt}}=L{\frac {di}{dt}}+i{\frac {dL}{dt}}\,$

In most physical cases, the inductance is constant with time and so

${\frac {d\Phi }{dt}}=L{\frac {di}{dt}}$

By Faraday's_law_of_induction of Induction we have:

${\frac {d\Phi }{dt}}=-{\mathcal {E}}=v$

where ${\mathcal {E}}$ is the Electromotive force (emf) and $v$ is the induced voltage. Note that the emf is opposite to the induced voltage. Thus:

${\frac {di}{dt}}={\frac {v}{L}}$

or

$i(t)={\frac {1}{L}}\int _{0}^{t}v(\tau )d\tau +i(0)$

These equations together state that, for a steady applied voltage v, the current changes in a linear manner, at a rate proportional to the applied voltage, but inversely proportional to the inductance. Conversely, if the current through the inductor is changing at a constant rate, the induced voltage is constant.

The effect of inductance can be understood using a single loop of wire as an example. If a voltage is suddenly applied between the ends of the loop of wire, the current must change from zero to non-zero. However, a non-zero current induces a magnetic field by Ampère's law. This change in the magnetic field induces an emf that is in the opposite direction of the change in current. The strength of this emf is proportional to the change in current and the inductance. When these opposing forces are in balance, the result is a current that increases linearly with time where the rate of this change is determined by the applied voltage and the inductance.

Multiplying the equation for $di/dt$ above with $Li$ leads to

$Li{\frac {di}{dt}}={\frac {d}{dt}}{\frac {L}{2}}i^{2}=iv$

Since iv is the energy transferred to the system per time it follows that

$\left(L/2\right)i^{2}$

is the energy of the magnetic field generated by the current.