Inductor Energy Storage Analyzer

Physics Electricity and Magnetism • Electromagnetic Induction and Inductance

Written by STEM Calculators Team Published January 18, 2026 Updated February 24, 2026

9. Inductor Energy Storage Analyzer

Computes inductor energy \(U=\tfrac12 L I^2\), and field energy density \(u=\tfrac{B^2}{2\mu_0}\). Optional time mode plots \(I(t)\) and \(U(t)\) for simple current ramps (growth/decay) and shows how energy builds up in the magnetic field.

Units: henry (H), ampere (A), tesla (T), joule (J), seconds (s), meters (m). Inputs accept 1e-3, pi, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Mode + inputs

Mode:

Inductor (lumped)

Inductance \(L\) (H):

Energy stored (ideal): \(U=\tfrac12 L I^2\).

Current \(I\) (A):

Use magnitude with sign if you want sign tracking; energy uses \(I^2\).

Solenoid model (approx.)

Turns density \(n\) (turns/m):

Inside long solenoid: \(B\approx \mu_0 n I\).

Current \(I\) (A):

Field energy density uses this \(I\).

Cross-sectional area \(A\) (m²):

Energy in volume \(V \approx A\,\ell\): \(U \approx uV\).

Length \(\ell\) (m):

Long-solenoid approximation works best when \(\ell \gg \sqrt{A/\pi}\).

Also compute equivalent inductance estimate

For an ideal long solenoid: \(L \approx \mu_0 n^2 A\,\ell\). This links the field picture to the lumped \(U=\tfrac12 L I^2\).

Time evolution (current and energy)

Process:

Inductance \(L\) (H):

Energy plotted: \(U(t)=\tfrac12 L I(t)^2\).

Time constant \(\tau\) (s):

For LR: \(\tau=L/R\). Here you can enter \(\tau\) directly.

Total time \(T\) (s):

Plot window. More time shows the asymptote.

Time step \(\Delta t\) (s):

Smaller \(\Delta t\) gives smoother curves (more points).

Growth: \(I_\infty\) (A):

Used in growth mode only.

Decay/Ramp: \(I_0\) (A):

Decay starts at \(I_0\). Ramp starts at \(I_0\).

Ramp slope \(\alpha\) (A/s):

Used in linear ramp mode only.

Marker time \(t_\*\) (s):

Highlights \(I(t_\*)\) and \(U(t_\*)\) on the plot.

Plot + view

Plot: Graph text size:

Pan/zoom: drag to pan • mouse wheel/trackpad/pinch to zoom in/out • “Reset view” restores default.

Ready

Steps

Choose a mode and click Solve.

Plot

Interactive plot (pan/zoom enabled)

What you’re seeing

computed curve
marker at \(t_\*\) (time mode)
axes with units + ticks

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Theory

Inductor energy: where \(U=\tfrac12LI^2\) comes from

An inductor resists changes in current. The induced voltage is \(v_L = L\,\dfrac{dI}{dt}\). The instantaneous power going into the inductor is \(p = v_L I\).

\[ \begin{aligned} p &= v_L I = \left(L\frac{dI}{dt}\right)I = L I \frac{dI}{dt} \end{aligned} \]

Integrate power over time while current rises from 0 to \(I\):

\[ \begin{aligned} U &= \int p\,dt = \int_0^{I} L I\, dI = \frac{1}{2}LI^2 \end{aligned} \]

Useful identity: \(\dfrac{dU}{dI}=LI\). Also \(\lambda = LI\) is the flux linkage (often written \(N\Phi\)).

Magnetic energy density \(u=\dfrac{B^2}{2\mu_0}\)

In vacuum/air, magnetic field energy can be expressed per unit volume:

\[ \begin{aligned} u &= \frac{B^2}{2\mu_0}, \qquad \mu_0 = 4\pi\times 10^{-7}\ \mathrm{H\cdot m^{-1}} \end{aligned} \]

If the field is approximately uniform in a region of volume \(V\), a rough total energy estimate is \(U\approx uV\). Real coils have non-uniform fields and fringing, so this is an approximation.

Ideal solenoid formulas (field + inductance)

For a long solenoid (ideal model), the interior field is approximately uniform:

\[ \begin{aligned} B &\approx \mu_0\mu_r n I \end{aligned} \]

With \(N=n\ell\) total turns, the ideal inductance estimate is:

\[ \begin{aligned} L &\approx \mu_0\mu_r\frac{N^2A}{\ell} \end{aligned} \]

In the calculator, we show both \(U=\tfrac12LI^2\) (using the inductance estimate) and \(U\approx u(A\ell)\) (field-volume estimate) as two idealized viewpoints.

LR time behavior (optional)

In a simple series LR circuit with a DC step \(\varepsilon\), current grows as:

\[ \begin{aligned} \tau &= \frac{L}{R}, \\ I(t) &= \frac{\varepsilon}{R}\left(1-e^{-t/\tau}\right) \end{aligned} \]

After disconnecting the source, current decays as:

\[ \begin{aligned} I(t) &= I_0 e^{-t/\tau} \end{aligned} \]

Because \(U=\tfrac12LI^2\), energy decays like \(U(t)\propto e^{-2t/\tau}\) during decay.