Energy Stored in a Magnetic Field

To establish a magnetic field, energy must be supplied—even though no energy is needed to maintain it once formed. Consider the example of the exciting coils of an electromagnet:

• A portion of the supplied energy is lost as I²R loss (heat), which is non-recoverable.
• The remaining portion is stored in the magnetic field as potential energy, much like the potential energy stored in a raised object.

Just as raising a weight $W$ to a height $h$ stores energy as Wh, the energy used to establish a magnetic field is stored and can be recovered when the field collapses.

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Understanding the Energy Stored in Magnetic Fields

When current in an inductive coil increases gradually from 0 to a maximum value $I$, a self-induced electromotive force (e.m.f.) opposes this change. The energy used to overcome this opposition is stored in the magnetic field.

Method 1: Calculus-Based Derivation

Let:

• i = instantaneous current
• e = induced e.m.f. = L × (di/dt)​

Work done in time $dt$ to overcome this e.m.f.:

$$dW = e \cdot i \cdot dt = L \frac{di}{dt} \cdot i \cdot dt = L i \, di$$

Integrating from 0 to $I$:

$$W = \int_0^I L i \, di = \left[\frac{1}{2} L i^2\right]_0^I = \frac{1}{2} L I^2$$

✅ Energy stored in the magnetic field:

$$E=\frac{1}{2}L{I}^2\text{ joules}$$

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Method 2: Using Average Power

If the current increases uniformly from 0 to I over time t:

Average current = $\frac{I}{2}$
Induced e.m.f. = $e=\frac{LI}{t}$

Average power absorbed:

$$P_{avg} = e \cdot I_{avg} = \frac{L I}{t} \cdot \frac{I}{2} = \frac{L I^2}{2t}$$

Total energy absorbed:

$$E = P_{avg} \cdot t = \frac{L I^2}{2}$$

Thus, both methods yield the same result:

✅ Final Formula:

$$\boxed{E = \frac{1}{2} L I^2 \text{ joules}}$$

Energy Stored in Magnetic Field of Multiple Coils

🔹 Series-Aiding Coils:

For two inductors $L_1$ and $L_2$ with mutual inductance $M$:

$$E = \frac{1}{2} (L_1 + L_2 + 2M) I^2$$

🔹 Series-Opposing Coils:

$$E = \frac{1}{2} (L_1 + L_2 - 2M) I^2$$

✅ Key Takeaways

• Energy is needed to establish a magnetic field, not to maintain it.
• Stored magnetic energy is fully recoverable when the field collapses.
• The energy is proportional to the inductance L and the square of the current I.

$$E = \frac{1}{2} L I^2$$

This principle is widely used in applications like transformers, inductors, motors, and energy storage systems.